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UBC Theses and Dissertations

Qualitative analyses of ecological models : an automated dynamical systems approach Van Coller, Lynn 1995

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Q U A L I T A T I V E A N A L Y S E S OF E C O L O G I C A L -AN  AUTOMATED DYNAMICAL SYSTEMS  MODELS APPROACH  By L y n n van Coller B. Sc. University of N a t a l ( U N P ) , South A f r i c a , 1990 B. Sc. Hons. University of N a t a l ( U N P ) , South A f r i c a , 1991 M . Sc. University of N a t a l ( U N P ) , South A f r i c a , 1992  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY  in T H E FACULTY OF G R A D U A T E STUDIES INSTITUTE OF APPLIED MATHEMATICS D E P A R T M E N T OF MATHEMATICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  December 1995 © L y n n van Coller, 1995  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It  is understood that copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  MfrfW&AjfrTlC  The University of British Columbia Vancouver, Canada Date  OE-6 (2/88)  to  i>ecm&e<i  Abstract  Ecological models and qualitative analyses of these models can give insight into the most important mechanisms at work in an ecological system. However, the mathematics required for a detailed analysis of the behaviour of a model can be formidable. In this thesis I demonstrate how various computer packages can aid qualitative analyses by implementing techniques from dynamical systems theory. I analyse a number of continuous and discrete models to demonstrate the kinds of results and information that can be obtained. I begin with three fairly simple predator-prey models in order to introduce the terminology and techniques and to demonstrate the reliability of the computer software. I then look at a more practical system dynamics model of a sheep-pasture-hyrax-lynx system and compare the techniques with a traditional sensitivity analysis. A ratio-dependent model is the focus of the next chapter. The analysis highlights some of the biological implausibilities and mathematical difficulties associated with these models. Two discrete population genetics models are considered in the following chapters. The techniques are able to deal with the complex nonlinearities and lead to insights into the conditions under which stable homomorphisms and polymorphisms occur. The final example is a complicated discrete model of the spruce budworm-forest defoliating system. The mechanisms responsible for insect outbreaks and the relative effects of dispersal and predation are studied. In all the cases the techniques lead to a better understanding of the interactions between various processes in the system than was possible using traditional techniques. In two cases the results suggest improvements in the formulations of the models. The  11  techniques also identify parameters or processes which are crucial for determining model behaviour. All these results are obtained fairly easily with the use of the computer packages and do not require an extensive mathematical knowledge of dynamical systems theory or intensive mathematical manipulations.  111  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  x  List of Figures  xi  Acknowledgements  xxi  1 Introduction  2  1  1.1  General overview  1  1.2  Research objectives  1  1.3  Thesis outline  2  1.4  Motivation  6  1.5  The bigger picture  8  Preliminary Example  10  2.1  Introduction  10  2.2  Basic model  11  2.3  Adding intraspecific competition among prey  13  2.4  Adding intraspecific competition among predators  22  2.5  Conclusion  27  iv  3  Sheep-Hyrax-Lynx Model  30  3.1  Introduction  30  3.2  Dynamic models and systems analysis:—some background  31  3.3  Model equations  33  3.4  Technical details  38  3.5  Model analysis  39  3.5.1  Reference parameter values  39  3.5.2  Understanding model relationships  41  3.5.3  Adding density-dependence to pasture growth  47  3.5.4  A summary of the effects of culling both hyrax and lynx  51  3.5.5  Biological interpretation of results  54  3.6 4  • • •  57  Ratio-Dependent M o d e l  58  4.1  Introduction  58  4.2  Background  60  4.3  Model equations  62  4.4  Nondimensionalisation  64  4.5  Model analysis  70  4.5.1  One-parameter studies  70  4.5.2  Combining plant and herbivore dynamics  92  4.5.3  The role played by the isocline configurations  4.6 5  Conclusion  Conclusion  102 107  Population Genetics M o d e l I  112  5.1  Introduction  112  5.2  New terminology  114 v  5.3  Background  115  5.4  Model equations  117  5.5  Model analysis  118  5.5.1  Approach  118  5.5.2  One-parameter bifurcation diagrams  119  5.5.3  Two-parameter bifurcation diagram  124  5.5.4  Orbits of period four (and higher)  128  5.6 6  7  Conclusion  • 136  Population Genetics Model II  138  6.1  Introduction  138  6.2  Background  140  6.3  Fitness functions  140  6.4  Model analysis  141  6.4.1  Approach  141  6.4.2  (fen,rn)-parameter space  142  6.4.3  Criteria for polymorphisms  146  6.4.4  Stable period-2 polymorphisms  152  Spruce Budworm Model  164  7.1  Introduction  164  7.2  Background  165  7.3  Model equations  167  7.3.1  Foliage  167  7.3.2  Branch surface area  7.3.3  Budworm  7.4  . .  168 169  Model analysis  174 vi  7.5 8  7.4.1  Preliminaries  174  7.4.2  The effects of small larval dispersal  177  7.4.3  The effects of adult dispersal . . . . -  181  7.4.4  Biological interpretation  186  7.4.5  What causes outbreak cycles?  189  7.4.6  The effects of the other processes  194  7.4.7  The effects of predation  196  Conclusion  203  Conclusion  204  8.1  Main results  204  8.2  Limitations  205  8.3  Future possibilities  206  Bibliography  207  Appendices  217  A  D y n a m i c a l systems theory  217  A.l  217  Introduction  A.2 Basic concepts  217  A.2.1  Bifurcation diagram  217  A.2.2  Bifurcation point  218  A.2.3 Chaos  219  A.2.4  Continuation branch  219  A.2.5 Domain of attraction  219  A.2.6  221  Equilibrium point  A.2.7 Hard loss of stability  221 vii  A.2.8 Heteroclinic orbit  222  A.2.9 Homoclinic orbit  223  A.2.10 Hopf bifurcation  224  A.2.11 Hysteresis  226  A.2.12 Limit cycle  227  A.2.13 Limit point  228  A.2.14 Local stability  229  A.2.15 Parameter  229  A.2.16 Period-doubling bifurcation  230  A.2.17 Phase portrait  230  A.2.18 Pitchfork bifurcation  230  A.2.19 Qualitative behaviour  232  A.2.20 Saddle point  233  A.2.21 Sink  234  A.2.22 Soft loss of stability  234  A.2.23 Source  235  A.2.24 State variable  235  A.2.25 Transcritical bifurcation  236  A.3 Some mathematical details  237  A.3.1 Introduction  237  A.3.2 Equilibrium points and local stability  238  A.3.3 Global bifurcations  245  A.3.4 Periodic orbits  248  A.3.5 Maps (systems of difference equations)  252  A.3.6 Stability of bifurcations under perturbations  256  A.3.7 Multiple degeneracy  258 viii  A. 4 Conclusion  258  B Numerical details  259  B. l  Introduction  B.2 Theory  259 :  260  B.2.1  Continuation methods  260  B.2.2  Detection of bifurcations  262  B.2.3  Stability  263  B.3 Available computer packages  264  B.3.1  AUT086  265  B.3.2  Interactive AUTO  269  B.3.3  XPPAUT  270  B.3.4  AUT094  . 273  B.3.5  DSTOOL  274  B.3.6  Other packages  275  B.4 Using the packages  276  B. 5 Pointers and warnings  281  C Mathematical details for the sheep-hyrax-lynx model C. l  Modelling delays in system dynamics models  C. 2 Rescaling model equations  285 285 286  D Mathematical details for the budworm-forest model  287  D. l Derivation of new foliage equation in spruce budworm model  287  D.2 Summary of model equations  288  Listings  290  ix  L i s t of Tables  3.1  Table showing the processes affecting each state variable and some of the abbreviations used in the sheep-hyrax-lynx model equations  37  4.1  Table showing the reference parameter values for the ratio-dependent model. 71  7.1  Table of standard parameter values for the budworm-forest model  x  173  L i s t of Figures  2.1  One-parameter bifurcation diagram obtained by varying a in the basic system of equations with a = 0.6, b — 0.3, c = 0.4 and d = 0.2  13  2.2  Phase portraits for the basic model  14  2.3  Two-parameter bifurcation diagram of (e, a)-parameter space, (b) Phase portraits corresponding to regions (i), (ii) and (iii) in part (a)  2.4  15  One-parameter bifurcation diagram obtained by varying e i n B a z y k i n ' s prey competition model w i t h a — 0.6, b = 0.3, c = 0.4, d = 0.2 and a = 0.3.  2.5  Two-parameter continuation of the Hopf bifurcation shown i n the previous figure  2.6  18  Two-parameter bifurcation diagram of (e, a)-parameter space for the prey competition model with a = 0.6, b = 0.3, c = 0.4 and d = 0.2  2.7  17  19  Phase portraits corresponding to the points marked w i t h *'s in the previous figure  20  2.8  T i m e plots corresponding to the phase portraits in the previous figure.  .  2.9  (a) and (b) Two-parameter bifurcation diagrams of (a, /^-parameter space,  21  (c) Phase portraits corresponding to regions (i), (ii), (iii) and (iv) i n parts (a) and (b)  23  2.10 One-parameter bifurcation diagram obtained by varying a i n the predator competition model w i t h a = 0.6, b = 0.3, c = 0.4, d = 0.2 and [i — 0.06. . .  24  2.11 D i a g r a m showing the period of the l i m i t cycle oscillations i n the previous figure as a function of a  25  xi  2.12 A two-parameter bifurcation diagram of (a, ,u)-space for a = 0.6, b — 0.3, c = 0.4 and d = 0.2  25  2.13 Phase portraits corresponding to the points marked with *'s in the previous figure  26  2.14 Time plots corresponding to the phase portraits in the previous figure.  .  27  2.15 One-parameter bifurcation diagrams for (a) p = 0.1, (b) LI = 0.074 and (c) n = 0.02  28  3.1  The lynx predation multiplier, LPM, as a function of prey abundance, Ap.  36  3.2  Three one-parameter bifurcation diagrams with revenue plotted as a function of HCN  40  3.3  One-parameter bifurcation diagram of revenue versus LCN for i/cw=0.35.  41  3.4  One-parameter bifurcation diagrams obtained from varying SFCN  42  3.5  One-parameter bifurcation diagrams obtained from varying Lp^  43  3.6  The grazing multiplier function (GM)  3.7  The pasture multiplier (PM) as a function of pasture availability (PA)- • -  3.8  One-parameter bifurcation diagrams obtained from varying SFCN for the  a s  a  function of pasture availability.  new model which includes a pasture limiting multiplier 3.9  46 48  49  One-parameter bifurcation diagrams obtained from varying LFN for the new model which includes a pasture limiting multiplier  50  3.10 Time plots obtained using (a) the original model and (b) the modified model with SFCN — 0-5  51  3.11 One-parameter bifurcation diagram of revenue as a function of LCN for the modified model  52  3.12 Two-parameter continuation of the limit point in the previous figure.  xii  . .  53  3.13 Two-parameter bifurcation diagram of the HCN and LCN parameter space for the modified model.  54  3.14 (a) Surface plot and (b) contour plot of revenue as a function of the hyrax and lynx culling normals  55  4.1  Isoclines in the M\M  and M2M3 planes for the original model  4.2  One-parameter bifurcation diagrams obtained by varying 73 in (a) the  2  68  original model (a;=0, i — 1,2,3) and (b) the modified model (a =0.001, t  i = 1,2,3). The state variable Mi is plotted on the y-axes 4.3  72  One-parameter bifurcation diagrams obtained by varying (f> in the original 3  model 4.4  74  One-parameter bifurcation diagrams obtained by varying </>3 in the modified model  4.5  75  Time plots of (a) M  (b) M and (c) M for <f> = 0.16. All the other  u  2  3  3  parameter values are as in the reference parameter set 4.6  Isoclines in the M M X  2  76  plane with 4> = 0.06 in (a) and </> = 0.05 in (b) 3  3  and (c). 4.7  78  One-parameter bifurcation diagrams obtained by varying $73 in the original model  4.8  80  One-parameter bifurcation diagrams obtained by varying fi in the modi3  fied model 4.9  81  One-parameter bifurcation diagrams showing the effects of varying cf> for 2  the original model  83  4.10 One-parameter bifurcation diagrams showing the effects of varying </> for 2  the modified model  84  xiii  4.11 One-parameter bifurcation diagrams showing the effects of varying 71 for (a) the original model and (b) the modified model  85  4.12 Two-parameter bifurcation diagrams showing the effects of varying both 71 and fa on the positions of the l i m i t points and Hopf bifurcations i n the previous figure  86  4.13 One-parameter bifurcation diagrams showing the effects of varying 72 for (a) the original model and (b) the modified model  87  4.14 Two-parameter bifurcation diagrams showing the effects of varying both 72 and </>3 on the positions of the Hopf bifurcations in the previous figure.  88  4.15 Two-parameter bifurcation diagram of the Hopf bifurcation continuations for the modified model w i t h a =0.002 (i = 1,2, 3)  89  t  4.16 T i m e plots corresponding to the points that are marked with *'s i n the two-parameter diagrams for the original model and the modified model w i t h ai^O.OOl (z = 1, 2,3) and a,=0.002 (i = 1,2,3)  90  4.17 One-parameter bifurcation diagrams showing the effects of varying O i for (a) the original model and (b) the modified model 4.18 One-parameter bifurcation diagrams showing the effects of varying 0  91 2  for  (a) the original model and (b) the modified model  92  4.19 Two-parameter bifurcation diagram obtained using the modified model w i t h a,-=0.001, (i = 1,2,3) and  7 l  = 0.4  94  4.20 Two-parameter bifurcation diagram obtained using the modified model w i t h a -=0.001, (i = 1,2,3) and 71 = 0.6 t  95  4.21 Two-parameter bifurcation diagram obtained using the modified model w i t h a -=0.001, (i = 1,2,3) and t  7 l  = 1.2  96  4.22 Two-parameter bifurcation diagram obtained using the modified model w i t h a,-=0.001, {i = 1,2,3) and 71 = 1.8 xiv  97  4.23 Two-parameter bifurcation diagram obtained using the modified model w i t h a;=0.001, (i = 1,2,3) and  7  l  = 2.4  98  4.24 One-parameter bifurcation diagrams obtained by varying 72 w i t h and (a)  fa = 0.07, (b) fa = 0.17 and  (c)  fa  7  l  = 0.4  = 0.25  99  4.25 One-parameter bifurcation diagram obtained by varying 72 w i t h 71 — 1.2 and  fa = 0.2  101  4.26 Examples of isocline configurations showing different possibilities for the position of the tritrophic equilibrium  104  4.27 Examples of isocline configurations at different points i n (71,^2)-space.  .  105  4.28 Isocline configurations w i t h (a) a; = 0, (b) a,- = 0.001 and (c) a - = 0.005 2  (i = 1,2,3) together w i t h the reference parameter set  106  4.29 One-parameter bifurcation diagrams when a; = 0.005 (i = 1,2,3)  108  4.30 P l a n t isoclines for (a) the original and (b) the modified model  109  5.1  Dynamics in the (p, JV)-plane for a n = 2.1, a i = 1.9, a 2  2 2  = 1-1, fen = 1.0,  612 = 0.904 and fe = 0.54  119  22  5.2  One-parameter bifurcation diagram w i t h an = 2.1, a 6n = 1.0 and b  12  5.3  = 0.906. K  u  = 2.1 and K  = 1.9, a  r  12  12  2 2  = 1.1,  = 2.097  12  D i a g r a m of the (p, A )-plane for a n = 2.1, a b  = 1.1, 121  2  12  2 2  = 0.904 obtained using A U T O  One-parameter bifurcation diagram w i t h a n = 2.1, a\ 611 = 1.0 and b  5.4  = 1-9, a  i2  = 1.9, a  123 2 2  = 1.1, bu = 1.0,  = 0.906 and 622 = 0.52 showing the domains of attraction for the stable  phenomena 5.5  124  Two-parameter bifurcation diagram w i t h a n = 2.1, a  1 2  = 1.9, a  = 1.1  and fe = 1.0 obtained using A U T O  125  Two-parameter bifurcation diagram including additional curves  126  n  5.6  2 2  xv  5.7  One-parameter bifurcation diagram with a n = 2.1, a i  = 1-9, a  2  22  = 1-1,  611 = 1.0 and &i2 = 0.908 obtained using A U T O 5.8  Dynamics in the (p, AQ-plane with a n = 2.1, a  12  and various combinations of b  and 6  12  22  127 = 1.9, a  = 1.1, &u = 1.0  22  which correspond to regions A to  H i n the two-parameter figure 5.9  129  One-parameter bifurcation diagram with a  = 1.9, a  12  = 1.1, Wi = 1-0,  22  612 = 0.905 and fe = 0.525 obtained using A U T O  130  22  5.10 Dynamics i n the (p, 7V)-plane for a 6  = 0.525 and (a) a  22  n  12  = 1.9, a  = 2.68, (b) a  n  = 1.1, 6  22  X1  = 1.0, &i2 = 0.905,  = 2.69 and (c) a  n  = 2.75. In (a)  there is a period-8 attractor at p — 1. In (b) this changes to a period-16 attractor and in (c) we have what appears to be a chaotic attractor. . . . 5.11 One-parameter bifurcation diagram with an = 2.1, a  22  = 2.1, bu = 1.0,  612 = 0.908 and & = 0.53 obtained using A U T O  133  22  5.12 Dynamics in the (p, AQ-plane for a n = 2.1, a  i 2  = 3.8, a 2 = 2.1, bn = 1.0, 2  &i = 0.908 and 622 = 0.53 showing a stable period-4 polymorphism. . . . 2  5.13 Two-parameter continuation of the period-doubling bifurcation w i t h a n 2.1, a  22  =  2.1, bu = 1.0 and b  134  5.14 Examples of complex dynamics, (a) A n interior period-8 orbit for a n a  = 2.9, a  12  22  = 2.5, fen = 1.0, 612 = 0.908 and 6  chaotic attractor for an = 2.6, a i = 3.1, a 2  22  22  = 2.3,  = 0.95. (b) A n interior  = 2.5, &n = 1.0, 612 =  0.908  and fe = 0.95  135  22  6.1  t£> 2 w i t h r22 = 0.8 and  6.2  A one-parameter bifurcation diagram obtained by varying k\\ w i t h r n  2  fixed at 0.7 ( r  22  133  =  = 0.908  12  132  AC22  = 0.6  = 0.8 and k  22  142  = 0.6)  xvi  143  6.3  Diagrams of the (p, iV)-plane for a number of different values of ku r  n  with  fixed at 0.7  144  6.4  Diagrams of the ( r C n , r n ) - p a r a m e t e r space  145  6.5  Diagrams of the (p, JV)-plane corresponding to the regions A to H in the two-parameter diagram  6.6  147  Diagrams of the (p, AQ-plane corresponding to the regions N to P in the two-parameter diagram  6.7  148  Examples of the fitness functions for parameter values corresponding to (a) a stable p o l y m o r p h i s m ( r n = 0.7,ku = 2.0, r  = 0-8,k = 0.6) and  22  22  (b) an unstable polymorphism ( r n = 0.4,fen = 2.0, r  22  6.8  Curves given by w\ = w  2  22  = 0.6). .  22  = 0.8, k  22  = 0.6).  (a) Fitness functions and (b) (p, iV)-plane for r n = 1.3, ku = 0.5, r 22  . . . 151 = 7.5  22  and k  149  and w — 1 for parameter values corresponding  to a stable p o l y m o r p h i s m ( r n = 0.7, ku = 2.0, r 6.9  = 0.8,/c  = 4.57  153  6.10 A p a r t i a l one-parameter bifurcation diagram obtained by varying r . 22  . .  154  6.11 Two-parameter continuations of the period-doubling bifurcation H B * i n the previous figure obtained by varying (a) r n , (b) ku addition to r  and (c) k  22  in 155  22  6.12 (a) Fitness functions and (b) (p, AQ-plane for r n = 0.2, ku = 5.0, r and £:  22  22  = 0.3  = 0.4  156  6.13 One-parameter diagrams obtained by varying ku using (a) the original model and (b) the second iterate of the model  157  6.14 A bifurcation diagram showing the two-parameter continuation of the period-doubling at ku  = 4.226 and the subsequent period-doubling at  ku = 5.358 i n the previous figure  158  xvii  6.15 An example of an interior chaotic attractor obtained for r hi  = 5.0, r  2 2  = 0.3 and k  22  n  =  0.18,  = 0.4  159  6.16 An example of a fitness function configuration where u>n is always superior to u>  160  22  6.17 The (kn, r )-parameter space showing the region of higher order stable 11  polymorphic behaviour and the region corresponding to fitness function configurations of the type shown in the previous figure 6.18 w  = 2.5  161  6.19 The new two-parameter space showing the same regions as before  161  7.1  Graph of G versus H  172  7.2  One-parameter bifurcation diagram of budworm larval density versus dsL- 177  7.3  Diagrams of budworm larval density versus foliage for (a) dsL = 0.2, (b)  22  d  SL  7.4  with r  2 2  = 0.25 and k  160  = 0.35, (c) d  22  SL  = 0.8 and (d) d  SL  = 0.9  178  Time plots of (a) budworm larval density, (b) foliage density and (c) branch surface area density versus time for dsL = 0.35  7.5  179  One-parameter bifurcation diagram of budworm larval density versus for d  SL  A  t  h  r  = 0.45  181  7.6  Two-parameter bifurcation diagram of  versus dsL  7.7  Diagrams of budworm larval density versus foliage for the regions marked  Athr  A-I in the previous figure  182  184  7.8  Simplified two-parameter bifurcation diagram of  7.9  Two-parameter bifurcation diagram of  A  t  h  r  t  h  r  versus d$L  187  versus dsL for the simplified  model which only includes dispersal 7.10 Isoclines of recruitment versus budworm density  xvm  A  191 194  7.11 Two-parameter bifurcation diagram of A hr t  versus dsL for the predation  model which includes dispersal as well as predation 7.12 One-parameter bifurcation diagram for p  max  with d$L = 0.4 and  196 A h t  r  —  5. 198  7.13 Time plots of budworm larval density for (a) p x = 0.6 x 23 000 and (b) ma  Pmax = 3.4 x 23 000  200  7.14 Two-parameter bifurcation diagram of p  max  A.l  versus dsL  201  Time plots and a phase portrait showing the domain of attraction of an equilibrium point  219  A.2 Phase portrait showing the domains of attraction of equilibrium points in two dimensions  220  A.3 Hard loss of stability  221  A.4 Example of a heteroclinic orbit  222  A.5 Example of a homoclinic orbit  223  A.6 A bifurcation diagram of a Hopf bifurcation and phase portraits corresponding to different parameter values  224  A.7 Bifurcation diagram of a Hopf bifurcation where the periodic orbits are unstable  225  A.8 Bifurcation diagram of hysteresis  226  A.9 (a) Time plot and (b) phase portrait of limit cycle behaviour  227  A. 10 (a)Bifurcation diagram showing a limit point and (b)a phase portrait corresponding to p = p,\  228  A.11 (a) Period-doubling bifurcations at Ai and A . (b)Behaviour over time for 2  the state variable Xi  230  A. 12 Derivation of a phase plane showing the time-dependent behaviour of two variables, Xi and x  231  2  xix  A. 13 (a)Bifurcation diagram of a pitchfork bifurcation and phase portraits corresponding to different parameter values  232  A. 14 Example of a saddle point  233  A. 15 Phase portraits and time plots of a stable node and a spiral attractor. . .  234  A. 16 Phase portraits and time plots of an unstable node and a spiral repeller.  235  A. 17 (a) A bifurcation diagram of a transcritical bifurcation and phase portraits corresponding to different parameter values  236  A. 18 A summary of the local stability behaviour near an equilibrium point of the continuous system when m = 2  240  A. 19 (a)Two-parameter bifurcation diagram showing a cusp point and the positions of the two limit points associated with the hysteresis as both /J, and A are varied, (b)One-parameter bifurcation diagrams corresponding to different, fixed values of A in part (a) and with fi as the bifurcation parameter  243  A.20 (a)Phase portrait of a saddle connection or heteroclinic orbit. (b)Phase portrait of a saddle loop or homoclinic orbit  244  A.21 (a)Phase portraits for parameter values near a saddle connection or heteroclinic orbit. (b)Phase portraits for parameter values near a saddle loop or homoclinic orbit  245  A.22 Phase portraits near a saddle loop or homoclinic orbit showing possible behaviour near the second equilibrium point  246  A.23 Schematic representation of a Poincare section and a limit cycle in three dimensions  248  A.24 Examples of Hopf bifurcations  249  A.25 State space diagrams of a spiral sink for (a) a continuous model and (b) a discrete model  252 xx  A.26 (a)One-parameter bifurcation diagram showing period-doubling bifurcations at /it =  n*i  and  fx  =  LI*  2  for a discrete system.  (b)State space  diagrams showing the dynamics at LI = fix, LI = \i and LI = u.3  253  2  A.27 Time plot of a period-2 orbit  254  A.28 (a)State space diagram showing an invariant circle, (b) Time plot of the situation in (a) in terms of Xx  255  A.29 Possible results of perturbing transcritical and pitchfork bifurcations.  xxi  . .  256  Acknowledgements  I could not have researched and written this thesis without the help of numerous people. Foremost among these I would like to thank my supervisor, D o n L u d w i g , most sincerely for his invaluable suggestions and guidance. I benefited greatly from his experience i n both m a t h e m a t i c a l and ecological disciplines. I would also like to thank Gene N a m k o o n g for suggesting the population genetics models and for giving me both direction and encouragement while I studied them. I a m indebted to B a r d Ermentrout for writing the package X P P A U T and making it freely available. His suggestions and interest in the project were also greatly appreciated. I am also grateful to C o l i n C l a r k , Wayne Nagata, Gene N a m k o o n g and James Varah for reading a previous version of the manuscript and for their numerous helpful comments and suggestions. There are many others who responded to questions and email messages or provided general encouragement: Leah Edelstein-Keshet, John Hearne, Johan Swart, and also P e ter A b r a m s , Eugene Allgower, A l a n B e r r y m a n , Eusebius Doedel, Lev G i n z b u r g , J o h n Guckenheimer, Andrew Gutierrez, Alexander K h i b n i k , L i x i n L i u , Jesse Logan, Peter T u r c h i n , Graeme Wake and Peter Yodzis. T h a n k - y o u for your time and your interest. T h a n k - y o u also to my fellow graduate students i n the Mathematics department for e m pathising w i t h me through the frustrations which accompany most graduate studies. A very special thank-you to m y family and non-mathematics friends for your endless encouragement and interest. Your support was a continual source of strength to me. F i n a l l y , I would like to acknowledge the E m m a S m i t h , Oppenheimer and I.W. K i l l a m Trusts for their financial support which made my studies at U . B . C . possible.  xxn  Chapter 1  Introduction  1.1  G e n e r a l overview  In this thesis I demonstrate how techniques from dynamical systems theory can be applied to ecological models i n order to study their qualitative behaviour. T h e techniques allow one or two parameters to be varied across ranges of values so that a comprehensive picture of their effects on the behaviour of the model can be determined. Since computer software is used to take care of the mathematical details, both mathematicians and ecologists can make use of these techniques. I hope to reach the latter group i n particular, by showing how d y n a m i c a l systems theory can increase our understanding of the behaviour of a model considerably and thus help us formulate more plausible models. B o t h continuous and discrete models are considered. In the next section I outline m y objectives more formally.  I then describe how I  go about fulfilling these aims w i t h specific references to later chapters i n the thesis. I conclude this introduction w i t h a discussion of why I chose this topic and its place amongst current areas of research.  1.2  Research objectives  Ecological models and qualitative analyses of these models can give insight into the most important mechanisms at work i n an ecological system. However, the mathematics required for a detailed analysis of the behaviour of a model can be formidable. Because  1  Chapter 1.  Introduction  2  of the uncertainty associated with parameter values in nature, solving a system of model equations for a fixed parameter set is insufficient. A more informative approach is to study the behaviour of a model for ranges of parameter values, but this requires complicated mathematical techniques. It would be of considerable interest, particularly to ecologists, if some of these techniques could be applied without the requirement of further formal mathematical training. W i t h the above in mind the main aims of my thesis are twofold: • to provide examples of the usefulness of dynamical systems theory in analysing the behaviour of ecological models—in particular those techniques which describe the effects of varying parameters across ranges of values, and • to demonstrate how certain computer packages can aid the analysis by taking care of the mathematical details. In applying these aims I uncovered biological implausibilities in two models and improved on previously obtained approximate results in another. The techniques also highlighted some limitations of more traditional methods of analysis. The  computer packages I used are D S T O O L [10], Interactive A U T O [117] and X P -  P A U T [35]. Descriptions of their capabilities, as well as suggestions regarding their use, are included in appendix B together with references to a few other packages that are available.  1.3  Thesis outline  My approach to achieve the above aims was to analyse ecological models—both continuous and discrete—to demonstrate the kinds of results and information that can be obtained using dynamical systems techniques.  Chapter 1.  Introduction  3  I begin i n chapter 2 w i t h the analysis of three fairly simple predator-prey models which have already been studied by B a z y k i n [14]. This chapter is intended as an introduction to some of the terminology and techniques of dynamical systems theory as well as to the use of the available computer packages. B a z y k i n [14] studied the models analytically. T h e computer packages allowed me, a novice, to reproduce and i n fact improve upon his results. The chapter also illustrates how the computer packages can encourage an iterative approach to modelling which may aid the development of more plausible models. H a v i n g demonstrated the reliability of some of the computer software i n chapter 2, I wanted to apply the techniques to a few more recent models from the literature. Since a number of theoretical models have been studied using dynamical systems techniques (for example, [5, 24, 26, 29, 33, 102]), I wanted to look at a more practical example. In chapter 3 a system dynamics model of a sheep-pasture-hyrax-lynx system is analysed. T h e model is a large one consisting of 10 ordinary differential equations and numerous parameters. E v e n the most knowledgeable theoretician would find an analysis of this model using pencil and paper a formidable task.  T h e dynamical systems techniques  prove to be a useful alternative to the sensitivity analyses which are traditionally used when studying these models. In particular, an improvement to the formulation of the model is suggested as a result of the analysis. Chapter 4 returns to a more theoretical model describing a plant, a herbivore and a predator.  T h e model is an example from a controversial area of current research  known as ratio-dependent modelling. The analysis in this chapter highlights some of the biological implausibilities and mathematical difficulties associated w i t h ratio-dependent models which may be important for guiding future research. A modification to the model equations is analysed i n conjunction w i t h the original model and reveals that the latter is structurally unstable. M a n y systems in nature (for example, insects having nonoverlapping generations) are  Chapter 1.  Introduction  4  better represented by discrete models than by continuous ones. However, discrete models tend to exhibit more complex behaviour than continuous ones because of the inherent time delays i n the equations [84]. As a result detailed analyses have been restricted m a i n l y to one-dimensional examples (see, for example, [82, 83, 84]) although there are some two-dimensional examples (see [15, 94]). In chapters 5 and 6 I consider two population genetics models which are two-dimensional. B o t h models have fairly simple m a t h e m a t i c a l formulations involving only two alleles but they are capable of displaying complicated dynamics. I focus on the dynamics of the heterozygote. The model i n chapter 5 has been partially studied using pencil and paper and numerical simulation techniques. However, a more detailed analysis was restricted by the need for more complicated methods to take care of the complex nonlinearities. The dynamical systems techniques demonstrate the theoretical results fairly easily and also show the relative frequency w i t h which different types of qualitative behaviour can be expected to occur. The chapter focusses on periodic dynamics as this behaviour is the most difficult to study by hand. T h e model i n chapter 6 is a modification of that in chapter 5 but it has not been studied in detail before. T h i s is not surprising since it is not even possible to find explicit expressions for the equilibrium points. Computers are particularly useful in such situations. For this model it is found that there is always the possibility of one of the alleles being excluded and that the threat of extinction is high for many parameter sets. C r i t e r i a for determining the existence and stability of polymorphic equilibria are given and periodic dynamics are also studied. To round off the thesis I wanted to see how the dynamical systems techniques would fare in the context of a more practical, and hence more complicated, discrete model. Chapter 7 considers a model of a defoliating insect system, namely the spruce budwormforest system. Despite the complexity of the dynamics, useful insights are obtained into the mechanisms responsible for insect outbreaks and the relative effects of dispersal and predation. Outbreaks are found to occur for a wide range of parameter values and regions  Chapter 1.  Introduction  5  of m u l t i p l e stable states are also located. Specific conclusions relating to the particular examples are included at the end of each chapter. M o r e general conclusions are summarised i n chapter 8.  Special mention  should be made of two of the appendices. A p p e n d i x A contains a glossary of the basic d y n a m i c a l systems concepts which are used i n the m a i n body of the thesis, as well as a brief introduction to some of the underlying mathematical theory. Diagrams are used wherever possible so as to keep the mathematical details to a m i n i m u m as the appendix is intended for those who may have had little prior exposure to dynamical systems concepts. A p p e n d i x B describes how computers can be used to implement the d y n a m i c a l systems techniques. Descriptions of the capabilities and relative advantages and disadvantages of the packages that I used, as well as procedures for obtaining time plots, phase portraits and bifurcation diagrams, are given.  Some pointers and warnings regarding  their use are also included. Examples of computer listings for the various models are placed after the appendices. I must emphasise that it is not my a i m to provide a comprehensive structure whereby every detail of a system of equations can be understood. This would be an impossible task. Instead I want to develop a procedure which can be applied to a wide variety of practical situations. I would like to emphasise the word practical since it is very easy for a model analysis to become more of a mathematical exercise than one of biological relevance.  Detailed mathematical analyses may require more complicated techniques  than I have used in order to study complex phenomena. W h i l e these phenomena may be of intellectual interest, they are often of little practical use. M y viewpoint is summarised by the following quote from Adler and Morris [2]:  Chapter 1.  Introduction  6  O n l y by avoiding the unthinking use of familiar and mathematically convenient models and by having the discipline to ignore interesting but dynamically unimportant interactions, can we ever hope to develop predictive ecological theory.  1.4  Motivation  M a t h e m a t i c a l models have been used to describe ecological systems for many decades. However, the interdisciplinary nature of the field has led to some conflict i n opinions. M a n y experimental ecologists argue that theoretical models are too simple to adequately describe natural systems, but complicated models are often intractable to mathematical analysis. According to Holling et al.  [62] "a simple but well-understood model is the  best interface between a complex system and a complex range of policies." However, the complexity of ecological systems and the perceived added realism of larger, more complex models has led many ecologists to favour the latter. Because of the inevitable uncertainty associated w i t h the parameter values in an ecological model [40, 53, 115, 121], it is not sufficient to merely simulate the model equations over time and observe the behaviour. A different set of parameter values may give rise to very different dynamics. According to Walker et al. [121] many parameters of ecological models are really variables. They are chosen to be constants for convenience, simplification or because information regarding the relevant dynamics is lacking. Hence, it is important to know whether altering parameter values w i l l significantly affect the predictions of the model. This is not a t r i v i a l task when large numbers of parameters are involved. A s has already been mentioned, we can vary parameters across ranges of values using techniques from dynamical systems theory. In this way we can obtain information  Chapter 1.  Introduction  regarding the presence and nature of attractors  7  1  in the system. Whereas transient dy-  namics vary w i t h the i n i t i a l values of the state variables and the time period over which solutions are calculated, the techniques in this thesis are concerned w i t h the behaviour of the system once the i n i t i a l transients have died away. The attractors determine this long-term behaviour. M y viewpoint is that these qualitative analyses of ecological models are indispensable if we hope to use the models to gain insight into real ecological systems. T h e application of the dynamical systems techniques to a system of nonlinear equations can be a formidable task for a mathematician, let alone a non-mathematician. It is also time-consuming when a large number of parameters is involved.  In such cir-  cumstances computer programs can be of great assistance. In fact, Seydel [111] asserts that "the extensive application of numerical methods is indispensable for practical b i furcation and stability analysis". Although analytical methods can provide remarkable results, they have two strong limitations [111]. F i r s t , in many cases numerical methods are needed to evaluate analytical expressions anyway.  A n d secondly, analytical results  are generally local and only hold for 'sufficiently small' distances where 'sufficiently small' is left unclarified. Fortunately a number of computer packages have become available in recent years to aid the analysis. A few of these have already been mentioned. It is one matter to do the analysis but we also need to convey the results effectively. Edelstein-Keshet [34] comments that pictures derived from qualitative analyses are often more informative than mathematical expressions. In this thesis the traditional time plots and phase portraits are used to display results as well as bifurcation diagrams. T h e latter diagrams provide a concise way of summarising the effects of different parameter values on the behaviour of the system. 1  S e e section A . 2 . 5 for a definition of this t e r m .  Chapter 1.  1.5  Introduction  8  T h e bigger picture  Qualitative analyses are not new and can be traced back at least to P o i n c a r e . In eco2  logical circles names such as L o t k a and Volterra [73, 120], Rosenzweig and M a c A r t h u r [76, 103], H o l l i n g [61], M a y [79, 80, 82, 83] and G i l p i n [44], and many others, are wellknown for their qualitative analyses of various models.  Most studies have involved  predator-prey models [14, 44, 79, 80, 98] but other systems have also been analysed [50, 70, 74, 83, 95]. However, all these models are fairly simple theoretical models because of the mathematical difficulties encountered with more complicated models. T h e introduction of various computer packages since the mid-1980's has allowed dynamical systems techniques to be applied w i t h greater ease as well as to more complicated models. However, there are relatively few examples where ecological models have been studied using these packages and most of the papers in this category are very technical and require considerable mathematical knowledge [5, 26, 29, 33, 48, 86, 87, 102]. Few people have heeded the suggestion by Oster and Guckenheimer [97] that less exhaustive analyses but of more meaningful models (from a biological viewpoint) would be more useful and of greater interest to biologists. T h e papers by Collings [23], Collings and W o l l k i n d [24], Collings et al. [25] and W o l l k i n d et al. [127] study a fairly practical biological control model of mite interactions. However the mathematics is still complicated and difficult for the reader w i t h little prior exposure to dynamical systems theory.  Of particular relevance to this thesis is that  these papers demonstrate the power of the computer package A U T 0 8 6 [28] and illustrate how conclusions regarding model sensitivity and resilience can be drawn from bifurcation diagrams. T h e y also derive meaningful ecological implications from their results. A l l four A H o p f b i f u r c a t i o n (see section A . 2 . 1 0 ) , w h i c h is an i m p o r t a n t concept i n d y n a m i c a l systems theory, is also k n o w n as a P o i n c a r e - A n d r o n o v - H o p f bifurcation a n d the P o i n c a r e - B e n d i x o n t h e o r e m is fundam e n t a l to q u a l i t a t i v e analyses. F u r t h e r details can be found i n A r n o l d [7] a n d W i g g i n s [124]. 2  Chapter 1.  Introduction  9  papers note that many of their conclusions would not have been obtained without the use of A U T 0 8 6 [28]. In particular, Collings and W o l l k i n d [24] obtained three previously undiscovered possibilities for qualitative behaviour using a predator-prey model of the type studied by B a z y k i n [14]. In this thesis I hope to take the road less travelled by showing how d y n a m i c a l systems techniques can lead to biologically useful and meaningful results without the requirement that the user have an extensive mathematical background in the field. I begin i n the next chapter w i t h a few models that have already been studied using pencil and paper and show what I was able to accomplish with the aid of computer packages.  Chapter 2 Preliminary Example  2.1  Introduction  T h i s chapter is for readers for whom concepts such as bifurcations and bifurcation d i agrams are relatively new as well as for those who are sceptical about the accuracy and reliability of computer packages such as D S T O O L [10], Interactive A U T O [117] and X P P A U T [35]. D y n a m i c a l systems techniques are applied to three fairly simple predatorprey models to show how certain parameter values affect the qualitative behaviour of the models. B o t h one- and two-parameter bifurcation diagrams are used to summarise the results. Behaviour i n different regions of these diagrams is explained using phase portraits. T h e three models differ from one another by the addition or subtraction of only one or two terms. This chapter therefore exemplifies an iterative approach to m o d e l l i n g — the relative ease w i t h which qualitative analyses may be done using the abovementioned computer packages allows a fairly quick determination of the effects of model alterations. T h i s can facilitate the formulation of more plausible models. These predator-prey models have already been studied analytically by B a z y k i n [14]. I have included his results for comparison with those obtained by the computer software. The latter are i n fact more accurate in certain situations and the results can be obtained without a detailed knowledge of the underlying mathematical techniques. I begin the chapter with a description of the first model which is a basis for the other two. I use X P P A U T to obtain a one-parameter bifurcation diagram and show how  10  Chapter 2. Preliminary  Example  11  this summarises Bazykin's results. T h e next two sections discuss two modifications to the basic model. M a n y of the phenomena that occur i n later chapters of the thesis are introduced here. For those who plan to read this chapter, a quick reading of the first part of appendix A (namely, section A . 2 ) may prove useful. This section is non-technical but introduces a l l the basic terminology as well as the conventions I use i n the figures.  2.2  Basic model  One of the first predator-prey models to be proposed and extensively studied was the model developed independently by L o t k a and Volterra i n the 1920's (see [34] for a description of the model and its analysis). The model equations are  x = ax — bxy V = ~cy + dxy where x represents prey density and y predator density. and positive.  (2-1) A l l the parameters are real  T h e term ax describes the exponential growth of the prey population  i n the absence of predators and — cy describes the exponential decline i n the predator population i n the absence of prey. The terms — bxy and dxy describe the interaction between predator and prey. F r o m a biological viewpoint, this linear dependence of the rate of predation and predator reproduction on the number of prey is considered to be a rather unrealistic approximation [14] . Also, from a mathematical viewpoint, the system is structurally unstable since an arbitrarily small perturbation to the model can change its qualitative dynamics. For example, replacing the exponential growth of the prey w i t h logistic growth changes the dynamics from (neutral) cycles to a stable e q u i l i b r i u m (see [34]). A number of modifications to this model have been studied since the 1920's. In particular, B a z y k i n ' s [14] work is well-known among ecologists because of his comprehensive  Chapter 2.  Preliminary  12  Example  qualitative analyses of the models and his accompanying diagrams, which summarise the different possible behavioural regimes. One of Bazykin's modifications to the LotkaVolterra equations (2.1) is given by the system  y =  + &  (-) 2 2  He justified using Michaelis-Menten interaction terms by analogy w i t h the mechanism of enzyme reactions. The denominators of these terms prevent u n l i m i t e d predation of prey and u n l i m i t e d growth of the predator population with the growth of prey density, respectively. T h e H o l l i n g type II functional response term is very similar to these terms and is based on biological mechanisms [59, 60]. Before we can apply the dynamical systems techniques we need to choose parameter values. I chose a = 0.6,6 = 0.3, c = 0.4, d = 0.2 and a = 0.1 but any other reasonable values would do. Using X P P A U T ( D S T O O L could also have been used) I found that there is only one non-trivial (that is, non-zero) equilibrium point (see section A . 2 . 6 for an explanation of this phenomenon) corresponding to these parameter values and that it is unstable (see section A.2.14 for an explanation of this term).  Using this equilib-  r i u m point as a starting value for A U T O (either Interactive A U T O or X P P A U T can be used—see appendix B ) I varied a to obtain the one-parameter bifurcation diagram (see section A.2.1) shown i n figure 2.1. As can be seen from this figure, there are no bifurcations (see section A.2.2) and the equilibrium point remains unstable as a is varied. B u t the figure does show how the equilibrium value of x changes w i t h a. We can also view this diagram in terms of y and a using X P P A U T . The diagram is exactly the same as figure 2.1 because of the symmetry of the equilibrium point w i t h respect to x and y. Using D S T O O L or X P P A U T we can generate phase portraits (see section A.2.17) for different values of a. These are qualitatively the same as those obtained by B a z y k i n [14]  Chapter 2. Preliminary  Example  13  12 9  x  6 3 0  Figure  2.1:  .0  0.1  One-parameter  bifurcation  0.2  a  0.3  0.4  diagram obtained  by varying  0.5  a i n s y s t e m (2.2) w i t h  a = 0.6, b = 0.3, c = 0.4 a n d d = 0.2. T h e state v a r i a b l e x is p l o t t e d o n the y - a x i s .  (see figure 2.2) and verify that figure 2.1 summarises Bazykin's [14] results. T h e phase portraits i n figure 2.2 show that x increases indefinitely for a l l values of a. T h i s is an obvious shortcoming of the model and led B a z y k i n to introduce further modifications.  2.3  A d d i n g intraspecific competition among prey  To improve the model B a z y k i n added a term to the prey equation to take into account intraspecific competition among prey. Here competition refers to a decrease i n reproduction or an increase i n death rate with an increase i n prey density. T h e assumption that competition is linearly dependent on prey density results i n the system of equations • .  X  — dX '  V  =  bxy  -.1+ax ,  2  £ X  T^-x-  (2-3)  We can create one- and two-parameter bifurcation  diagrams (see section A . 2 . 1 for a de-  -cy +  scription of these terms) by varying a and e to see what effects these additional terms have on the behaviour of the model. I chose the same values for a, 6, c and d as before.  14  Chapter 2. Preliminary Example  x  Figure 2.2:  x  P h a s e p o r t r a i t s o b t a i n e d by B a z y k i n for m o d e l (2.2). Q u a l i t a t i v e l y s i m i l a r d i a g r a m s can  be o b t a i n e d by t a k i n g a = 0.6, 6 = 0.3, c = 0.4, d = 0.2 a n d (a) a = 0.1, (b) a = 0.2, (c) a = 0.3 a n d (d) a = 0.55.  Bazykin's [14] results for this system are shown infigure2.3. There are three regions in (e, a)-parameter space, each corresponding to a different form of qualitative behaviour. The phase portraits indicate the dynamics that occur in these regions. Since there are two equilibrium points of interest, A and B, in regions (i) and (ii) and one equilibrium point of interest, B, in region (iii) we expect a bifurcation to occur as the line with 1  negative slope in figure 2.3(a) is crossed from regions (i) and (ii) into region (iii). In crossing from region (i) to region (ii) point A changes from stable to unstable and a limit cycle (see section A.2.12) is initiated. Thus, we expect a curve of Hopf bifurcations (see T h e o r i g i n is also an e q u i l i b r i u m p o i n t i n b o t h cases b u t only n o n t r i v i a l e q u i l i b r i u m p o i n t s h a v i n g x > 0 a n d y > 0, (x, y) ^ (0, 0) are considered i n d e t a i l . In region ( i i i ) y < 0 at the e q u i l i b r i u m p o i n t A a n d hence this p o i n t is not o f b i o l o g i c a l interest. J  Chapter 2. Preliminary  Figure 2.3:  Example  15  T h e f o l l o w i n g d i a g r a m s are adapted from B a z y k i n [14]. (a) T w o - p a r a m e t e r b i f u r c a t i o n  d i a g r a m of (e, o ) - p a r a m e t e r space, (b) Phase p o r t r a i t s c o r r e s p o n d i n g to regions (i), (ii) and ( i i i ) i n part (a).  Chapter 2.  Preliminary  Example  16  section A.2.10) to divide these regions. In order to use A U T O to reproduce Bazykin's results we need a starting point which must be an equilibrium point.  B y choosing values of 0.3 for a and 0.1 for e, we can  either determine such a point analytically (as B a z y k i n did) or we can use D S T O O L or X P P A U T to perform the'task numerically. The latter choice involves integrating the equations forward i n time u n t i l we are near the equilibrium point.  Using  XPPAUT  this is done by choosing the menu option I N T E G R A T E followed by G O . Choosing the S I N G u l a r P O I N T option then finds the equilibrium point and indicates whether it is stable or unstable. A separate window appears with this information. T h e state variable values at the equilibrium point are then entered as the i n i t i a l point in the i n i t i a l point window. Since B a z y k i n plotted e on the x-axis in figure 2.3(a) I vary this parameter first.  In  A U T O this is done by choosing e to be the m a i n parameter in the A X E S menu. After choosing the R U N - S T E A D Y S T A T E commands, A U T O locates a transcritical  bifurcation  (see section A.2.25) at e = 0.12 (see figure 2.4). I then made the value of D S i n the N U M E R I C S window negative so that A U T O would decrease e, chose the point labelled 1 (that is, chose our original starting point) using the G R A B c o m m a n d , and then chose R U N again. A U T O finds a Hopf bifurcation at e = 0.045 in this case. B y generating periodic orbits (limit cycles) from this latter point (choose the Hopf bifurcation point using G R A B and then R U N - P E R I O D I C O R B I T ) we can see that there are stable l i m i t cycles surrounding an unstable equilibrium point.  Since a is fixed i n figure 2.4, this  one-parameter diagram describes the dynamics along a horizontal line at, say, a  = cx\  i n figure 2.3(a). In figure 2.4 I have labelled the continuation  branches A and B to indicate which  e q u i l i b r i u m point corresponds to which branch (see section A . 2 . 4 for a description of a continuation branch).  Notice that the x-coordinate of A does not vary w i t h e but the  Chapter 2. Preliminary  Example  17  x  0.09  Figure 2.4:  One-parameter  bifurcation  diagram obtained  0.15  by varying  e  i n system  (2.3) w i t h  a = 0.6,6 = 0.3, c = 0.4, d = 0.2 a n d a = 0.3. T h e labels A a n d B m a r k the c o n t i n u a t i o n branches c o r r e s p o n d i n g to the e q u i l i b r i u m points given i n equations 2.4, H B stands for H o p f b i f u r c a t i o n a n d B P for b i f u r c a t i o n p o i n t ( t r a n s c r i t i c a l i n this case).  E x p l a n a t i o n s o f the various line types c a n be f o u n d  i n section A . 2 . 1 . I n p a r t i c u l a r , the curves of solid circles m a r k the m a x i m a a n d m i n i m a o f stable l i m i t cycles.  x-coordinate of B does. We can check this observation with the analytical forms of the equilibrium points which are given by A A  , x  =  c i — >  • da(d — ac) — ec . y  d — etc B  ( x = ±y  Tu—V~  =  = 0).  o  [d — acy (2.4)  These are obtained by setting the right hand sides in equations (2.3) equal to zero and solving for x and y. As expected, e does not appear in the x-coordinate for A but does appear in that for B. Figure 2.4 summarises the information given by the phase portraits in figure 2.3. For 0 < e < 0.045 point A is unstable (a source—see section A.2.23) and B is a saddle point (see section A.2.20). There is also a stable limit cycle surrounding A. Hence, these values of e correspond to region (ii) in figure 2.3. For 0.045 < e < 0.12 A is a stable equilibrium  Chapter 2. Preliminary Example  18  _i  i  0.1  0.2  I 0.3  e  F i g u r e 2.5:  T w o - p a r a m e t e r c o n t i n u a t i o n of the H o p f b i f u r c a t i o n shown i n figure 2.4.  point and B is again a saddle point. This configuration corresponds to region (i) i n figure 2.3. For e > 0.12 B is now stable and A is a saddle point, but the numerical output f r o m A U T O shows that the y-coordinate for A is negative for these values of e. Figure 2.3(b)(iii) represents the corresponding dynamics for positive x and y. W e would also like to reproduce B a z y k i n ' s two-parameter bifurcation diagram shown i n figure 2.3(a).  A U T O can be used to continue the Hopf bifurcation at e = 0.045 in  a as well as e (see section B.4 for an explanation of how to generate a two-parameter bifurcation diagram). T h e result is figure 2.5. T h e first observation we can make from this diagram is that the curve of Hopf bifurcations is very different from B a z y k i n ' s straight line i n figure 2.3(a). I w i l l return to this point shortly. A second observation is that, for the given values of a,b,c  and d, a Hopf  bifurcation (and hence l i m i t cycle behaviour) is only possible if e < 0.0515 and a < 0 . 5 , 2  A U T O slows d o w n considerably as a increases t o w a r d 0.5 a n d e tends to 0 a n d never a c t u a l l y reaches t h i s p o i n t , a l t h o u g h the curve does get very close i f A U T O is left to r u n for a sufficiently l o n g t i m e p e r i o d . It can be verified a n a l y t i c a l l y t h a t the curve does pass t h r o u g h (0,0.5). However, c o m p l e x b e h a v i o u r a l changes occur at this p o i n t w h i c h is w h y A U T O has c o m p u t a t i o n a l difficulties. 2  Chapter 2. Preliminary Example  19  0.5 0.4 0.3  a 0.2 0.1 °0  U U U 1 U  0.1  0.2  0.3  e  F i g u r e 2.6:  T w o - p a r a m e t e r bifurcation d i a g r a m of (e, a ) - p a r a m e t e r space for m o d e l (2.3) w i t h a = 0.6,  6 = 0 . 3 , c = 0 . 4 a n d d — 0.2. T h e regions (i), (ii) and (iii) correspond to those i n figure 2.3.  which is a fairly small region of parameter space. It is not possible to continue a transcritical bifurcation i n two parameters using A U T O (see page 267 for an explanation). However, if a is fixed at a number of different values and one-parameter bifurcation diagrams similar to figure 2.4 are created by varying e i n each case, then the values corresponding to transcritical bifurcations can be recorded. A n approximation to the two-parameter curve can then be drawn through these points. Figure 2.6 shows the resulting curve together w i t h the Hopf bifurcation continuation. M o d e l (2.3) is simple enough for the curves i n figure 2.6 to be determined analytically although the algebra is rather messy. It can be shown that transcritical bifurcations occur along the straight line  a =  d + a c e  and Hopf bifurcations occur along the curve  (—ac )a + (acd — ec )a — edc = 0. 2  2  2  (I used M A P L E [122] for some of the algebraic manipulations required to obtain these  Chapter 2. Preliminary Example  20  results.) These are exactly the curves shown infigure2.6 as can be verified by substituting points from the curves calculated by AUTO into the above equations. Hence, AUTO's results are more accurate than those given by Bazykin in figure 2.3(a) for the Hopf bifurcation curve. Bazykin did not have a symbolic package such as M A P L E available and made an approximation in calculating this curve. The new regions (i), (ii) and (iii) are shown in figure 2.6. I obtained phase portraits corresponding to the points marked with *'s using DSTOOL (see figure 2.7). XPPAUT could also have been used. These phase portraits are qualitatively the same as Bazykin's diagrams infigure2.3(b).  0  6  10  x Figure 2.7:  15  20  0  5  10  15  20  x  P h a s e p o r t r a i t s corresponding to the points m a r k e d w i t h *'s i n figure 2.6.  It is also informative to view the temporal dynamics of a model. Time plots corresponding to figure 2.7 for initial values x = 10 and y = 3 are shown infigure2.8. These were also obtained using DSTOOL. Time plots are useful for indicating the speed with  Chapter 2.  Preliminary  Example  21  which the stable equilibrium or limit cycle is attained and the period of the limit cycle if applicable. If a system takes a long time to approach an attractor (see section A.2.14) then the transient dynamics may be of greater practical importance than the long-term behaviour. (a)  (b)  10  10  x 5  x 5  100 Time  (c)  (d)  10  10  x 5 -  x 5  Figure 2.8:  200  100  100  Time  Time  T i m e plots corresponding to the phase p o r t r a i t s i n figure 2.7. T h e i n i t i a l p o i n t x — 10, y = 3  was used i n each case.  This section has shown how to rederive Bazykin's work [14] on the system of equations (2.3) more accurately and without having to understand the complicated mathematical techniques involved. In the next section I look at another of Bazykin's models in which it is not feasible to do much of the mathematical analysis by hand. Fairly accurate results can be obtained using AUTO.  Chapter 2. Preliminary  2.4  Example  22  Adding intraspecific competition among predators  Suppose that instead of having intraspecific competition among prey we have intraspecific competition among predators. The new system of equations is then x  =  ax — ^  V  =  -cy + T^- -W .  y  x  (2.5)  2  In this model predator population growth is limited even when there is an excess of prey. Bazykin's results for this system are shown in figure 2.9. Again there are two nontrivial fixed points, A and C, but in this case both have positive coordinates. Whether we have the situation in figure 2.9(a) or (b) depends on the parameter values for a, b, c and al. Using numerical experimentation Bazykin postulated that both variations are possible [14]. He managed to find an analytical approximation to one of the lines 0J_, 0G_ but had not found an approximation to the second at the time of writing his paper. Setting a = 0.6,6 = 0.3, c = 0.4 and d = 0.2 as before and choosing p =  0.06  and a — 0.1, I used D S T O O L to locate equilibrium points and A U T O to generate a one-parameter bifurcation diagram by varying a. The result was figure 2.10.  In this  case we have a Hopf bifurcation at a = 0.0977, a limit point (see section A.2.13 for an explanation of this bifurcation point) at a = 0.176 and it appears as if the periodic orbit collides with the saddle point C suggesting a homoclinic  bifurcation  (see section A.2.9).  All three phenomena can be investigated in two parameters using A U T O . The first two are straightforward two-parameter continuations. Since a homoclinic bifurcation is not actually detected, we need to calculate an approximation to this curve. As the periodic orbit in figure 2.10 approaches the saddle point it can be seen from A U T O ' s numerical output that the period of the oscillations increases fairly rapidly. It is possible to plot the period as a function of a if X P P A U T is used (see figure 2.11).  Chapter 2. Preliminary Example  Figure 2.9:  These d i a g r a m s are adapted from  of ( a , / z ) - p a r a m e t e r space, (a) a n d (b).  23  [14].  (a) and (b) T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m s  (c) Phase p o r t r a i t s corresponding to regions (i), ( i i ) , ( i i i ) and ( i v ) i n p a r t s  Chapter 2. Preliminary  Example  24  60  x  0  0.04  0.08  0.12  0.16  0.2  a Figure 2.10:  O n e - p a r a m e t e r bifurcation  d i a g r a m o b t a i n e d by  varying  a  i n s y s t e m (2.5)  with  a = 0.6, 6 = 0.3, c = 0.4, d — 0.2 and fi = 0.06. T h e labels A a n d C m a r k the c o n t i n u a t i o n branches for the two n o n t r i v i a l e q u i l i b r i u m p o i n t s . H B m a r k s the H o p f bifurcation and L P the l i m i t p o i n t .  Such a steep increase i n period suggests that a homoclinic orbit is being approached as these orbits have infinite period. To approximate the curve of homoclinic bifurcations we can set a U S Z R function i n A U T O to locate an orbit of high period. T h e required approximation is obtained by continuing this orbit of fixed period in /J, as well as a. T h e resulting two-parameter diagram i n (a, ^)-space is shown in figure 2.12. (I would have liked to have chosen an orbit of period greater than 30 to approximate the curve of homoclinic bifurcations but for this particular model A U T O had difficulty w i t h larger periods. However, one-parameter bifurcation diagrams at different fixed values of \i show that the curve corresponding to a period of 30 provides a fairly good approximation to the required curve.) This figure can be compared w i t h figure 2.9(a). Phase portraits and time plots corresponding to the points marked with *'s in figure 2.12 are shown i n figures 2.13 and 2.14 respectively. These results agree qualitatively with Bazykin's but the approximations to the curves OG_ and OJ_ are more accurate. In particular, region (iii) corresponding to stable l i m i t  Chapter 2. Preliminary  Figure d =  0.2.  2.12:  Example  25  T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m of ( a , p)-space for a = 0.6, 6 = 0.3, c = 0.4 a n d  H B m a r k s the H o p f b i f u r c a t i o n c o n t i n u a t i o n , L P m a r k s the l i m i t p o i n t c o n t i n u a t i o n a n d  p e r i o d = 3 0 m a r k s the c o n t i n u a t i o n of the orbit o f fixed p e r i o d .  Chapter 2. Preliminary Example  26  (a)  (b)  (c)  (d)  y  Figure 2.13:  5 -  P h a s e p o r t r a i t s corresponding to the points m a r k e d w i t h *'s in figure 2.12.  cycle behaviour is very small for these parameter values. A s a and p decrease, the curve of homoclinic bifurcations approaches the Hopf bifurcation curve and almost coincides with it so that for small p and a the region of l i m i t cycle behaviour is negligible. We can investigate figure 2.12 further by generating one-parameter bifurcation diagrams for different values of p. Setting p = 0.1 gives figure 2.15(a) which corresponds to the horizontal dotted line in figure 2.12 at p, — 0.1. A s expected from figure 2.12 there is no Hopf bifurcation or homoclinic bifurcation in this case. For p = 0.074 we obtain figure 2.15(b) which corresponds to the horizontal dotted line at p = 0.074. A s expected from figure 2.12 there are two Hopf bifurcations in this case but no homoclinic bifurcation. Setting p = 0.02 results in figure 2.15(c). In this case the curve of periodic orbits is very steep, becomes unstable and then A U T O fails to be able to calculate further and signals  Chapter 2. Preliminary  (a)  Example  27  (b)  10  x  * 5 www\AAAA/W  5  100 Time (c)  F i g u r e 2.14:  200  100 Time  (d)  10  x  10  10  x  5  5  T i m e plots corresponding to the points m a r k e d w i t h *'s in figure 2.12. T h e i n i t i a l p o i n t  x = 5, y = 3 was used i n each case.  non-convergence.  This is not unexpected (see the horizontal dotted line in figure 2.12  at p — 0.02) as the Hopf bifurcation and homoclinic bifurcation curves are very close together for this value of p.  2.5  Conclusion  T h i s chapter describes how qualitative analyses of three predator-prey models may be done using various computer software. The numerical results are compared with analytical results obtained by B a z y k i n [14]. For model (2.3) which has intraspecific competition among prey, the numerical results are more accurate than Bazykin's approximate analytic results. This conclusion is possible as I obtained an exact analytical expression for  Chapter 2. Preliminary Example  (a)  28  60  x  (b)  60  (c)  F i g u r e 2.15:  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s for (a) fi = 0.1, (b) fx = 0.074 a n d (c) / i = 0.02.  These c o r r e s p o n d to the h o r i z o n t a l d o t t e d lines i n figure 2.12.  Chapter 2. Preliminary  Example  29  the two-parameter Hopf bifurcation curve for which the discrepancy i n results arises. The analysis of model (2.5) results in a numerical approximation to a curve which B a z y k i n did not describe analytically. The position of this curve results in a very small two-parameter region corresponding to l i m i t cycle behaviour. A knowledge of the relevant mathematical techniques, such as centre manifold theory and normal form theory (see page 241), is not required to obtain the above results. Computer packages such as A U T O take care of the mathematical details. T h i s is especially useful for models which are too difficult to study by hand, such as model (2.5). It also allows accurate and fairly quick qualitative analyses of models to be done thus facilitating an iterative approach to modelling since modifications of model equations can be investigated fairly easily. In the next chapter I look at a model which has not been studied before using dyn a m i c a l systems techniques. In this case the analysis suggests an improvement i n the formulation of the model equations.  Chapter 3  Sheep-Hyrax-Lynx  3.1  Model  Introduction  T h i s chapter investigates a more complicated model having 10 state variables and a large number of parameters. A n a l y t i c a l work done by hand and isocline analyses are of little use i n such situations. Traditionally computers have been used to obtain numerical solutions corresponding to a fixed parameter set and to implement sensitivity analyses . 1  I show how dynamical systems techniques can be used to increase our understanding of the relationships between different components in the model. In particular, bifurcation diagrams give more information than sensitivity analyses. These diagrams also highlight an incomplete relationship i n the model and lead to an improvement i n the formulation of the equations. T h e model I have chosen is an example of a system dynamics model and has four m a i n components—sheep, hyrax, lynx and pasture.  I begin i n section 3.2 w i t h some  background to the systems modelling approach for those who may be unfamiliar w i t h it. I also discuss the traditional methods that have been used to solve and analyse such models. In section 3.3 I describe the formulation of the model followed by a few technical details which are required i n order to use X P P A U T to analyse the dynamics.  Section  3.5 contains the model analysis. I begin by studying the effects of various parameters and density-dependent functions on the behaviour of the model.  This analysis shows  the limitations of a traditional sensitivity analysis and highlights dynamics which are 1  S e e section 3.2 for a d e s c r i p t i o n of s e n s i t i v i t y analyses.  30  Chapter 3. Sheep-Hyrax-Lynx Model  31  biologically implausible, namely that pasture growth is u n l i m i t e d when sheep densities are low. A modification to the pasture growth term is discussed i n section 3.5.3. T h e analysis is completed by a two-parameter study of the effects of culling rates on farmers' revenue. T h i s is followed by section 3.5.5 which interprets the m a i n results of the analysis from a biological viewpoint.  3.2  D y n a m i c models and systems analysis—some background  T h e systems approach to modelling was made popular by Forrester [39] i n the early 1960's. T h i s approach involves dividing a system into a large number of very simple unit components (Watt [123]) and then using equations to describe the processes affecting each of these components. T h e methodology was originally applied to industrial, urban, and world population systems but its utility has been extended to ecological applications by a number of researchers (see Jeffers [64] and Watt [123]). Patten [99] summarises the advantages of these dynamic or simulation models i n the ecological context: T h e formulation of the models allows for considerable freedom from constraints and assumptions, and allows for the introduction of the non-linearity and feed-back which are apparently characteristic of ecological systems. T h i s ease of formulation and flexibility are important for modelling ecological systems. A variety of aspects such as age structure, developmental rates and density-dependent relationships can be included explicitly, thus increasing the realism of the model. K o w a l [69] notes that an analysis of dynamic models can provide approximations to ecosystem dynamics long before traditional experimental approaches can provide more detailed conclusions. Insights can also be obtained into aspects of the system which may otherwise be obscured by its complexity.  Chapter 3.  Sheep-Hyrax-Lynx  Model  32  T h e problem w i t h complex models comes at the time of analysis. D y n a m i c models usually involve a large number of equations (generally ordinary differential equations) and parameters which makes their behaviour difficult to predict (Patten [99]). W e need to find suitable ways of analysing the dynamics of these models. T h e traditional approach has been to use numerical routines to obtain solutions over t i m e for a given set of parameter values. Optimisation routines are also often employed ( M a y n a r d S m i t h [85]) to determine the 'best' possible strategy w i t h respect to a cost or revenue function. These routines are implemented using computers. A computer's speed of computation and ability to provide rapid access to large quantities of data makes it particularly suitable for analysing these large models (Jeffers [64]). A description of the basic routines involved can be found in Patten [99] as well as in any introductory textbook on numerical routines for systems of ordinary differential equations (for example, [43, 63]). W h i l e these methods are useful, their results depend on the particular parameter set used.  Intuitively a sufficiently small variation in the parameter values should lead to  an arbitrarily small change i n the solution given by the model if we are to have any faith i n the predictions of the model (Hadamard [53]). This corresponds to Hadamard's concept of a well-posed problem w i t h respect to partial differential equations [53] and led to the development of sensitivity  analyses. For ordinary differential equations this  method is based on the ideas of Tomovic [118] and involves changing the values of the input variables and parameter values by a small amount (say 1 percent or 10 percent) and seeing whether these changes produce large or small variations in the predictions of the model (Jeffers [64]). A good description of the basic theory involved, as well as some examples, is given i n B r y l i n s k y [18]. Sensitivity analyses are often used when studying system dynamics models and do give some idea of the robustness of model predictions, but the information is l i m i t e d in that only a single, small perturbation of each parameter is considered. T h i s chapter  Chapter 3. Sheep-Hyrax-Lynx Model  33  shows how additional information can be obtained using dynamical systems techniques to vary parameters across ranges of values. The next section describes a particular example of the systems approach to modelling which I w i l l use to illustrate the latter techniques.  3.3  M o d e l equations  Swart and Hearne [116] developed a dynamic model to study the impact of hyrax (a type of rock rabbit) and lynx on sheep farming in a region in South A f r i c a . T w o m a i n problems were identified. The first involves competition for pasture between hyrax and sheep; hyrax encroach on farm land when the hyrax population exceeds the carrying capacity of wilderness areas i n the region. The second problem is the predation on sheep by l y n x . T h e principal food for lynx is hyrax, but from time to time l y n x prey on sheep. It is the latter problem that is of direct concern to f a r m e r s — t h e y tend to be more tolerant of the competition w i t h hyrax. T h e model i n [116] was developed to increase understanding of the problems caused by the spillover of hyrax and lynx from their predator-prey system into the sheep-pasture system, and to determine the effects of different culling strategies for hyrax and l y n x . There are 10 state variables in the model. The sheep, hyrax and l y n x populations are each divided into three classes—juveniles, female adults and male a d u l t s — a n d there is one variable representing pasture. T h e quantity of most interest to farmers is revenue. T h i s auxiliary variable is a function of the state variables and is made up of wool sales, m u t t o n sales, the value of sheep stock, and the cost of culling hyrax and l y n x . T h e differential equation for each state variable is formulated by adding and subtracting quantities representing the processes affecting that variable. For example, the equation for hyrax juveniles is as follows:  Chapter 3. Sheep-Hyrax-Lynx  Rate of change of ^  hyrax juveniles  34  Model  births - maturation - deaths - predation - culling j  where • b i r t h s depend on the number of hyrax female adults and decrease w i t h increasing hyrax density, • m a t u r a t i o n represents the number of juveniles that mature to become adults i n a given year and is a constant fraction of the number of hyrax juveniles, • d e a t h s are a proportion of the number of hyrax juveniles and increase w i t h i n creasing hyrax density, • p r e d a t i o n (by lynx) varies with the relative number of hyrax and l y n x (that is, predation increases as hyrax abundance increases), and • c u l l i n g is a constant fraction of the number of hyrax juveniles and is determined externally by the farmer or an environmentalist. In m a t h e m a t i c a l terms the above equation becomes dHj —-  dt  JJ  TT  — tijB — tijM  u II — njr) — Hjp  u — tljc  where Hj is the number of hyrax juveniles, HJB is the number of hyrax births i n a given year, HJM is the number of hyrax that mature to become adults during the year, HJD is the number of hyrax juvenile deaths during the year, Hjp  is the number of hyrax  juveniles killed by lynx during the year, and HJC is the number of hyrax juveniles that are culled.  A full description of the mathematical formulation of these terms can be  found i n [116]. B y way of example,  Chapter 3. Sheep-Hyrax-Lynx  Model  HJM  where HJMN  35  — Hj  X HJMN  (hyrax juvenile maturation normal) is the fraction of juveniles that become  adults each year and (3.1) where • Lj = LJR x Lj + Lp + LM is the total number of lynx (a lynx juvenile ratio  (LJR)  converts lynx juveniles into equivalent adult units, for example, 1 juvenile = 0.5 adults), • LPN is the lynx predation normal which is the average number of sheep k i l l e d per lynx per year, • LPM is the lynx predation multiplier (lynx functional response) which is an increasing function of prey abundance, Ap, and • Ap =  j  is an index of the availability of hyrax as prey for the lynx population.  It is a ratio of the total number of hyrax to the total number of lynx relative to a ' n o r m a l ' ratio,  representing the usual level of abundance under typical  environmental conditions. T h e last t e r m i n equation (3.1) adjusts the total amount of predation so that only the number of juveniles killed is taken into account i n this equation. A possible choice for the lynx predation multiplier is shown i n figure 3.1. A n S-shaped functional response is used as the lynx can only eat a l i m i t e d amount even when prey  Chapter 3. Sheep-Hyrax-Lynx  Model  36  2  1.5  LpM  1  0.5  0 0  1  2  3  A  P  Figure 3.1: The lynx predation multiplier (LPM) as a function of prey abundance (Ap). abundances are very high, and when abundances are low lynx have difficulty  finding  hyrax. T h e other equations i n the model are formulated i n a similar manner to the above example for hyrax juveniles. Table 3.1 shows the processes affecting each a n i m a l group. Predation is by lynx and culling is done by the farmer (in the case of sheep) or controlled by environmentalists (in the case of hyrax and lynx). Three quantities i n the model are averaged using first order delays. A description of how this is included in the model is given i n appendix C. Swart and Hearne [116] used traditional methods to study this model. T h e y used optimisation routines to find the hyrax and lynx culling rates which gave m a x i m u m profitability i n terms of revenue. They also investigated the sensitivity of the system to parameter perturbations. T h e y found that lynx culling is essential and that substantial increases i n both sheep numbers and revenue are possible by simultaneously culling hyrax and lynx.  O p t i m a l culling rates in terms of revenue are around 30 percent per annum  Chapter 3. Sheep-Hyrax-Lynx  Model  State variable  G r o w t h processes  Hyrax juveniles  births  (HJ)  (H ) JB  Hyrax female adults (Hp) Hyrax male adults (HM)  juvenile maturation  Lynx juveniles  births ( L j )  (Lj)  (HJM)  D e a t h processes deaths (Hjp>),maturation (HJM), predation (ifjp),culling (HJC) deaths (HFD),predation (Hpp), culling (H ) deaths (ifM.o),predation (HMP), culling (H c) FC  juvenile maturation (HJM)  M  B  deaths (LJD),maturation culling (L )  (LJM),  JC  Lynx female adults (Lp) Lynx male adults (LM)  juvenile maturation  Sheep juveniles  births (SJB)  (SJ)  37  (LJM)  juvenile maturation (LJM)  Sheep female adults (SF) Sheep male adults (SM)  juvenile maturation  Pasture (P)  production (Pp)  deaths (LFD),culling  (Lpc)  deaths (ZMB),culling (LMC)  deaths (SJD),maturation (SJM), predation (Sjp),culling (SJC) deaths (S^£>),culling (SFC)  (SJM)  juvenile maturation  deaths (5M£»),culling (SMC)  (SJM)  grazing (Pq)  Table 3 . 1 : T a b l e s h o w i n g the processes affecting each state variable a n d some o f the a b b r e v i a t i o n s used i n the m o d e l equations.  for b o t h hyrax and lynx. F r o m a policy point of view the model is robust w i t h respect to small parameter variations. The purpose of this chapter is not to redo the work done by Swart and Hearne [116]— their model and analysis have accomplished their aims. Instead I want to illustrate the usefulness of the dynamical systems software i n this setting.  Chapter 3.  3.4  Sheep-Hyrax-Lynx  38  Model  Technical details  A few minor modifications to the model are required to facilitate the use of this software. T h e first involves scaling the state variables so that they all have the same order of magnitude.  C o m b i n i n g quantities of very different magnitude may lead to computer  round-off errors [43]. I chose values close to the i n i t i a l values in [116] as scaling constants. In other words, I replaced each state variable  V{  by the quantity  SjUj  where  is the scaling  constant for V{. Now v,- takes on values between say 0 and 10. In order to calculate the magnitude of the i population we can multiply this new Vi by s;. To prevent the scaling th  from altering the dynamics of the model, the differential equation for u - is divided through t  by S{. T h e above manipulations are made clearer i n appendix C. Secondly, i n order for the computer packages to generate continuous bifurcation diagrams, all functions i n the model need to be continuous. T h e original model represents farmers' sheep culling strategies using two step functions. I replaced these w i t h continuous functions having steep slopes in the region of the step. In the original model pasture production and fecundity rates vary seasonally. Since this complicates the dynamics considerably when it comes to parameter studies and since the present study is more concerned with long-term equilibrium behaviour than w i t h day to day variations, I did not include the seasonality functions.  Solving the system of  equations numerically over time is still the best way to study seasonal variation i n most cases. T h e various versions of A U T O only allow a state variable or the _L -norm of the state 2  variables to be displayed on the y-axis of the bifurcation diagrams they generate. Z-2-norm of a vector v = ( u . . . , v ) l 5  m  is given by  The  Chapter 3.  Sheep-Hyrax-Lynx  Model  39  However, the quantity of most interest to farmers is the revenue corresponding to different management strategies. In order to have direct access to revenue values it would be most convenient if revenue were a state variable. The following suggestion by B a r d Ermentrout makes this possible. Let v be the vector of existing state variables and let h(v)  be the revenue function.  W e can add the equation  dR _ ~dt ~  -R+h(v) T  '  where r is a small parameter, to the original system. R is the variable that we want to represent revenue. This ordinary differential equation w i l l not affect system equilibria since at these points R = h(v)  and hence ^  = 0 (as required for an e q u i l i b r i u m value).  T h e existence and stability of phenomena such as periodic orbits (limit cycles) are also not affected. Since r acts as a delay time, we would like it to be small so that R is a close approximation to revenue.  However, care must be taken i n the choice of r as  very small values can give rise to computer truncation errors. For the current problem I used r = 0.05. Larger values of r resulted in R values which were less satisfactory approximations to revenue while smaller values of r gave hardly any change in the R values. We are now in a position to begin the analysis.  3.5 3.5.1  M o d e l analysis Reference p a r a m e t e r values  We need to choose an i n i t i a l set of parameter values before we can determine the effects of varying parameters across ranges of values. Most of the values are given i n [116]. O n l y values for the hyrax and l y n x culling normals (HCN and LCN) need to be chosen. It seems most natural to choose those values which, give the m a x i m u m revenue at equilibrium. In  Chapter 3.  Sheep-Hyrax-Lynx  Model  40  5 ^£CAT=0-15,0.2,...  4 3 Revenue  2 1 0  0  0.35  0.7  1.05  1.4  HCN  Figure 3.2:  T h r e e one-parameter b i f u r c a t i o n d i a g r a m s w i t h revenue p l o t t e d as a f u n c t i o n of  E a c h curve corresponds to a different (fixed) value of  HCN-  LCN-  order to do this I first fixed LCN at 0.1, chose a value of 0.2 for HCN (any reasonable values would do just as well), and used a numerical solver i n X P P A U T to integrate the system of equations u n t i l an equilibrium point was reached. Using this e q u i l i b r i u m point as the i n i t i a l point, I then used the A U T O interface to vary HCN-  T h i s is done by  choosing HCN to be plotted on the x-axis and one of the state variables (I chose revenue) for the y-axis.  Using the R U N and G R A B  2  commands a parameter diagram can be  generated. T h i s diagram shows how the equilibrium revenue changes as HCN varies (see figure 3.2). T h e above exercise was repeated for a few different values of LCN and the resulting parameter diagrams were plotted on the same pair of axes to give figure 3.2. It can be seen from this figure that a value of 0.35 for HCN is close to o p t i m a l (in terms of revenue) for all values of LCN-  I chose this value for HCN and then used X P P A U T  to vary LCN i n order to find the corresponding optimal value for LCN- T h i s gave figure 2  A c o m p l e t e d e s c r i p t i o n of the available c o m m a n d s can be f o u n d i n the X P P A U T d o c u m e n t a t i o n as  w e l l as i n the i n t e r a c t i v e t u t o r i a l t h a t is available—see a p p e n d i x B .  Chapter 3. Sheep-Hyrax-Lynx  Model  41  5 4 3 Revenue 2 1 0 0  0.1  0.2  0.3  0.4  LCN  F i gure 3.3: One-parameter bifurcation diagram of revenue versus LCN for -ffc;v=0-35. 3.3. Lower values for HCN resulted i n revenue curves lying below that i n figure 3.3 and higher values resulted i n curves almost identical to that for HCN=0.35. For the reference parameter set I chose values of 0.35 for HCN and 0.15 for LCN as these values are close to o p t i m a l and have the added advantage that small perturbations w i l l not have much effect on revenue, since the revenue surface appears to be fairly flat i n a region surrounding these values. A t these reference values e q u i l i b r i u m revenue equals 3.94 which is slightly greater than 3.91, the value when LCN = 0.3 and HCN = 0.3. (Note that the scaling factor for revenue is 10 m i l l i o n R a n d so the above values need to be m u l t i p l i e d by this factor to get the true revenue values.)  3.5.2  Understanding model relationships  The effects of culling sheep and of lynx fecundity There are many parameters i n this model which could be used to illustrate the d y n a m ical systems techniques. Those affecting population growth rates are likely to have the  Chapter 3. Sheep-Hyrax-Lynx  42  Model  Revenue  (b)  0.8  L  0.4  F  0.2  0.4  0.6  0.2  SFCN  (c)  0.4  0.6  SFCN  10  P  5 -  0.2  0.4  0.6  SFCN  Figure 3.4: O n e - p a r a m e t e r bifurcation d i a g r a m s o b t a i n e d f r o m v a r y i n g SFCN ( n o m i n a l value = 0.28/yr) with L f j v = 0 . 7 / y r .  In (a) revenue is p l o t t e d o n the y - a x i s , i n (b) the state variable for l y n x  females is used, a n d i n (c) the state variable for pasture is used.  greatest influence on the dynamics. I have chosen to study two such parameters i n this c h a p t e r — t h e sheep female culling normal SFCN,  which is the average number of ewes  culled by a farmer per year, and the lynx fecundity normal LFN, which is the average number of offspring produced per female lynx per year. A s before I used X P P A U T to integrate the system numerically, using the reference parameter values, u n t i l an equilibr i u m was reached. Using these equilibrium values for the state variables as the starting point, I employed X P P A U T ' s A U T O interface to produce a bifurcation diagram. T h e results are shown i n figures 3.4 and 3.5. Once a bifurcation diagram has been generated using X P P A U T , it is easy to switch the variable on the y-axis. T h e effects of varying a parameter w i t h respect to different state variables can then be seen. For each parameter I have chosen three diagrams. One  Chapter 3.  Sheep-Hyrax-Lynx  Model  43  Revenue  (c)  1  1  o  o  o o o o  :_ :  §•••••  1 1  puudj) 2 LFN  Figure 3.5: O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s o b t a i n e d f r o m v a r y i n g LFN ( n o m i n a l value = 0 . 7 / y r ) w i t h S'jrc'jv=0.28/yr. I n (a) revenue is p l o t t e d on the y-axis, i n (b) the state v a r i a b l e for l y n x females is used, a n d i n (c) the state variable for pasture is used.  shows the effect of the parameter on equilibrium revenue, the second the effect on the e q u i l i b r i u m number of lynx females, and the t h i r d the effect on the e q u i l i b r i u m amount of pasture. A number of observations can be made from figures 3.4 and 3.5. T h e first is that A U T O encounters points beyond which it cannot calculate. In figure 3.4 such a point is SFCN=0-54:  and in figure 3.5 there are two points, Z^JV=0.39 and LFN=2.78.  Using  X P P A U T to solve the system of equations numerically for parameter values on either side of these l i m i t i n g points, the causes of these difficulties can be determined. O u t p u t from a numerical integration is shown in X P P A U T ' s data window and from this it was seen that the sheep population dies out for SFCN > 0.54 or LFN > 2.78, and the l y n x population dies out for LFN < 0.39. The latter conclusion could also be drawn from figure 3.5(b).  Chapter 3.  Sheep-Hyrax-Lynx  Model  44  Beyond the l i m i t i n g values there is no equilibrium at which all three populations are present and, hence, A U T O cannot continue the equilibrium branch any further. A second observation can be made by looking at the bifurcation diagram for pasture i n figure 3.4(c).  A s the equilibrium number of sheep declines (as SFCN  increases and  approaches the value corresponding to sheep extinction), the equilibrium value for pasture increases unchecked.  Clearly this is unrealistic and suggests that some modification  should be made to the model equations to l i m i t pasture growth.  I return to this in  section 3.5.3. In figure 3.5 there is a threshold value at LFN=0.76  above which revenue declines as  LFN increases and below which revenue remains fairly constant as LFN is varied. A g a i n I chose parameter values on either side of this threshold and used X P P A U T to integrate the system of equations numerically u n t i l an equilibrium was reached. V i e w i n g the numerical output i n the data window lent some insight into the behaviour of the m o d e l . I found that as LFN increases through 0.76, prey abundance at equilibrium passes through a 3  threshold value above which lynx begin to supplement their diet w i t h lambs. T h i s loss of lambs explains the decrease in revenue. A n important point to note is that if a traditional sensitivity analysis had been done using o p t i m a l equilibrium values for the state variables and a n o m i n a l value of L  F i V  = 0 . 6 / y r , then no change in equilibrium revenue would have been seen for a 10 percent  increase i n LFN- However, using a nominal value of 0.7/yr, a 10 percent increase places LFN  at 0.77/yr. T h i s is above the threshold point and thus a decrease i n e q u i l i b r i u m  revenue occurs. T h e only stability change in figures 3.4 and 3.5 occurs at 1 , ^ = 2 . 0 6 . A t this point a Hopf bifurcation occurs (see section A.2.10), which means that the stable e q u i l i b r i u m point becomes unstable and a periodic orbit is initiated. However, oscillations are only 3  S e e the e x p l a n a t i o n under e q u a t i o n (3.1).  Chapter 3. Sheep-Hyrax-Lynx  Model  45  associated with the sheep-pasture subsystem. T h e lynx and hyrax populations do not cycle.  In this case the l i m i t cycles associated with the Hopf bifurcation are unstable  and thus not of practical interest.  In such a case it is more enlightening to examine  the temporal dynamics of the system for 2.06 < LFN < 2.78. Using X P P A U T for this purpose it was found that the sheep population declines to zero i n this range of parameter values while the hyrax and l y n x populations reach steady states. This trend continues for LFN  > 2.778 but the sheep population dies out much faster and A U T O fails to converge. Before modifying the equations to l i m i t pasture growth as suggested earlier, it would  be informative to study the roles of the existing density-dependent functions. T h i s can be viewed as another form of sensitivity analysis i n which we investigate the system's response to whole functions instead of single parameters.  The effects of density-dependence There are a number of functions i n the model which modify growth and death rates as conditions change. A l l these functions are normalised to take the value 1 when the quantities on which they depend are at their reference values. For example, the equation governing pasture (P) dynamics is given by dP —jj-  —  , . pasture production — pasture grazing  =  A.PPN  — TSSU-GN-GM(PA)  (3.2)  where A is the area of the farming region under study, PPN is the pasture production normal (average pasture growth rate), Tssu is total small stock units (a representative value for the number of sheep), GN is the grazing normal (average amount of pasture grazed per unit stock) and GM is the grazing multiplier which is a function of P ^ , the pasture availability index. PA is given by P/P where P is an average pasture density. T h e grazing multiplier has the form shown i n figure 3.6.  Chapter 3.  Sheep-Hyrax-Lynx  Model  46  1.5  0  0  1  2  3  PA  Figure 3.6:  T h e g r a z i n g m u l t i p l i e r f u n c t i o n (GM)  as a f u n c t i o n of pasture a v a i l a b i l i t y .  In order to remove a density-dependent function from the model, the simplest approach is to replace it by a constant function. For example, we can set GM = 1- T h i s was done for each multiplier function in turn and bifurcation diagrams obtained f r o m the altered model were compared with those from the original model in each case. R e m o v i n g the grazing multiplier, GM, had the greatest effect on model dynamics. E v e n after altering a number of parameter values the system did not reach equilibrium. T h e sheep population either increased indefinitely or decreased to extinction for each parameter set that was tried. This is not surprising since the grazing multiplier affects pasture grazing and sheep fecundity as well as sheep juvenile deaths. W i t h o u t GM the density-dependence of pasture grazing and sheep dynamics on pasture availability (GM is a function of PA) is removed. The results show that this density-dependence is critical for regulating the sheep-pasture subsystem. T h e effects of the other functions in the model were quantitative rather than qualitative. T h a t is, removing them from the model tended to decrease the range of parameter  Chapter 3.  Sheep-Hyrax-Lynx  Model  47  values over which sheep, hyrax and lynx coexist at equilibrium, but did not alter the qualitative dynamics. In relative terms however, the effects of the fecundity multiplier functions were more noticeable than those of the death and predation multipliers. The above comments do not i m p l y that the latter multiplier functions play an i n significant role i n the model. T h e y have a regulatory effect and reduce the impact of parameter changes on model behaviour. This resilience to disturbances is very desirable [61] and is expected of many natural systems. Thus, although these functions may not be critical in determining model behaviour, they are important for m a k i n g the model more realistic. It was observed earlier that the model lacks a feedback relationship that would l i m i t pasture growth when sheep densities are very low. We also know that pasture availability has a significant influence on pasture grazing and sheep fecundity and hence on the predictions of the model. Thus modifying equation (3.2) to include density-dependent growth may have a considerable effect on the behaviour of the model. T h i s modification is discussed in the next section.  3.5.3  A d d i n g density-dependence to pasture growth  T h e reader w i l l probably have noticed that equilibrium pasture values only become u n realistic for extreme parameter values. However, it is desirable to have a model which can describe a variety of situations instead of one that is only suitable for a small range of values. A l s o , improving model realism in extreme regions may affect the dynamics corresponding to more normal values and so play an important part i n understanding system behaviour. The formulation of a model also affects statistical parameter fitting routines. If important relationships are left out then these routines may give misleading results or fail to converge. In the sheep-pasture, hyrax-lynx model pasture growth occurs at a fixed rate and  Chapter 3. Sheep-Hyrax-Lynx  Model  48  is independent of existing pasture density (see equation 3.2). In order to l i m i t pasture growth, I included a pasture multiplier (PM) i n equation (3.2) as follows:  (3.3) where PM is a function of pasture availability, PA- For high values of PA we expect pasture growth to slow down and saturate. We also expect a decline i n growth when PA is very low following the principle that 'growth promotes growth'. A function having this general form is the Ricker function. Using this type of relationship I introduced a density-dependent function having the shape shown i n figure 3.7.  1  0  0  2  4  PA  Figure 3.7:  T h e pasture m u l t i p l i e r (PM) as a f u n c t i o n o f pasture a v a i l a b i l i t y (PA)-  To test the effects of this new function I generated a number of bifurcation diagrams and compared them with those from the original model. Figures 3.8 and 3.9 show those diagrams which correspond to figures 3.4 and 3.5 respectively.  Chapter 3.  Sheep-Hyrax-Lynx  49  Model  SFCN  Figure 3.8: O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s o b t a i n e d f r o m v a r y i n g SFCN ( n o m i n a l value = 0 . 2 8 / y r ) for the new m o d e l w h i c h includes a pasture l i m i t i n g m u l t i p l i e r .  In (a) revenue is p l o t t e d  o n the y - a x i s , i n (b) the state variable for l y n x females is used, and i n (c) the state v a r i a b l e for pasture is used.  In figure 3.8(a) total revenue declines to zero as SFCN increases. This is more m a t h ematically satisfactory than figure 3.4(a), where the curve stops abruptly at a positive revenue value, as it indicates clearly where a positive sheep population is no longer possible. Figure 3.8(c) shows the m a x i m u m pasture density which occurs when a positive sheep e q u i l i b r i u m (and hence a positive value for revenue) is impossible. T h i s density is lower than i n figure 3.4(c) but depends on the exact nature of the pasture multiplier. Another observation from figure 3.8 is that instead of A U T O being unable to converge at very low SFCN  (sheep female culling normal) values, a Hopf bifurcation occurs and  the bifurcation diagrams show that no stable equilibrium at which all three populations  Chapter 3.  Sheep-Hyrax-Lynx  Model  50  (a)  Revenue  0.5  1  1.5  LFN  (c)  Figure 3.9: O n e - p a r a m e t e r bifurcation diagrams obtained f r o m v a r y i n g LFN ( n o m i n a l value = 0 . 7 / y r ) for the new m o d e l w h i c h includes a pasture l i m i t i n g multiplier. In (a) revenue is p l o t t e d o n the y - a x i s , i n (b) the state variable for l y n x females is used, and i n (c) the state variable for pasture is used. coexist is possible for SFCN < 0.059. For these values there is insufficient pasture to support the high sheep population. Thus, introducing the pasture multiplier has improved the dynamics at low values of SFCN as well as high values and has solved the problem of revenue increasing rapidly as i n figure 3.4(a). In figure 3.9 a l i m i t point (see section A.2.13) has replaced the Hopf bifurcation of figure 3.5. T h e l i m i t point bifurcation clearly shows that for LFN > 1-491 no e q u i l i b r i u m at which all three populations coexist is possible. A g a i n , using X P P A U T to integrate the system numerically gives insight into the dynamics corresponding to the different regions i n the bifurcation diagrams. In particular, the temporal dynamics show that the pasture multiplier slows and limits pasture growth as desired. Comparing the dynamics  Chapter 3.  Sheep-Hyrax-Lynx  Model  51  of the original and modified models at SFCN = 0.5 shows that l i m i t i n g pasture growth has a stabilising influence (see figure 3.10). The oscillatory approach to e q u i l i b r i u m by the original model (figure 3.10(a)) is replaced by a smooth approach i n figure 3.10(b).  40  60  Time  40  60  Time  Figure 3.10: T i m e plots o b t a i n e d using (a) the o r i g i n a l m o d e l and (b) the m o d i f i e d m o d e l w i t h SFCN = 0.5.  T h e l i m i t i n g value for pasture in figure 3.8(c) is still rather high but, since I had no experimental data on which to base the form of the pasture multiplier, I did not think it worthwhile to fiddle with the function to obtain a more plausible value. T h e effects of introducing the multiplier have already been adequately demonstrated. A closer look at the behaviour exhibited by the model may suggest further modifications to the equations. The above is just one example of how bifurcation diagrams can help i n the process of model building. Another example can be found i n chapter 4. T h e next section describes how two-parameter studies can be used to obtain useful summaries of model behaviour.  3.5.4  A s u m m a r y of the effects of c u l l i n g b o t h h y r a x and l y n x  T h e model i n this chapter was originally developed to study the effects of culling hyrax and l y n x . A s was done in [116] and earlier in this chapter, o p t i m a l values (with respect to  Chapter 3.  Sheep-Hyrax-Lynx  Model  52  5  4 3 Revenue 2 1 0 0 Figure 3.11:  0.1  0.2 LCN  0.3  0.4  One-parameter bifurcation diagram of revenue as a function of LCN for the modified  model. The change in stability (denoted by the change from a solid to a dotted line) occurs at a limit point.  revenue) were found for the hyrax and l y n x culling normals. However, only one parameter was varied at a time. We can obtain a two-parameter diagram to summarise the effects of varying both parameters simultaneously as follows. Using the modified model the bifurcation diagrams for the lynx culling n o r m a l LCN have the f o r m shown i n figure 3.11. For LCN below the l i m i t point the sheep population dies out as there is too much predation by lynx. Using A U T O the l i m i t point can be continued i n HCN, the hyrax culling normal, as well as LCN- T h a t is, we can see how the position of this l i m i t point varies as a function of both HCN and LCN • T h i s gives the twoparameter bifurcation diagram i n figure 3.12. F r o m this figure it can be seen that sheep become extinct as a result of the combined effect of lynx predation and competition w i t h hyrax for pasture since both LCN and HCN are low in the region where sheep die out. Note that figure 3.12 could not have been produced using the original model as there was no l i m i t point i n the corresponding LCN bifurcation diagrams (see figure 3 . 3 ) — A U T O  Chapter 3.  Sheep-Hyrax-Lynx  Model  53  0.4  0.3  H  CN  0.2 Sheep die out  0.1  0 0  0.1  0.2  0.3  0.4  LCN  Figure 3.12:  T w o - p a r a m e t e r c o n t i n u a t i o n of the l i m i t p o i n t i n figure 3.11. ( T o d e t e r m i n e the b e h a v i o u r  c o r r e s p o n d i n g to a p a r t i c u l a r region i n this two-parameter d i a g r a m , choose values for HCN t h i s region and then use X P P A U T to integrate the s y s t e m  a n d LCN  in  numerically.)  signalled non-convergence and stopped calculating. C o m b i n i n g figure 3.12 with the observation that the l y n x population dies out for LCN  > 0.37 for all values of HCN gives figure 3.13. This was determined by generating  bifurcation diagrams for LCN for a number of different (fixed) HCN values.  Conversely,  varying HCN for a variety of fixed LCN values does not produce any parameter ranges where the hyrax population dies out. Figure 3.13 shows that all three populations coexist at equilibrium for a large set of culling rates. T h e diagram would be of even greater use if we knew the revenue value corresponding to each point in this two parameter space. T h i s can be done by recording information given by X P P A U T and using some other graphics package to plot a threedimensional surface. Using the modified model developed in the previous section I fixed the value of HCN, chose LCN = 0.15 (say) and used X P P A U T to find the equilibrium point numerically. I  Chapter 3.  Sheep-Hyrax-Lynx  Model  54  0.4  A l l three p o p u l a t i o n s coexist at e q u i l i b r i u m  HCN  Lynx die " out  0.2 |-  Sheep die out  0.1  0.2  0.3  0.4  LCN  Figure 3.13: modified  T w o - p a r a m e t e r bifurcation  d i a g r a m of the  HCN  and LCN  p a r a m e t e r space for  the  model.  then used the A U T O interface to vary LCN i n both directions. Using the G R A B feature of X P P A U T to move along the branch of equilibrium points, I recorded the revenue values at regular intervals along the curve. I did this for a number of HCN values and plotted the results using the public domain graphics package G N U P L O T  [125]. A surface plot  and corresponding contour plot are shown i n figure 3.14. As can be seen from the figure there is a large region of parameter space over which revenue does not vary much indicating that the model is very robust to changes i n the culling rates i n this region. This is a desirable property when it comes to developing management strategies.  3.5.5  B i o l o g i c a l i n t e r p r e t a t i o n of results  T h e analysis of the previous sections has led to a number of insights into the sheeppasture, h y r a x - l y n x system. Figure 3.8 shows that altering the number of ewes that are  Chapter 3.  Sheep-Hyrax-Lynx  Model  55  Figure 3.14: (a) Surface plot and (b) contour plot of revenue as a function of the hyrax and lynx culling normals. The arrow in (b) indicates the direction of increasing revenue.  Chapter 3.  Sheep-Hyrax-Lynx  Model  56  culled only affects the sheep-pasture subsystem. However, the effects on this subsystem are considerable. C u l l i n g too many ewes w i l l obviously cause sheep numbers to decline. M o r e important is the effect on revenue.  Significant increases i n revenue are possible  if the farmer culls fewer ewes as there is a greater return if these sheep are allowed to reproduce than if they are taken to market. (This is provided that the sheep stock is not too large for the pasture to support it, that is, provided SFCN > 0.059.)  However,  there is a trade-off as culling fewer ewes results i n a lower cash flow. Another trade-off results from the decrease i n pasture availability which accompanies a larger sheep stock. T h i s is already reflected i n the model by the dependence of sheep fecundity on pasture availability. However, an additional quantity representing the quality of sheep may be useful as this w i l l affect the returns from wool and m u t t o n sales and hence revenue. This presents another opportunity for improving the model. Figure 3.9 summarises the effects of lynx fecundity on the system. If l y n x fecundity is very high (LFN > 1-5) then the sheep population w i l l not be able to survive.  However,  if l y n x fecundity is sufficiently low (LFN < 0.78) then the lynx population does not need to prey on sheep as it can be supported by the hyrax population. i n l y n x fecundity at these values have no effect on revenue.  Further decreases  For intermediate values  (0.78 < LFN < 1.5) .considerable increases in revenue are possible if l y n x fecundity is decreased. T h i s favours the culling of lynx females i n particular. In section 3.5.2 we found that density-dependence of sheep and pasture dynamics on pasture availability is critical for regulating the sheep-pasture subsystem. In fact this encouraged the modification of pasture growth to include density-dependence. T h i s m o d ification restricts both pasture and revenue values from increasing indefinitely (compare figures 3.4 and 3.8) and also stabilises the temporal dynamics (see figure 3.10). F i n a l l y , the effects of culling both hyrax and l y n x were summarised using twoparameter diagrams. In particular, figure 3.14 shows that the model is robust to changes  Chapter 3.  Sheep-Hyrax-Lynx  Model  57  i n culling rates provided these rates are sufficiently high.  3.6  Conclusion  T h i s chapter has illustrated a number of potential uses of bifurcation analyses using packages such as X P P A U T . F i r s t , models having a large number of state variables and parameters can be analysed in greater depth than was previously practical. For system dynamics models bifurcation diagrams give more information than traditional sensitivity analyses as they summarise the behaviour of the model across a range of parameter values instead of being restricted to a single, fixed perturbation. A s a result these diagrams can indicate where model relationships are incomplete and can thus aid i n model formulation. Another example of this can be found in chapter 4. T h e analysis also showed that the model is quite robust i n a qualitative sense—the stability of the system is not greatly affected by parameter variations. However, revenue magnitudes are sensitive to certain parameters.  This is an important observation for  farmers as they seek to maximise their revenue.  Trade-offs between higher long-term  revenue and lower cash flows as well as higher revenue and lower sheep quality were also noted. T h e dynamics of the model turned out to be fairly simple from a bifurcation viewpoint. Other similar models may not be quite so robust. A n analysis similar to the one i n this chapter can be useful for uncovering regions of more complex behaviour i n such cases.  Chapter 4  Ratio-Dependent M o d e l  4.1  Introduction  Despite having a large number of state variables and parameters, the system dynamics model i n the previous chapter turned out to have fairly simple dynamics. This chapter focusses on a more theoretical model having only three state variables but whose dynamics are more complex.  In addition to describing how an analysis of such a model m a y  be approached using dynamical systems techniques, a dual a i m of the chapter is to highlight some of the difficulties associated with ratio-dependent models. These models are currently a topic of considerable controversy i n ecological circles. T h e example that I have chosen is a tritrophic model of a plant, herbivore and predator system developed by Gutierrez et al. [52]. It is a general model and is physiologically b a s e d — a property which the authors c l a i m makes estimation of parameter values from experimental data fairly straightforward.  However, there are a number of correction  factors i n the model whose function is to scale potential rates to realised rates. These factors complicate parameter estimation considerably.  Nevertheless, having appeared  in a leading journal, this model is sure to receive attention and further analysis of its dynamics m a y be of interest. In the next section I summarise the arguments for and against ratio-dependent models. Following this I describe the model equations and the technique of nondimensionalisation that I used to scale the equations and reduce the number of parameters i n the model. I  58  Chapter 4. Ratio-Dependent  Model  59  also introduce a small modification to the ratio-dependent terms. In the analysis that follows I consider both the original and this modified model in order to highlight some of the difficulties associated w i t h ratio-dependent models. After choosing a set of parameter values which give rise to a stable tritrophic equilibr i u m (that is, a stable equilibrium at which the plant, herbivore and predator populations are all nonzero) I begin the analysis by varying each parameter value i n turn to see what effect it has on the dynamics and to determine the range of behaviour that the model can exhibit.  Two-parameter bifurcation diagrams summarising the effects on system  behaviour of the plant and herbivore respectively complete the preliminary analysis. Having identified those parameters which have the greatest influence on the dynamics, I obtain a series of two-parameter diagrams using the modified model. These diagrams illustrate the combined effects of the lower two trophic levels on the behaviour of the model.  Of particular interest are parameter combinations which give rise to m u l t i p l e  stable states. In some cases a stable tritrophic equilibrium coexists w i t h a stable l i m i t cycle suggesting the possibility of an abrupt change i n the behaviour of the system if it is sufficiently perturbed (see section A.2.7). To complete the study the limits of isocline analysis i n a three-dimensional setting 1  are demonstrated. Gutierrez et al. [52] used this technique in their analysis of the model. A l t h o u g h isocline analyses have been employed in many settings and w i t h considerable success [36, 38, 44, 56, 74, 92, 103], in more complicated higher dimensional models for which the categorisation of variables as slow versus fast is not possible, their application 2  is l i m i t e d . A n isocline analysis allows at most two variables to vary simultaneously. T h i s means that for the current model one variable is held fixed which results in a partly static * A d e s c r i p t i o n of this technique together w i t h examples can be found i n [34]. I f the state variables i n a m o d e l v a r y on different t i m e scales it is often possible to a p p r o x i m a t e the s y s t e m b y a t w o - d i m e n s i o n a l m o d e l representing either the slow or the fast d y n a m i c s . A n isocline analysis can t h e n be done using the reduced s y s t e m . 2  Chapter 4.  Ratio-Dependent  Model  60  representation of the dynamics. The dynamical systems techniques allow all three state variables to vary simultaneously thus permitting a more accurate analysis.  4.2  Background  Ratio-dependent models assume that the functional response terms depend on ratios of the state variables rather than on absolute, values or products of variables as is the case for classical models. Although not a new idea, the concept of ratio-dependence in predator-prey interactions has been approached w i t h fresh interest i n ecological theory i n recent years (Berryman [17]). A m o n g the advantages of these types of models are that they prevent the paradoxes of enrichment and biological control predicted by classical models [17]. E x p e r i m e n t a l 3  4  observations of A r d i t i and Saiah [6] suggest that prey-dependent models are appropriate in homogeneous situations and ratio-dependent models in heterogeneous situations. In support of this G i n z b u r g and A k c a k a y a [45] and M c C a r t h y et al. [88] conclude f r o m their work that natural systems are closer to ratio-dependence than to prey-dependence and Gutierrez [51] develops a physiological basis for the theory. Gleeson [46], however, questions the assumptions of ratio-dependent models and notes that direct density-dependence, or self-regulation, i n the top consumer is sufficient to preclude the paradox of enrichment from classical models. F r o m his work on whether patterns among trophic levels are a reliable way of distinguishing between prey- and ratiodependence, Sarnelle [107] concludes that the ratio-dependent approach should only be applied when the predator and prey are the top two trophic levels i n an ecosystem. A b r a m s [1] argues that patterns and experimental results that have been used in support C l a s s i c a l m o d e l s p r e d i c t t h a t enriching a s y s t e m w i l l cause an increase i n the e q u i l i b r i u m density of the p r e d a t o r b u t not the prey and w i l l destabilise the c o m m u n i t y e q u i l i b r i u m (see B e r r y m a n [17]). C l a s s i c a l m o d e l s predict t h a t it is not possible to have b o t h a very low a n d a stable pest (prey) e q u i l i b r i u m density ( B e r r y m a n [17]). 3  4  Chapter 4.  Ratio-Dependent  Model  61  of ratio-dependent predation are consistent with numerous other explanations and that these other explanations do not suffer from pathological behaviours and a lack of plausible mechanism as do ratio-dependent models. Lundberg and Fryxell [75] note that it may be difficult to distinguish between competing hypotheses without a proper mechanistic understanding of the processes involved. In a recent paper Akgakaya et al. [3] respond to some of the above criticisms. T h e argument relevant to the current chapter concerns their refutation of the pathological behaviour of ratio-dependent models.  Freedman and Mathsen [41] note that ratio-  dependent models are invalid near the axes (that is, where the state variables are close to zero) as the ratios tend to infinity i n these regions. A s a result even when prey (resource) densities are very low, ratio-dependent models predict a positive rate of predator (consumer) increase provided that predator densities are low enough, since the number of prey available per predator increases to infinity as predator density declines to zero [1, 46]. In terms of isoclines, the problem stems from the fact that the predator isocline passes through the origin in ratio-dependent models which means that, even at low prey densities, a sufficiently small predator population can increase. According to H a n s k i [55] this is against intuition and many field observations. It also means that ratio-dependent models cannot be used to study extinction of species. However, Akgakaya et al. [3] state that the above problems near the axes are only pathological i n a mathematical sense and that in biological terms the result would be that both species increase initially and then predators consume all the prey and both species become extinct. Since prey-dependent models cannot predict this outcome they are pathological i n a biological sense. However, I show below that ratio-dependent models do not necessarily predict this outcome either. The ratio-dependent model that I w i l l describe i n the next section predicts oscillations of large amplitude i n these 'pathological' regions (see figure 4.3). W h i l e these large amplitude oscillations may be interpreted as  Chapter 4. Ratio-Dependent  Model  62  signalling extinction from a practical viewpoint, this is not the prediction of the ratiodependent model. A k c a k a y a et al. [3] state that: A realistic model of prey-predator interactions should be able to predict the whole range of dynamics observed in such systems in nature.  A ratio-  dependent model can have stable equilibria, l i m i t cycles, and the extinction of both species as a result of overexploitation. However, a few sentences later they agree that ratio-dependent models are not valid at very low densities (which are a precursor of extinction) and earlier i n the paper they state that: ...we do agree that it is at the extremes of low and high densities that strict ratio dependence may not be valid. In an attempt to clarify some of the arguments i n this debate, I introduce a small m o d ification to the ratio-dependent model of Gutierrez et al. [52] and study this modified model i n conjunction w i t h the original one. The analysis given below shows that the original model is structurally unstable as a small perturbation to the ratio-dependent terms substantially alters the dynamics.  4.3  M o d e l equations  T h e model equations are functionally homogeneous (that is, all three equations contain the same basic terms) as the authors argue that the same generalised functional and numerical responses must describe the search, acquisition and conversion of all organisms as they seek to satisfy their metabolic requirements. Details of the formulation of the model can be found i n Gutierrez et al. [52]. The final equations are:  Chapter 4.  Ratio-Dependent  Model  63  '  dMx  <xxM " 0  rDxMx. )  dt  Dx - rxMx  h  Mx  DM 2  dM  2  2  D  -rM *  M  -rM  M  b  2  dt  2  2  2  DM 3  dM  3  D  3  3  dt where Mx is the biomass of plants, M  2  3  b3 3  (4.1)  3  is the biomass of herbivores, and M  3  is the biomass  of predators. T h e parameters 9{ represent assimilation rates corrected for the efficiency of biomass conversion;  represents the proportion of the resource that is available to  its consumers (that is, its apparency); D{ is the per unit demand of the consumer for resources; r; is the base respiration rate corrected for the efficiency of biomass conversion; and hi is the degree of self-limitation. MQ represents a biomass equivalent of the light energy incident i n the growing space of the plants, and (f> x and d) x (which both lie Vt  u>  between 0 and 1) scale the potential photosynthesis rate to the realised rate. T h e respiration t e r m , r M , 4  1 + f >  ' , requires further explanation. Respiration usually i n -  creases w i t h population density [52] and thus should be an increasing function of M,-. However, introducing such a functional dependence increases the complexity of the model and, since the effect is usually small, the authors chose the simpler formulation r , - M ,  1+6  '  where bi has a value between 0.02 and 0.05. T h e disadvantage of this choice is that r; must have rather unusual units which depend on bi so that the t e r m r ; M / 5  same units as  (namely, g.day  -1  + b  ' has the  where g are the units of Mi). T h i s dependence of  the units on b{ is not satisfactory from a mathematical viewpoint but since bi is small I chose to ignore this initially. In the next section I discuss a small alteration to the model 5  T h e u n i t s of ri are g~ 'day~ where g are the u n i t s of M ; . b  1  Chapter 4. Ratio-Dependent  Model  64  which takes care of the difficulty. Gutierrez et al. [52] use parameter values corresponding to a cassava -mealybug 6  7  parasitoid system i n A f r i c a and claim that their analysis demonstrates that the para8  sitoid Epidinocarsis whereas Epidinocarsis  lopezi (De Santis) can control the mealybug (except on poor soils) diversicornis (Howard) and native natural enemies cannot.  M y first a i m was to reproduce the results in the paper [52]. However, the parameter values for the cassava-mealybug-parasitoid system given to me by the authors (only values for 6i,cti,Di are reported in the paper) did not yield the isocline configurations or the behaviour that they described. O n l y after changing some of the parameter values by several orders of magnitude did I succeed in producing qualitatively similar diagrams. T h i s haphazard approach of fiddling w i t h parameter values is not satisfactory. Scaling the equations would give a better idea of the relative magnitudes of the parameters. T h e procedure involved is discussed in the next section.  4.4  Nondimensionalisation  T h e technique of nondimensionalising or scaling is commonly used to simplify a system of equations as it has a number of other advantages. It illuminates which parameters are most important i n determining the dynamics of the model (Edelstein-Keshet [34]) and gives insight into the relative magnitudes of the parameters required to produce biologically reasonable behaviour.  Also, the state variables are scaled so that they all  have the same order of magnitude, say between 0 and 1. T h i s is important when solving the equations numerically as very different magnitudes can lead to computer roundoff errors (Gerald and Wheatley [43]). In the cassava-mealybug-parasitoid system the  Manihot escuhnta C r a n t z Phenacoccus manihoti M a t . - F e r r .  6  7  T h e larvae of a p a r a s i t o i d feed o n l i v i n g host tissue such t h a t the host is not k i l l e d u n t i l l a r v a l development is finished. 8  Chapter 4. Ratio-Dependent  Model  65  biomass of cassava is much larger than that of the mealybug and the parasitoid (an average cassava plant has a mass of about 2kg whereas the mealybug and parasitoids have average masses around 2mg). Scaling M i , at least, is thus v i t a l . N a t u r a l scalings for the state variables are given by their carrying capacities when 9  resources are abundant. Gutierrez et al. [52] calculate these to be  Ki = Replacing Mi by KiMi  'BiDi  i = 1,2,3.  (i=1,2,3) and t by rt* (here Ki has the dimensions of Mi and r  the dimensions of t and hence Mi (i=l,2,3) and t* are dimensionless variables) transforms equations (4.1) into system (4.2) . 10  dMi  dMx dMx dt  dt*  dM  dt dt*  x  =  a  —T ^ 1 — exp 2  ~d~F  2  DKM  3  dt*  2  aKM 2  r  3  3  1  r {K M )  b2  2  2  3  2  2  M  2  D3K3M  2  3  K  DKM 3  2  2  aKM  2  3  aKM 3  1 - exp  6D  1  DKM  3  dM  2  2  1 — exp  Mi  x  2  —r  r^ExMiP  cx K Mx  T (e D (I - exp 2  0  DxKxMx  2  dM  M  x  T (OxDx (l - exp  2  2  - r (K M ) * b  3  DKM 3  3  3  3  M. 3  (4.2)  3  Note that riK.\* = 6iDi. Choosing the dimensionless combinations of parameters ji  =  <f>i =  9  2 = 2,3  TOLi  etiKi-x  n  a  i — 1,2,3  rOiDi  -  =  t  t  1  !  =  1  '  2  '  3  F o r a n i n t r o d u c t i o n to n o n d i m e n s i o n a l i s i n g systems o f o r d i n a r y differential equations see [34].  I replaced the p r o d u c t #i</>r),i^aj,i b y 0 \ since a l l three p a r a m e t e r s have the s a m e effect o n the d y n a m i c s a n d , thus, do n o t need t o be considered separately. 1 0  Chapter 4.  where K  Q  Ratio-Dependent  Model  66  = M , gives the nondimensionalised equations (4.3) where I have replaced Mi 0  by Mi and t* by t for convenience. dMi dt dM 2  dt dM 3  dt  7i ^ 1 - e x p 72^1 -exp  =  7s ( ( 1 - exp  2  0i  "~M\  n Mx 2  1  1  3  exp  b  2  n  - M  h 3  M  nM 3  2  M  3  M  3  3  2  Mo  M  - M * )M -^-(l2  M £l M  X  2  2  =  0 M  M )M -^-[l-exp  (4.3)  1 M  3  3  T h e choice of dimensionless parameters is not unique.  Other combinations would  have led to slightly different final equations, however the above choices lend themselves to biological interpretation. For example, 7 ; can be thought of as the potential per unit biomass growth rate [52] or as the conversion efficiency of the consumer i n converting , the resource into biomass. fa can be thought of as the availability of the resource to the consumer or perhaps the nutritional value of the resource. 0 ; is made up of a ratio of quantities. T h e numerator can be thought of as the m a x i m u m amount of resource available to the consumer and the denominator as the m a x i m u m demand of consumers for resource. M o r e simply, Cli gives a measure of the ratio of supply to demand. T h e results i n [52] are based on the relationship between fa and 7,- since r is just a scaling factor, and hence results using equations (4.3) are comparable w i t h those i n [52]. I. mentioned earlier that the parameters r,- i n the original model have units which depend on bi. T h i s can be prevented by replacing the terms r i M i  1 + b i  w i t h terms of the  form r i M ^ Y ) ' where Ti has the same units as Mi. T can be thought of as a threshold 1  6  value above which self-limitation becomes noticeable. W i t h this modification the units of r,- are day  -1  and r - can be interpreted as a respiration rate as was originally intended. t  T h e new carrying capacities are given by '9 iDi Ti  Ti ; 1 — 1 , 2 , 3 .  Chapter 4. Ratio-Dependent  Model  67  It can be shown that setting the K^s equal to these new values and scaling the equations as above results i n system (4.3) once again. Thus the problems w i t h the respiration term can be ignored i n the rest of the analysis. H a v i n g scaled the equations we need to choose values for the parameters. A n advantage of scaling is that there are now 11 parameters instead of the original 18. For comparison w i t h the results of Gutierrez et al. [52] I wanted to find values which resulted i n isocline configurations similar to those in their paper. X P P A U T calculates and displays isoclines i n two dimensions and parameter values can be altered interactively. This proved useful for studying the effects of the different parameters on the isoclines. Since the competition effect is very small but difficult to quantify, I followed Gutierrez (personal communication) and chose bi = 0.02 ( i = l , 2 , 3 ) . Values of 71 = 2.0, 72 = 0.4, 73 = 0.1, f i i = 9.0, O2 = 8.0, O 3 = 10.0, 4>2 — 0.4, fa = 0.05 gave the isocline configurations shown in figure 4.1. These isoclines have similar shapes to those in [52]. A noticeable difference is that the M  3  isocline intersects the M  2  isocline to the left of the M - p e a k . 2  In fact, using the techniques in [52] it would not have been possible to conclude that the tritrophic e q u i l i b r i u m is stable for the isocline configuration shown i n figure 4.1 because of the position of the intersection point in the M2M3 plane. For the above parameter set (which I shall call the reference set) there is also a stable l i m i t cycle. T h e i n i t i a l values of Mi, M  2  and M  3  determine whether the system approaches the stable equilibrium or  the l i m i t cycle. Biological considerations suggest that the above values for the f2; are rather high and that the value for fa is rather low. However, in the absence of better information and since this parameter set has a nontrivial, stable equilibrium point, it is a convenient starting point. Before beginning an analysis of the model I would like to introduce a small m o d ification to the equations. Since the problems associated with ratio-dependent models  Chapter 4.  Figure fi  2  Ratio-Dependent  4 . 1 : Isoclines in the (a)  = 8.0, Q  3  = 10.0,  fa  = 0.4,  Mode]  MiM <j>  2  3  and  68  (b) M M 2  3  planes for 7! = 2.0,  = 0.05, and b -  j  2  = 0.4, 73 = 0.1, £2i = 9.0,  0.02 ( i = l , 2 , 3 ) . B o t h the stable e q u i l i b r i u m p o i n t  {  and the l i m i t cycle are s h o w n . M3 is fixed at 0.550 i n (a) and M\  is fixed at 0.602 i n (b). T h e s e values  correspond to the e q u i l i b r i u m p o i n t .  mentioned in section 4 . 2 involve low population densities, it would be interesting to know what effects a small modification to the model, which prevents the denominators of the ratios from getting too close to zero, would have on the dynamics. A b r a m s [l] states that modifications to ratio-dependent models cannot be made biologically realistic because the original models have no clear mechanistic derivation. However, some form of modification which prevents the ratios from tending to infinity may be useful for revealing any spurious behaviour near the axes which may result from the ratio-dependence. T h e ratios in model (4.3) have the form  Mi  '  T h e difficulties are experienced when Mi approaches zero.  A d d i n g a constant in the  denominator, that is replacing the ratio by  *T*< <'> a,- + Mi Although this addition may appear difficult to justify 4  would alleviate the problem.  4  Chapter  4. Ratio-Dependent  Model  69  biologically, Gutierrez [51] used an exponent of this form i n his functional response term. T h a t model is physiologically based as is the present one. In model (4.3) the ratio-dependent terms have the form n,-M,-_i-  k (1 — e x p  Mi  Mi  (4.5)  where k is a parameter or combination of parameters. W h e n the exponent is small we have k (1 — exp  M  Mi  «  t  (i-(i-9M^))  Mi  k  =  kiliMi-x.  In order to preserve this property when using the modified term (4.4), I replaced (4.5) by  ftM-i  k ( 1 — exp  (a; + Mi)  cii + Mi.  where a,- is a small constant, say 0.001. T h e resulting equations are: dMx dt  71 ^1 - exp  r-])(a + M ) - M ax + M ft Mx (a + M ) a + M n Mx I a + M ) - >y M * a + M 1  1  7  l  1  1  +  6  1  :  2  - j £  dM  2  ~dT  l  l  -  e  X  P  72 [1 - exp  2  2  2  3  ~dT  -rU  73 (1 - exp  3  2  2  2  (a + M ) 3  a + M nM 1 3  3  3  3  2  a + M\ 3  2  2  VL M  fa f, dM  1+b  2  2  1 _ e x p  2  2  a + 3  M )- M \ 1+b  3  73  3  (4.6)  3  If a = 0 (i = 1,2,3) then the above model is equivalent to system (4.3). W e are now i n a t  position to begin the analysis.  Chapter 4.  4.5  Ratio-Dependent  Model  70  M o d e l analysis  4.5.1  One-parameter studies  Since a partial qualitative analysis of the original model was done by Gutierrez et al. [52] but using different techniques (namely, isocline analysis), it w i l l be informative to compare some of the results. The choice of dimensionless parameters was done w i t h this in m i n d . I begin the analysis by varying each parameter in turn using A U T O (through X P P A U T ) to see how it affects the dynamics. This w i l l be done for all the parameters except the 6;'s due to the observation that intraspecific competition at the i  th  trophic  level now increases as bi decreases since Mj now lies between 0 and 1 as a result of the scaling. T h i s was overlooked i n the original model and can only be rectified by changing the formulation of the respiration term. Rather than modifying the model at this stage I chose instead to keep the 6,-'s fixed. (For a given value of bi respiration still increases w i t h biomass as required.) Since one of the m a i n conclusions i n Gutierrez et al. [52] concerns the relative efficacy of two parasitoids in controlling the cassava mealybug population, I begin by studying those parameters affecting the third trophic level, namely 73, (j>$ and fi . I then discuss 3  the remaining parameters. B o t h the original model corresponding to a = 0 (i = 1,2,3) in t  system (4.6) and the modified model with a,=0.001 (i — 1,2,3) are investigated. I chose this particular value for the a,-'s as it only affects the isocline configurations i n figure 4.1 at low values of the state variables which is where the difficulties are encountered. I also investigate a few other values. The reference values for the other parameters are summarised i n table 4.1. The results, together with possible biological interpretations, are described in the next section. For generality I w i l l refer to the plant, herbivore (or prey) and predator biomasses rather than the cassava, mealybug and parasitoid since the reference parameter set was not chosen from experimental data.  Chapter 4. Ratio-Dependent Model  Parameter 7i 72 73 <j>2  4>3  fi  x  6i b  2  b  3  Table 4.1:  Description potential growth rate per unit plant potential growth rate per unit herbivore potential growth rate per unit predator availability of plant to herbivore availability of herbivore to predator supply of resources/demand by plants supply of plants/demand by herbivores supply of herbivores/demand by predators degree of self-limitation for plants degree of self-limitation for herbivores degree of self-limitation for predators  71  Value 2.0 0.4  0.1 0.4  0.05 9.0 8.0 10.0 0.02 0.02 0.02  Reference p a r a m e t e r set. I n the subsequent figures i n this chapter, o n l y those p a r a m e t e r s  w h i c h are e x p l i c i t l y m e n t i o n e d have been altered. T h e values for a l l the other p a r a m e t e r s c o r r e s p o n d to the ones i n this t a b l e .  A n a l y s i s of the predator parameters T h e parameter 73 can be thought of as the potential predator biomass growth rate when prey are abundant, or as the predator's conversion efficiency i n the presence of abundant prey. A n important observation is that the value of 73 does not affect the isoclines. The M  3  zero isocline is given by  = 0 and the solution of this equation is independent of  73 (see equations (4.6)). Hence, an isocline analysis similar to that done i n [52] would not give any insight into how this parameter influences the behaviour of the system. Bifurcation diagrams showing the effects of varying 73 for both the original model and the modified model are shown in figure 4.2. Mi is plotted on the y-axis.  These  diagrams were obtained using A U T O through X P P A U T . The system was first integrated numerically u n t i l it was close to an equilibrium and then X P P A U T was used to find the exact location of the equilibrium point (singular point). This point was used as a starting point for A U T O . In both the diagrams in figure 4.2 the position of the equilibrium point does not vary  Chapter 4. Ratio-Dependent Model  (a)  1  72  (b)  0.8  LI  S3,oooYoo*LP  0.6  HB  Mi  Mi  8 0.4 _ LPs P D  0.2  i  < « « » «i'i*LP 0  i  0 0  0.05  0.1 73  0.15  0.2  F i gure 4.2: O n e - p a r a m e t e r bifurcation d i a g r a m s o b t a i n e d b y v a r y i n g  7 3 i n (a) the o r i g i n a l  model  ( a ; = 0 , i = 1,2,3) a n d (b) the modified m o d e l (a;=0.001, i — 1 , 2 , 3 ) . T h e state variable M\ is p l o t t e d o n the y-axes. H B denotes a H o p f bifurcation, LP a l i m i t p o i n t a n d PD a p e r i o d - d o u b l i n g  bifurcation.  w i t h 73, which agrees with the previous observation that 73 does not affect the isocline configuration.  However, 73 does affect the stability of the system. In both cases the  e q u i l i b r i u m point is unstable for very low values of 73 (low predator growth rate) and the stable attractor is a l i m i t cycle for these values.  For the original model we have  an example of hard loss of stability (see section A.2.7) so that for certain values of 73 there are two stable a t t r a c t o r s — a sink and a stable l i m i t cycle (such as i n figure 4.1). T h e i n i t i a l values of the state variables determine which final state is reached.  Also,  perturbations to the system may cause a j u m p from one stable attractor to the other if the disturbance is sufficiently large. For a very small range of 73 values there are two stable l i m i t cycles i n figure 4.2(a). T h e range of values is so small, however, that it is not of m u c h biological significance. Observing the temporal dynamics of the system (using X P P A U T ) for different values of 73 I found that larger values of 73 decrease the time taken to reach e q u i l i b r i u m . Thus, increasing 73, the potential growth rate of the predator, has a stabilising influence on the system. This seems biologically plausible as higher values of 73 suggest that the predator is better adapted to controlling its prey. It is interesting, however, that this trait does  Chapter 4. Ratio-Dependent Model  73  not affect any of the equilibrium biomasses. T h e modified model has a much smaller range of parameter values over which cycles occur and the amplitudes of these cycles are smaller than for the original model (see figure 4.2). T h u s , even though the a,-'s have small values, they appear to have a stabilising influence on the dynamics. Another parameter which directly affects the predator is fa, the availability (or nutritional value) of the herbivore to the predator.  Bifurcation diagrams for the original  and modified models respectively and for all three state variables are shown i n figures 4.3 and 4.4. F r o m these figures we can see that as fa increases there is a general increase i n the Mi e q u i l i b r i u m value or l i m i t cycle m a x i m u m . The larger fa the greater the availability of the herbivore to the predator and the easier it is for the predator to control the herbivore. Obviously, the lower the herbivore population the higher the plant e q u i l i b r i u m . A s the Mi e q u i l i b r i u m value approaches the Mi carrying capacity in the original m o d e l , a Hopf bifurcation (see section A.2.10) occurs at fa = 0.09 (see figure 4.3). T h e periodic orbit associated w i t h this Hopf bifurcation undergoes a number of period-doubling bifurcations (see section A.2.16) which leads to more complicated cycling behaviour. A n example of the temporal dynamics when fa — 0.16 is shown in figure 4.5. There are two complete cycles i n these diagrams. It is interesting that the predator dynamics are less variable than those of the plant and the herbivore.  This cycling behaviour also contrasts with  that described by Akcakaya et al. [3] as being biologically plausible for low herbivore and predator values (see section 4.2). A s fa increases above the upper Hopf bifurcation the m i n i m a of the M  2  i n particular get very small (of the order of 1 0  - 1 5  and M% cycles  and lower according to X P P A U T ' s data  window). F r o m a practical viewpoint these populations would be considered extinct due to statistical variation i n which case the plant population would increase to its carrying  Chapter 4. Ratio-Dependent  74  Model  0  0.05  0.1  0.15  1  1  <f>3  (b)  0.6 |  1  0.15  (c)  0.15 fa  Figure  4.3:  One-parameter  state variables M\,  M  2  b i f u r c a t i o n d i a g r a m s o b t a i n e d by v a r y i n g  <f>  3  i n the o r i g i n a l m o d e l . T h e  a n d M3 are p l o t t e d on the y-axes i n (a), (b) a n d (c) respectively.  Only  the  p o s i t i o n s o f the H o p f bifurcations have been i n d i c a t e d . T h e changes i n s t a b i l i t y o f the l i m i t cycles occur at l i m i t p o i n t s or p e r i o d - d o u b l i n g bifurcations b u t these have not been m a r k e d i n the d i a g r a m s .  Chapter 4.  Ratio-Dependent  Model  Figure 4.4: One-parameter bifurcation diagrams obtained by varying <f>3 in the modified model. state variables Mi, M and M are plotted on the y-axes in (a), (b) and (c) respectively. 2  3  75  The  Chapter  4. Ratio-Dependent  Model  76  (a) 1.2  (b)  Time (c) 10.8 0.6 M3 0.4  Time  F i g u r e 4.5: Time plots of (a) M i , (b) M and (c) M for <p = 0.16. A l l the other parameter values 2  are as in the reference parameter set.  3  3  Chapter  4. Ratio-Dependent  77  Model  capacity for these values of fa. This is exactly the case for the modified model (see figure 4.4).  Thus the upper Hopf bifurcation may be an artifact of ratio-dependent models.  T h i s w i l l be discussed in more detail later. Another problem w i t h these low m i n i m a is that they lead to numerical difficulties. T h i s occurs as a result of the way in which the model is f o r m u l a t e d — t h e dependence of many of the terms on the ratio  i n particular. These ratios become difficult to  evaluate numerically as M ; approaches zero causing the ratio to tend to infinity. X P P A U T cannot calculate zero isoclines and crashes while A U T O often enters an infinite loop if such a situation arises and may crash. Setting the total number of steps for a continuation fairly low sometimes allows A U T O to break out of the loop and signal non-convergence. M a n u a l l y stopping a continuation when one of the state variables gets very close to zero also prevents the package from crashing. This explains why the l i m i t cycles in figure 4.3 are only calculated up to fa — 0.12. The above problems do not occur when using the modified model. It may have been noticed that 4> has a significant effect on the equilibrium values of 3  all three state variables. This is in agreement w i t h Gutierrez et al. [52]. However, the way in which they arrive at this conclusion is not entirely correct. In Gutierrez et al. [52] it is stated that a less efficient parasitoid has a wider C-shaped M2-isociine. It is true that if fa is decreased the M2-isocline widens (see figures 4.6(a) and (b)). provided that M  3  is constant. If the system is integrated and M  B u t this is  is allowed to vary u n t i l  3  a new e q u i l i b r i u m is reached and the isoclines are plotted w i t h this new e q u i l i b r i u m  M  3  value, then the final M2-isocline may in fact have a narrower C-shape than before (see figure 4.6(c)). T h e t h i r d parameter which directly affects the predator is Q . It is more difficult to 3  interpret this parameter biologically but it can be thought of as the ratio of the 'supply' of herbivore to the 'demand' of the predator when both populations are at their carrying  Chapter 4.  Ratio-Dependent  Model  78  (a)  1 0.8 0.6 M2 0.4  / - -  0.2  -  0, 0 Ml  (b)  1 0.8  -  0.6 M2  "A  0.4 0.2 0. 0  OA  03.  OS  01  0.6  0.1  Ml  (c)  0.2  0.4 Ml  Figure 4.6:  Isoclines i n the M i M  2  plane.  3  stable t r i t r o p h i c e q u i l i b r i u m p o i n t i n (a) when equilibrium point when  <f> — 3  0.05.  <f>  I n (a)  other p a r a m e t e r values are fixed. In (a) a n d (b) M  3  = 0.06 a n d i n (b) a n d (c)  <f> = 0.05. 3  A l l the  has the same value—the value c o r r e s p o n d i n g to the  <j> = 3  0.06. I n (c)  M  3  has the value c o r r e s p o n d i n g to the  Chapter 4. Ratio-Dependent  79  Model  capacities. Hence it reflects how l i m i t i n g resources are to the predator. Figure 4.7 shows the bifurcation diagrams corresponding to the original model for all three state variables. For most values of fi there is no change in the qualitative behaviour which i n this 3  case is given by stable l i m i t cycles. E v e n the amplitudes of these cycles do not alter m u c h although those for M 3 decline slowly as f i is increased. It is only at low values of 0 3  3  that  a change i n dynamics occurs. It is also at these low values that D 3 has the greatest effect on e q u i l i b r i u m magnitudes. Low values of tts suggest a restricted supply i n relation to demand and are reflected i n both low M  2  and low M  3  equilibrium values. Since there  are relatively fewer herbivores available (in relation to predators) the predators are able to control them better but the lower herbivore equilibrium also restricts the number of predators that can survive. As expected, lower equilibrium values for M  2  correspond to  higher e q u i l i b r i u m values for M i . T h e corresponding bifurcation diagrams for the modified model are shown i n figure 4.8. These diagrams are very similar to those in figure 4.7 except that there is a second Hopf bifurcation resulting in first a decline i n the amplitudes of the cycles followed by a stable tritrophic equilibrium as ££3 is increased. A g a i n the introduction of the a,-'s has had a stabilising influence on the dynamics. F r o m figure 4.8 we can clearly see that high values of fi are, however, undesirable as the M 3  M  2  x  equilibrium value is low while that for  is relatively high. T h e above analysis has shown that the properties of the predator affect the stability  of the system as well as the equilibrium magnitudes of the herbivore (directly) and the plant (indirectly). The extent of these effects depends on the properties of b o t h the plant and the herbivore. In the next section the parameters affecting these lower two trophic levels are examined in more detail.  Chapter 4. Ratio-Dependent  Figure 4.7:  Model  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s o b t a i n e d by v a r y i n g  state variables M\,  M  2  80  Q  3  i n the o r i g i n a l m o d e l . T h e  a n d M 3 are p l o t t e d on the y-axes i n (a), (b) a n d (c) respectively.  Chapter 4.  Ratio-Dependent  Model  81  Figure 4.8: One-parameter bifurcation diagrams obtained by varying ^ 3 in the modified model. The state variables M i , M and M 3 are plotted on the y-axes in (a), (b) and (c) respectively. 2  Chapter 4.  Ratio-Dependent  82  Model  Analysis of the plant and herbivore parameters T h e parameter (f> can be thought of as the availability or the nutritional value of the 2  plant to the herbivore. T h e bifurcation diagrams for the original and modified models respectively and for Mi, M  2  and M  3  are shown i n figures 4.9 and 4.10.  For both models the values of M and M 3 at equilibrium remain constant for the most 2  part as (f) is varied (up to a l i m i t i n g point) while the Mi equilibrium value declines. A 2  possible explanation is that, due to the greater availability of the plant to the herbivore, a lower plant biomass can support the same biomass of herbivore at a higher <f> value. T h i s 2  enables the herbivore to have an even greater impact on the plant. If (f> is sufficiently 2  high then the herbivore can send the plant population to extinction. T h i s is suggested by the very low cycle m i n i m a for the original model and is even clearer for the modified model where the Mi equilibrium value is very low for (j> > 1.061 (the location of the Hopf 2  bifurcation). O f course, once the plant is extinct both the herbivore and the predator are forced into extinction as well. This can be checked by observing the temporal dynamics of the system using X P P A U T for (j) to the right of the upper Hopf bifurcation (4> > 1.061) 2  2  using the modified model. For the original model we again have the problem of very low cycle m i n i m a for the state variables. These are biologically unrealistic and cause numerical difficulties. T h e modified model does not have this problem. Suppose we vary 71, the assimilation or conversion efficiency of the plant when resources are abundant. T h e bifurcation diagrams are shown i n figure 4.11. C o m p a r i n g these diagrams w i t h figures 4.9 and 4.10 we see that they are almost mirror images. T h a t is, decreasing 71 has a very similar effect to increasing <f> . B o t h parameters can be 2  thought of as affecting the resistance of the plant to the herbivore. Decreasing 71 lowers the quality of the plant as it cannot convert resources as effectively.  A s a result the  Chapter 4.  Figure  4.9:  Ratio-Dependent  Model  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s s h o w i n g the effects of v a r y i n g  m o d e l . T h e state variables M i , M  2  83  <f>2  for the o r i g i n a l  a n d M 3 are p l o t t e d o n the y-axes i n (a), (b) a n d (c) respectively.  Chapter 4. Ratio-Dependent  Model  84  Figure 4.10: One-parameter bifurcation diagrams showing the effects of varying fa for the modified model. The state variables Mi, M and M 3 are plotted on the y-axes in (a), (b) and (c) respectively. 2  Chapter 4.  Ratio-Dependent  Model  85  (a)  (b)  Mi  Mi  Figure 4.11: O n e - p a r a m e t e r bifurcation d i a g r a m s showing the effects of v a r y i n g 71 for (a) the o r i g i n a l m o d e l a n d (b) the m o d i f i e d m o d e l . T h e state variable M i is p l o t t e d o n the y-axes. T h e l i m i t p o i n t s ( L P ) m a r k the e n d p o i n t s of the region of hysteresis (see section A.2.11).  detrimental effect of the herbivore on the plant is greater. Increasing fa, the availability (nutritional value) of the plant to the herbivore, achieves the same result but more directly. W e can generate two-parameter diagrams in ( i , 0 2 ) - s p a c e by continuing the l i m i t 7  points and the Hopf bifurcations in two parameters (see sections A . 2 . 1 and B.4).  The  results are shown i n figure 4 . 1 2 — s o l i d lines indicate Hopf bifurcation continuations and dotted lines indicate l i m i t point continuations. These diagrams show clearly that decreasing 0 2 or increasing 7 1 has a similar effect and that there is a transition between different types of qualitative behaviour as the region enclosed by the l i m i t point continuations is crossed. Part of this result could have been predicted from [52] since they note that it is the ratio of 7 1 and fa that determines the nature of the plant isocline. We thus expect this inverse relationship. However, the one-parameter bifurcation diagrams have given us the additional information that for certain parameter ranges l i m i t cycles and/or multiple stable states are possible. B o t h these phenomena are important biologically. W e can make a number of observations by comparing figures 4.12(a) and (b).  First,  Chapter 4.  Ratio-Dependent  Model  (a)  1  0.75  fa  1  E q u i l i b r i u m values for a l l three _ p o p u l a t i o n s tend to zero  0.5 Stable tritrophic equilibrium  0.25  86  -(low  plant  I  •"'  —  _. • V ^ ^ \ T w o stable . -'li* attractors  MX j . J Stable .jy^ / cycles^^  Stable t r i t r o p h i c equilibrium  /  ( h i g h p l a n t value)  value)  i  i  i  2 7i  (b)  1  0.75  fa  E q u i l i b r i u m values for a l l three _ p o p u l a t i o n s tend to zero  0.5 -  0.25  Stable tritrophic equilibrium -(low p l a n t value) _.  Stable tritrophic equilibrium (high p l a n t value)  7i F i g u r e 4.12:  T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m s s h o w i n g the effects o f v a r y i n g b o t h 71 a n d fa o n the  p o s i t i o n s o f the l i m i t p o i n t s a n d H o p f bifurcations i n figure 4.11. T h e d i a g r a m i n (a) corresponds to the o r i g i n a l m o d e l a n d t h a t i n (b) to the m o d i f i e d m o d e l . S o l i d lines i n d i c a t e H o p f b i f u r c a t i o n c o n t i n u a t i o n s a n d d o t t e d lines i n d i c a t e l i m i t p o i n t c o n t i n u a t i o n s .  Chapter 4. Ratio-Dependent  Model  87  whereas in (a) A U T O could not calculate beyond the point denoted by M X , this problem does not occur in (b). A closer investigation reveals that the equilibrium values for the state variables are close to zero i n the upper left triangle of the two-parameter space and this results i n numerical problems when using the original model. X P P A U T also has difficulty calculating zero isoclines i n this region and often crashes. However, figure 4.12(b) gives a more complete picture of the dynamics. There are three distinct regions in this diagram two of which correspond to stable tritrophic equilibria while stable cycles occur i n the other. One of the regions of stable equilibria has high e q u i l i b r i u m values of M i but the other has low equilibrium v a l u e s — a n important distinction ecologically. T h e parameters 72 and fa also appear to have inverse effects if we compare the average of the cycle m a x i m a and m i n i m a when 72 and <j) are varied independently (compare 3  figures 4.3, 4.4 and 4.13). We can investigate this relationship i n more detail by continuing the Hopf bifurcations in figure 4.13 in fa as well as 72. T h e results are shown i n figure 4.14.  0  0.25  0.5 72  0.75  1  0  0.25  0.5  0.75  72  Figure 4.13: One-parameter bifurcation diagrams showing the effects of varying 72 for (a) the original model and (b) the modified model.  A g a i n , i n general decreasing fa has a similar effect to increasing 72. However there are a few more Hopf bifurcations associated w i t h these parameters. A U T O stops at the  Chapter 4. Ratio-Dependent  Figure 4.14:  Model  T w o - p a r a m e t e r bifurcation d i a g r a m s s h o w i n g the effects of v a r y i n g b o t h  the p o s i t i o n s o f the H o p f bifurcations i n figure 4.13.  88  and  <p3  on  Chapter 4.  Ratio-Dependent  Model  89  point marked M X when using the original model since the equilibrium values for M M  3  2  and  are very small here. The complete continuation is shown in figure 4.14(b) using the  modified model. B y comparing the two diagrams in figure 4.14 it can be seen that the dynamics for the modified model, while very similar to those for the original m o d e l , are more stable in general. The regions corresponding to tritrophic equilibria are larger and the cycles i n the upper half of the two-parameter space are less complex. These claims are made clearer i n figures 4.15 and 4.16.  0.3  0.2 </>3  0.1  0 0  0.5  1  1.5  2  2.5  72  Figure 4.15: Two-parameter bifurcation diagram of the Hopf bifurcation continuations in (72,</>3)-space for the modified model with a;=0.002 (i = 1,2, 3). Figure 4.15 shows the (72,</>3)-space for the modified model w i t h a =0.002 [i = 1 , 2 , 3 ) . t  The regions of stable equilibria are even larger than in figure 4.14(b) resulting i n smaller regions of cycles.  The presence of the a,-'s seems to have a stabilising effect on the  dynamics of the system.  Chapter 4. Ratio-Dependent  Model  90  (a)  Time  Time  Figure 4.16: T i m e plots corresponding to the points marked with *'s in figures 4.14(a), 4.14(b) and 4.15. (a) These plots were obtained using the original model with (i) 72=0.4, <^>3=0.15 and (ii) -y =0-4, (^3=0.3. (b) These plots were obtained using the modified model with ai=0.001 (i = 1,2,3) and (i) 72=0.7, ^ 3 = 0 . 1 5 and (ii) 72=0.7, </>3=0.3. (c) These plots were obtained using the modified model with a « = 0 . 0 0 2 (i = 1,2,3) and (i) 7 =0.8, <p =0.2 and (ii) 72=0.8, <^ =0.3. 2  2  3  3  Chapter 4. Ratio-Dependent  Model  91  Figure 4.16 shows the time plots corresponding to points marked w i t h *'s i n the lefthand section of the upper region of cycles in figures 4.14(a),(b) and 4.15. T h e left-hand portion of this region is where the values of M  and M 3 are low and, hence, where the  2  nonzero a 's have most effect. Clearly nonzero a,-'s reduce the complexity of the cycles 8  (even for very small values) and increasing their value also reduces the cycle amplitude. A n additional point to note is that the cycles for the original model (a,=0, i — 1,2,3) undergo long periods of extremely low values which is unrealistic f r o m an ecological viewpoint. T h e only parameters that have not been discussed so far are f i i and fl . B i f u r c a 2  tion diagrams corresponding to O i are shown i n figure 4.17. O n l y the diagrams for Mi  (a)  (b)  Mi  Mi  10  20  30  40  50  10  20  30  40  50  fli Figure 4.17:  O n e - p a r a m e t e r bifurcation d i a g r a m s s h o w i n g the effects of v a r y i n g  fli  for (a) the o r i g i n a l  m o d e l a n d (b) the m o d i f i e d m o d e l . T h e state variable Mi is p l o t t e d o n the y-axes.  have been shown but those for M  2  and M 3 have the same s h a p e — t h e carrying capacity  equilibria are however different. A s can be seen from figure 4.17, i n both cases fli does not affect the stability of the equilibrium point and only affects e q u i l i b r i u m magnitudes when it drops to low values. Similar conclusions can be made regarding ft (see figure 4.18) except that at low Q 2  2  values there is a change i n dynamics to stable l i m i t cycles for both models although the  Chapter 4.  Ratio-Dependent  92  Model  range of parameter values giving rise to cycles is slightly smaller for the modified model.  4.5.2  C o m b i n i n g plant and herbivore dynamics  In the previous section it was seen that 71, fa, 72 and $3 all have significant effects on the dynamics of the model. The former two parameters determine the properties of the plant and the latter two the properties of the herbivore. T h e two-parameter diagrams i n figures 4.12 and 4.14 thus summarise the effects of the plant and herbivore trophic levels respectively provided that the dynamics of the other trophic levels are constant. It would be interesting to know how the behaviour of the model changes as both the plant and the herbivore dynamics are altered. This is the focus of the present section.  The  modified model w i t h a;=0.00T, (i = 1,2,3) is used to investigate the interaction of the two trophic levels as this model allows a more complete picture of the dynamics to be obtained than the original model. Figure 4.12 shows that increasing  7 l  or decreasing fa has a similar effect. Generating  two-parameter diagrams of the (72,^3)-space for different (fixed) values of one of these  Chapter 4.  Ratio-Dependent  Model  93  plant parameters, say 71, w i l l thus give an idea of the combined effects of the plant and herbivore.  Figures 4.19, 4.20, 4.21, 4.22 and 4.23 show the Hopf bifurcation and l i m i t  point continuations for five values of 71, namely 71 = 0.4, 0.6, 1.2, 1.8 and 2.4. Figure 4.14(b) corresponds to 71 = 2.0 and can be viewed i n conjunction w i t h these diagrams. A s can be seen from these diagrams, altering 71 affects the shape and/or the position of the Hopf bifurcation continuations (solid lines) and the l i m i t point continuations (dotted lines). Let us consider the l i m i t point continuations first. Figures 4.24(a), (b) and (c) show one-parameter bifurcation diagrams obtained by varying 72 w i t h 71 = 0.4 and <f>3 = 0.07, 0.17 and 0.25 respectively.  These diagrams correspond to the horizontal  dotted lines i n figure 4.19. The l i m i t points i n these diagrams demarcate the region of hysteresis, that is, the range of parameter values giving rise to multiple equilibria. We can see that this range of values increases as <f> increases. 3  T h e existence of a hysteresis phenomenon does not i m p l y the existence of two stable equilibria. This depends on other factors such as the occurrence of Hopf bifurcations. In figure 4.24(a) there are two stable equilibria for 72 between the two l i m i t points (0.18 < 72  < 0.30). One of these stable equilibria corresponds to a low e q u i l i b r i u m Mi  value  and the other to a high equilibrium Mi value. The unstable equilibrium intermediate to these stable equilibria demarcates their domains of attraction and indicates the extent of the perturbation (in terms of Mi) required to move the system from one stable attractor to the other (refer to section A.2.11). In figures 4.24(b) and (c) the region of two stable equilibria occurs between the lower l i m i t point and the Hopf bifurcation on the upper branch of equilibria (0.41 < 72 < 0.57 in (b) and 0.60 < 72 < 0.65 in (c)). Collating the information obtained from figure 4.24 allows us to classify the different regions i n figure 4.19 according to the qualitative behaviour that is found there. T h a t is, the region enclosed by the l i m i t point continuations (region of hysteresis) can be divided into a region of two stable equilibria and a region of only one stable e q u i l i b r i u m having a  Chapter 4.  Ratio-Dependent  94  Model  1  /  i  1  i  /Hysteresis / / (one stable-" /N  Stable e q u i l .  equil.)-'  ( h i g h plant .• value)  -  / Stable t r i t r o p h i c equilibrium (low p l a n t value)  -  0  Figure 4.19:  -  - - ^ j T w o stable equilibria  i  i  i  i  0.5  1  1.5  2  2.5  T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m o b t a i n e d u s i n g the m o d i f i e d m o d e l w i t h a , = 0 . 0 0 1 ,  (i = 1, 2, 3) a n d 71 = 0.4. T h e H o p f b i f u r c a t i o n continuations are i n d i c a t e d b y s o l i d lines a n d the l i m i t p o i n t c o n t i n u a t i o n s b y d o t t e d lines.  Chapter 4. Ratio-Dependent  Figure 4.20:  Two-parameter  Model  95  b i f u r c a t i o n d i a g r a m o b t a i n e d u s i n g the m o d i f i e d m o d e l w i t h 0^=0.001,  (i = 1, 2, 3) a n d 71 = 0.6. T h e H o p f b i f u r c a t i o n continuations are i n d i c a t e d by s o l i d lines a n d the l i m i t p o i n t c o n t i n u a t i o n s by d o t t e d lines.  Chapter 4. Ratio-Dependent  Figure 4.21:  Two-parameter  96  Model  b i f u r c a t i o n d i a g r a m o b t a i n e d u s i n g the m o d i f i e d m o d e l w i t h a = 0 . 0 0 1 , t  (i = 1, 2, 3) a n d 71 = 1.2. T h e H o p f b i f u r c a t i o n continuations are i n d i c a t e d by solid lines a n d the l i m i t p o i n t c o n t i n u a t i o n s by d o t t e d lines.  Chapter 4. Ratio-Dependent  Figure  4.22:  Two-parameter  (i = 1 , 2 , 3 ) a n d ji  Model  97  bifurcation d i a g r a m obtained using the modified m o d e l w i t h a , = 0 . 0 0 1 ,  = 1.8. T h e H o p f bifurcation continuations are i n d i c a t e d by solid lines a n d the l i m i t  p o i n t c o n t i n u a t i o n s by d o t t e d lines.  Chapter 4. Ratio-Dependent  Model  98  Figure 4.23: T w o - p a r a m e t e r bifurcation d i a g r a m o b t a i n e d using the modified m o d e l w i t h a;=0.001, (i = 1,2,3) and 71 = 2.4. T h e H o p f bifurcation continuations are indicated by s o l i d lines and the l i m i t p o i n t c o n t i n u a t i o n s by dotted lines.  Chapter 4.  Ratio-Dependent  Figure  One-parameter bifurcation  (a)  4.24:  <f>3 =  0.07, (b)  i n figure 4.19.)  c/>3  99  Model  d i a g r a m s o b t a i n e d by v a r y i n g 72 w i t h 71  = 0.17 a n d (c) ^ 3 = 0.25.  =  0.4 a n d  (These correspond to the h o r i z o n t a l d o t t e d lines  Chapter 4. Ratio-Dependent  low value for Mi.  Model  100  The regions outside the l i m i t point continuations have one e q u i l i b r i u m  point and it is stable. Those to the left of the hysteresis region have high e q u i l i b r i u m Mi values and those to the right have low equilibrium Mi values. T h e diagrams in figures 4.20-4.23 can also be divided into different qualitative regions using information from one-parameter studies. In these two-parameter diagrams the Hopf bifurcation continuations are not contained within the l i m i t point continuations.  The  one-parameter bifurcation diagram in figure 4.25 corresponds to the horizontal dotted line i n figure 4.21 (that is, 71 = 1.2 and fa = 0.2). In this case there are no parameter combinations which give rise to two stable equilibria but there is a region of stable l i m i t cycles.  Starting at 72 = 1.5, as 72 is decreased we have a single stable e q u i l i b r i u m  w i t h a low Mi value. A t the (lower) Hopf bifurcation this stable equilibrium is replaced by stable l i m i t cycles which increase in amplitude as 72 is decreased.  A s the upper  Hopf bifurcation is approached the cycles undergo some period-doubling bifurcations and then rapidly decrease i n amplitude. Due to a hard loss of stability associated w i t h this upper Hopf bifurcation, the region of stable cycles extends just beyond the Hopf bifurcation point creating a very small parameter range where both a stable equilibrium and stable cycles are present. This occurs near the upper Hopf bifurcation for the other two-parameter diagrams too but the regions are very small and have not been marked. Figure 4.25 shows which values of 72 give rise to stable l i m i t cycles when ^3 = 0.2. Using this information together w i t h results obtained from one-parameter diagrams at different fixed values of fa allows us to determine the regions i n figure 4.21 which give rise to stable cycles. Such regions also occur in figures 4.20, 4.22 and 4.23. A l t h o u g h figure 4.19 contains Hopf bifurcations, the cycles associated w i t h them are unstable (see, for example, figure 4.24) and occur over such a small parameter range that they are not of m u c h biological interest. Having classified the qualitative dynamics i n the various regions of the two-parameter  Chapter 4.  Ratio-Dependent  Model  101  1 0.8 Mi  0.6 0.4  •  -LP •  0.2 .  n  0  »  •  _  LP-iilB  »i »  0.5  1  1.5  72  Figure 4.25:  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m o b t a i n e d by v a r y i n g 72 w i t h 71 = 1.2 a n d  <f>  3  = 0.2.  ( T h i s corresponds to the h o r i z o n t a l d o t t e d line i n figure 4.21.)  diagrams i n figures 4.19-4.23 we can make a number of general observations. F i r s t , as 71 increases the l i m i t point continuation curves move closer together u n t i l i n figure 4.23 there are no l i m i t points at all. Hence, increasing 71 reduces the amount of hysteresis and the possibility of m u l t i p l e stable states. Also, the positions of the l i m i t point continuations in two-parameter space change as 71 is increased resulting i n relatively smaller regions of low M\ equilibrium values. Since higher 71 values correspond to faster plant growth rates this is not surprising. We have seen that stable cycles occur for higher values of 71 (figures 4.20-4.23). C o m p a r i n g these diagrams we can see that increasing 71 increases the area of the upper region of stable cycles thereby increasing the probability of finding cycles for a random choice of 72 and fa within the given two-parameter space. This occurs u n t i l 71 = 1.8. For larger values of 71 this upper region of cycles does not change position or shape (compare figures 4.14(b) and 4.23). There is also a second, smaller region of cycles which can be seen i n figures 4.21-4.23. The size of this region first increases and then decreases u n t i l , for sufficiently high 71 values, it ceases to exist.  Chapter 4. Ratio-Dependent  102  Model  We know that higher 71 values correspond to a faster plant growth rate and also that the herbivore population, and hence the predator population, depend on the availability of plants for survival.  F r o m the above observations we can deduce that when plant  resources are l i m i t e d (at low values of 71) the potential for having two stable equilibria is greater and the region corresponding to low M i equilibrium values is larger. W h e n plant resources are not as restricted (at higher values of 71) there is a greater chance of stable population cycles but less chance of metastability (multiple stable states). For 71 > 2 two-parameter diagrams of (7 ,</> )-space are very similar for all values of 71 (only the 2  3  lower region of cycles decreases i n area) suggesting that the system is no longer l i m i t e d by plant availability. Herbivore properties (as determined by 72 and ^ 3 ) have a greater influence on the behaviour of the system at these values of 71. In particular the properties of the herbivore determine whether stable cycling behaviour or a stable equilibrium occurs as well as the magnitude of the plant biomass at the equilibrium. Low values of fa (the availability or nutritional value of the herbivore to the predator) together with high values of 72 (the potential growth rate of herbivore biomass) are particularly detrimental to the plant while the reverse situation allows the plant to m a i n t a i n fairly high biomasses.  A t intermediate values of these parameters  population"cycles may occur provided the value of fa is sufficiently high. T h e preceding analysis has made use of various dynamical systems techniques to help us gain insight into the behaviour of the model. In the next section I discuss the results that can be obtained using a zero isocline analysis—the method used by Gutierrez et al. [52].  4.5.3  The role played by the isocline configurations  F r o m their isocline analysis of model (4.1) Gutierrez et al. [52] conclude that the parasitoid E. lopezi could control the cassava mealybug while E. diversicornis  could not.  Chapter 4.  Ratio-Dependent  Model  103  However, w i t h three state variables all having similar time scales, these deductions are not as straightforward as they may seem. F i r s t , the equilibrium isocline configuration in the MiM  2  (M M ) 2  3  phase plane depends  on the value of M 3 ( M i ) as well as on the parameter values. This was shown i n figure 4.6. T h u s , noting how an isocline changes as a parameter is varied does not give a complete picture. Secondly, it is not possible to tell from the qualitative structure of the isoclines which intersection point in the M\M  2  plane corresponds to a tritrophic equilibrium.  Figure 4.26 shows three possibilities. Two of these (namely, (b) and (c)) appear i n [52] but it was assumed that the equilibrium point i n (b) was unstable. E v e n if the exact position of the equilibrium point is known, it is not possible to tell from the qualitative structure of the isoclines whether this point is stable or unstable and whether or not l i m i t cycles occur. For example, although altering the parameter 73 has no effect on the isoclines, low values of 73 give rise to unstable fixed points and stable l i m i t cycles and high values to a stable equilibrium (see figure 4.2). T h u s , numerical computation is needed to determine the exact location as well as the local stability of an e q u i l i b r i u m point for the current model. T h e isocline configuration obviously has some effect on the behaviour of the system. Figure 4.27 shows the basic configurations for different points in figure 4.12(b).  The  qualitative structure of the isoclines changes as the diagonal lines (corresponding to a Hopf bifurcation and two l i m i t point continuations) are crossed.  In general, it is the  proximity of the tritrophic equilibrium point to the peaks of the M\ and M  2  the M\M  2  and M M 2  3  isoclines i n  planes respectively that is important for determining the robustness  of model behaviour with respect to parameter perturbations. If the e q u i l i b r i u m point is close to one of these peaks (as is the case near the diagonal lines in figure 4.27) then a small parameter perturbation may change the qualitative structure of the isoclines and hence the dynamics. However, to obtain this information the exact e q u i l i b r i u m isocline  Chapter 4. Ratio-Dependent  104  Model  (a)  M2  0  02  04  0  02  04  Ml  075  OS  1  06  08  1  0  02  04  0  02  OA  M2  06  08  1  06  08  1  (b)  Ml  (c)  M2  Figure 4.26:  M2  E x a m p l e s of isocline configurations s h o w i n g different possibilities for the p o s i t i o n o f the  tritrophic equilibrium.  Chapter 4.  Ratio-Dependent  Model  105  configuration for a given set of parameter values needs to be known.  0  1  2  3  4  7i Figure 4.27:  E x a m p l e s of isocline  configurations  at different points i n  (71 ,<?!>2)-space.  B y inference the above criticisms have all noted that if the exact positions of the isoclines and the tritrophic equilibrium were known i n both phase planes, then we could obtain a fair amount of information from them.  Using X P P A U T this is possible. In  particular we can study the effects of introducing nonzero values for the a,-'s. Figure 4.28 shows the results obtained using the reference parameter set for model (4.6) with a,- = 0, a; = 0.001 and a, = 0.005 (i = 1,2,3). C o m p a r i n g figures 4.28(a) and (b) we can see that introducing the a;'s prevents the M\ and M  2  isoclines from passing through the origin. Hence the equilibrium values for the  state variables do not approach zero as rapidly as for the original model and the modified model is more robust to parameter variations i n this region of low biomasses. Increasing the aj's from 0.001 to 0.005 reduces the humped shape of the M\ and M2 isoclines.  Chapter 4.  Ratio-Dependent  106  Model  (a)  M2  (b)  M3  1 0.8 M2  0.6 0.4 0.2 0,  02  OA  Ml  05  08"  (c)  0  1.4 1.2 1 0.8 M3 0.6 0.4 " 0.2 / / 0,0  M2  .  Figure 4.28:  02  04  M2  06  08  M  0T8  ,4 / /  /  /  /  /  / /  02  OA  M2  1  Isocline configurations for m o d e l (4.6) w i t h (a) aj = 0, (b) a< = 0.001 a n d (c) a - = 0.005  (z = 1 , 2 , 3 ) together w i t h the reference p a r a m e t e r set.  z  Chapter 4. Ratio-Dependent  Model  107  T h e result is an even more robust model.  The stabilising influence of increasing the  values of the aj's is illustrated by the bifurcation diagrams i n figure 4.29 corresponding to a;  =  0.005 (i = 1,2,3).  No regions of cycling behaviour are encountered.  Since  values of 0.005 are still small, this suggests that the model is structurally unstable and hence predictions from ratio-dependent models should be treated w i t h caution. Another consequence of introducing nonzero aj's is that we no longer get abrupt changes i n the qualitative shapes of the isoclines. 7i  Consider the plant isocline.  < 4>2 the plant isocline has the hump shape shown i n figure 4.30(a)(i).  For  If one or  both of these parameter values is altered so that the inequality is reversed, we get the asymptotic isocline in figure 4.30(a)(ii).  Using the modified model this abrupt change  does not occur. Instead there is a gradual change from the cubic curve i n figure 4.30(b) (i) to the asymptotic curve i n figure 4.30(b)(ii). The same applies to the herbivore isocline in the M M 2  4.6  3  plane.  Conclusion  In this chapter a partial analysis of a tritrophic ratio-dependent model has been done. Large differences i n the magnitudes of the state variables and uncertainty regarding the relative magnitudes of parameter values necessitated a scaling of the equations. This also reduced the number of parameters in the model. T h e bifurcation analysis revealed a number of cases of multiple stable states. M a n y of these phenomena occur over very small parameter ranges and are therefore not of m u c h biological interest. However, those occurring over larger ranges are important as they indicate the potential for sudden behavioural changes if the system is perturbed sufficiently.  M a n y of the instances of multiple stable states arise from a hard loss of  stability associated w i t h a Hopf bifurcation. This means that a stable equilibrium and  Chapter 4.  0  Ratio-Dependent  0.25  0.5  <f>2  Model  0.75  108  1  0  0.25  0.5  0.75  ^3  Figure 4 . 2 9 : O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s w h e n a; = 0.005 (i = 1 , 2 , 3 ) .  1  Chapter 4. Ratio-Dependent  109  Model  (a)  (ii)  (i)  2 1.6 1.2 M2  M2  0.8 0.4 02  04  06  0,  08  0  Ml  02  0 4 0 6 0 8 Ml  1  1.2  0 6 0 8 Ml  1  1.2  (b)  (ii)  2 1.6 1.2  M2  M2 0.8 0.4 0.  0  Figure 4.30:  02  OA  P l a n t isoclines for (a) the o r i g i n a l a n d (b) the m o d i f i e d m o d e l . T h e  i n (a)(i) a n d (b)(i) a n d the M 2 M 3 plane is s h o w n i n (a)(ii) a n d ( b ) ( i i ) .  M\M  2  plane is s h o w n  Chapter 4. Ratio-Dependent  Model  110  a stable l i m i t cycle can coexist for a given set of parameter values. If a parameter is varied (as conditions alter), there may be an abrupt change between l i m i t cycles of large amplitude and a stable equilibrium. A point worth noting is that if the Hopf bifurcation associated w i t h the l i m i t cycles had been studied analytically, the algebra required to identify the hard loss of stability would have been very time-consuming and possibly too difficult to do by hand. O n l y the i n i t i a l unstable cycles would have been located. Computers are invaluable in such circumstances. A l l the parameters were found to affect the dynamics of the model to some extent although the parameter ratios ^  and ^  nac  [  t  n e  m  o s t significant effect.  T h i s is in  general agreement w i t h Gutierrez et al. [52] (in terms of their model the ratios  and  ^j^ - were found to be important) although they did not explicitly describe these effects 2  as was done i n this study using bifurcation diagrams. In particular the two-parameter diagrams summarise the effects of the plant and the herbivore properties on the behaviour of the system. B o t h trophic levels affect the magnitude of M i at e q u i l i b r i u m , whether or not cycles occur, and the period and amplitude of these cycles. The consequences of a slowly growing plant were also shown, namely smaller regions of cycling behaviour, lower e q u i l i b r i u m M i values and larger regions of multiple stable states. It may seem that predator dynamics are not as important as those for the lower two trophic levels. However, it should be noted that increasing 73, the efficiency of the predator, can stabilise an oscillating system. T h e parameters affecting predator dynamics also affect the equilibrium biomasses. In the course of the analysis some previously noted criticisms of ratio-dependent m o d els were highlighted as they caused numerical difficulties and biological implausibilities. In particular, such models are not valid when a state variable which occurs in the denominator of a ratio approaches zero (that is, as a population approaches extinction) since a small perturbation to the model alters the dynamics in this region.  This was  Chapter 4. Ratio-Dependent  Model  111  shown using a modification to the ratio-dependent terms. W h i l e this modification had very little effect on the dynamics for parameter values corresponding to reasonable equil i b r i u m values of the state variables (provided the a,'s were sufficiently small), it did remove the numerical problems which occurred when one or more of the state variables approached zero (causing one or more ratios to tend to infinity).  It also revealed that  the ratio-dependence causes complex dynamics i n regions where the state variables are small.  E v e n the addition of very small terms (a,- = 0.001) reduced the complexity of  the cycles. T h e above results support the argument that ratio-dependent models exhibit pathological behaviour and that they are not valid near the axes. Thus they cannot be used to study extinction or situations where one of the state variables attains low values. However, the model by Gutierrez et al. (1994) that has been analysed in this chapter is a biological control model whose a i m is to suggest what k i n d of predator can keep herbivore numbers low. F i n a l l y , the zero isocline configurations in the M\M  2  and M2M3 planes were inves-  tigated and some limitations of the three-dimensional setting were discussed. In order to obtain useful information from the isoclines their exact equilibrium positions need to be determined so that the position and nature of the tritrophic e q u i l i b r i u m and its proximity to the isocline peaks can be found. Uncertainty regarding the parameter values for this model for a particular ecological system makes it difficult to interpret the results biologically except i n a very general sense. However, it is informative to see the k i n d of behaviour that the model can exhibit and the effects of ratio-dependence. It would be easy to apply the results and perform the same type of analysis if more accurate parameter values were obtained.  Chapter 5  Population Genetics Model I  5.1  Introduction  So far only continuous-time models have been considered. In the remaining chapters we w i l l look at discrete-time models. T h e model i n this chapter is a population genetics one i n which both population size and gene frequency are state variables. It is a single locus, two allele model w i t h density-dependent fitness functions and it has already been studied i n Asmussen [8] and Namkoong et al. [93]. Although the model equations are fairly simple, interesting dynamics arise as a result of the discreteness. N o new theoretical results are obtained i n this study but the dynamical systems techniques prove useful i n a number of ways: first, the theoretical results are demonstrated fairly easily and without having to struggle with the mathematical details. T h e dynamical systems techniques also provide a more systematic way of locating different kinds of behaviour when theoretical predictions are not possible or too difficult to obtain. Previous studies have tested various parameter combinations numerically using trial and error to try and locate the desired dynamics. In addition, while the theoretical results note the existence of various types of qualitative behaviour, they do not give information regarding the extents of the regions i n parameter space corresponding to these dynamics. In other words, they indicate the possibility of a certain type of behaviour occurring but not the relative frequency of occurrence.  T h e latter information is important from an ecological perspective as it  influences the amount of attention that is given to various possibilities. It w i l l be shown  112  Chapter 5.  Population  Genetics Model I  113  that bifurcation diagrams (two-parameter ones in particular) can be useful for indicating the sizes of regions i n parameter space corresponding to various types of behaviour. Another useful result is the location of stable polymorphic period-2 cycles.  It is  very difficult to predict the existence of these cycles intuitively or analytically as the bifurcation point at which they are initiated does not coincide with changes i n the relative carrying capacities of the genotypes.  Also, the period-2 cycle is initially unstable and  a further bifurcation is required before it becomes stable.  N u m e r i c a l techniques are  indispensable i n such situations. I begin i n the next section with a list of new terminology that is used i n this chapter as well as i n chapter 6. I then summarise some background information on population genetics models and the m a i n theoretical results relating to the particular model that is studied i n the rest of the chapter. Section 5.4 describes the model equations and is followed by the model analysis. The focus i n the latter section is on cycling or periodic behaviour as this behaviour is the most difficult to study by hand. T h e analysis begins w i t h one-parameter studies which investigate the effects of altering relative carrying capacities. A two-parameter bifurcation diagram is then obtained. T h i s diagram divides the two-parameter space into regions corresponding to different qualitative behaviour. T h e region of stable period-2 polymorphisms is larger than that for stable polymorphic equilibria for the particular case studied and intersects regions of heterozygote inferiority, superiority as well as regions of partial dominance. Finally, section 5.5.4 studies higher period cycles. P o l y m o r p h i c period-4 cycles are found but only for high genotypic growth rates. Of greater interest is the observation that the relative carrying capacities of the genotypes determine the location of attractors (boundary or interior) while the growth rates of the genotypes are responsible for the type of attractor (equilibrium or periodic cycle). Questions for further study are included i n the conclusion.  Chapter 5.  5.2  Population  Genetics Model I  114  New terminology  Some new terminology is required in this chapter as well as i n chapter 6 to explain both the discrete dynamics and the biological significance of the results. • p e r i o d - k cycle or o r b i t : For discrete models we do not get l i m i t cycles as in the continuous case. However, the values of the state variables may oscillate in a repetitive manner. If there are k points which are repeated then we refer to a period-k cycle or orbit. A period-1 point is the same as an e q u i l i b r i u m point (see section A . 2 . 6 ) . For further details on discrete models refer to section A . 3 . 5 . • p e r i o d - k sink: This is a period-k cycle which is locally stable (see sections A.2.14 and A.2.21). P e r i o d - k saddles and sources are period-k cycles which are locally unstable (see sections A.2.20 and A.2.23). • gene frequency: This is the number of gametes or individuals carrying a particular allele divided by the total number of gametes. • genotype: Suppose we have two alleles, A\ and A . T h e n there are three possible 2  genotypes: A\A\, A\A  2  and  AA. 2  2  • fitness: T h e fitness of a genotype is the contribution that it makes to the next generation's gene pool, that is, it is a measure of the successful survival and reproduction of that genotype [110]. • h o m o z y g o t e : The genotypes A-^Ai and A A 2  • h e t e r o z y g o t e : The genotype A\A  2  2  are homozygotes.  is a heterozygote.  • fixed or h o m o m o r p h i c e q u i l i b r i u m : This is an equilibrium point at which only one allele is present, that is, at which only a homozygote is present.  Chapter 5. Population  Genetics Model I  115  • p o l y m o r p h i c equilibrium: This is an equilibrium point where more than one allele is present, that is, where a heterozygote is present. • carrying capacity: This refers to the equilibrium population density corresponding to a particular genotype when only that genotype is present. It is denoted by K{j for the genotype AiAj i n this thesis. • heterozygote superiority: Heterozygote superiority or overdominance occurs when the heterozygote's carrying capacity is greater than those for the homozygotes, that is K\ > K\\,K for the case of two alleles. Heterozygote inferiority 2  22  or underdominance refers to the situation when the inequality is reversed. • partial dominance: This refers to the situation where the heterozygote is neither superior nor inferior, that is, K  n  < K  i2  < K  22  or K  22  < K12 < K\i for the case of  two alleles.  5.3  Background  In the past the theories of population dynamics and population genetics were considered to be separate pursuits since it was thought that evolution by natural selection proceeded on a much longer time scale than changes i n population size (Roughgarden [104]). However, once it was realised that gene substitution could occur i n the same length of time as that needed by a population to reach an equilibrium, the dangers of this separation were acknowledged. Since the late 1960's a number of models i n which both population size and gene frequency are variables have been studied. T h e classical one locus, two allele selection models i n this category used constant viabilities for the genes and predicted monotonic population convergence to a unique stable e q u i l i b r i u m (Asmussen [8]). However, incorporating density-dependent selection can have a dramatic effect on the  Chapter 5.  Population  Genetics Model I  116  d y n a m i c s — b o t h regular and chaotic cycles can arise. Studies of the effects of density-dependence for discrete generation organisms can be found i n [8, 9, 93, 104]. In particular, Asmussen and Feldman [9] and Asmussen [8] show that i n such situations local stability analyses may be inadequate to explain the global behaviour related to changes in gene frequency and population size. In addition to fixed and polymorphic stable equilibria, Asmussen [8] found regular and chaotic cycles when using monotone decreasing density-dependent fitness functions.  In certain situations  equilibria and cycles exist simultaneously. Asmussen [8] also shows that stable periodic polymorphisms may occur in the absence of heterozygote s u p e r i o r i t y — t h e latter condition being necessary for polymorphic equilibria when strictly decreasing fitness functions are used. In fact, using the same model Namkoong et al. [93] demonstrate the existence of an attracting polymorphic period-2 cycle for a case of heterozygote inferiority. T h u s , overdominance i n heterozygote carrying capacity is not necessary for the maintenance of genetic variation. Another important conclusion by Asmussen [8] is that an inherently stable genetic system can exert a stabilising influence on a model, allowing stable equil i b r i a and stable l i m i t cycles to persist for higher growth rates than would be possible w i t h the model's purely ecological counterpart. T h e above conclusions were arrived at analytically for linear (logistic) monotone decreasing fitness functions. However, when the density-dependence is modelled using exponential fitness functions, Asmussen [8] comments that the mathematics becomes very difficult. N u m e r i c a l solutions then become necessary. In [8, 93] it is also noted that when more complex, higher order behaviour (such as a cycle) is present, intuition and local stability analyses break down. A g a i n numerical techniques are required.  Chapter 5. Population  5.4  Genetics Model I  117  Model equations  Suppose we have a single population and two alleles, A  ±  size is denoted by N  t  genotype A{Aj  and A?. A t time t the population  and the frequency of allele A\ is denoted by p . T h e fitness of t  = 1,2) at time t is denoted by w\-. The marginal fitness of Ai is thus  ™\ = PMI + ( i - Pt)«4  (*' = 1> ) 2  and the mean population fitness is w  t  = p w\ + (1 -  p )w t  t  Differences in fitness among genotypes may be interpreted as the result of different responses to ecological pressures. Following Asmussen [8] and Namkoong et al. [93] exponential density-dependent fitness functions of the form , • = exp(a,j - bijN ) w•  a;j,6,j > 0, i,j = 1,2  t  (5.1)  are used i n this chapter. Such monotone decreasing functions of population density are often used to model the detrimental effects of population crowding [110]. If Hardy-Weinberg frequencies are assumed at each time t, then the recursion equations for p and N are: w\  Pt+i =  Pt —  N  w*N  w  l  =  t+1  t  (5.2)  where the region of practical significance is 0 < p < 1, N > 0. W i t h the above equations the carrying capacity  for genotype AiAj  acting alone is given by  K • =  ^  For a more detailed description of the model see Namkoong et al. [93].  Chapter 5.  Population  Genetics Model I  118  In studying this model I will be looking for attracting boundaries at p = 0 or p = 1, interior (polymorphic) equilibria having 0 < p < 1, period-2 and also higher period stable orbits. A t t r a c t i n g boundaries correspond to situations where one allele survives at the expense of the other, while at interior attractors both alleles persist thus m a i n t a i n ing genetic diversity.  Other phenomena of importance are the extents of the domains  of attraction corresponding to the stable phenomena (see section A.2.5) and the associated relative carrying capacities of the homozygotes and heterozygote. T h e methods of analysis and the results are discussed i n the next section.  5.5  Model analysis  5.5.1  Approach  E q u i l i b r i a and their associated stability properties have been studied analytically for models such as the one described above. These theoretical results help predict the conditions under which stable equilibria can be expected (see, for example, [8, 104]). However, periodic dynamics (particularly polymorphic cycles) are more difficult to study [8, 93]. I chose to concentrate on these more difficult phenomena to demonstrate the u t i l i t y of the available software.  I used D S T O O L to solve the system over time and to gener-  ate starting points for A U T O . The bifurcation diagrams were obtained using Interactive A U T O . T h e A U T O interface i n X P P A U T is not yet set up to deal w i t h discrete systems of equations. I used fairly small stepsizes when generating bifurcation diagrams for this model (ds=0.0001 and dsmax between 0.001 and 0.01) as the changes i n behaviour occur over small parameter ranges. Namkoong et al. [93] state that alleles that affect seedling survival can increase carrying capacity and simultaneously destabilise population growth dynamics. T h i s can be simulated by choosing a n > 2.0 so as to force the AiA\  genotypes to exhibit unstable  Chapter 5. Population  Genetics Model I  119  growth if they grow as a purely homozygous population. In order to investigate the effects of altering relative carrying capacities, I begin by setting a  n  = 2.1, a  1 2  = 1.9, a  2 2  = 1-1,  fen = 1.0, fei = 0.904 and varying fe . T h e results agree with those i n [93]. I then go on 2  22  to determine the effects of simultaneously varying the heterozygote parameter, fei , by 2  generating a two-parameter bifurcation diagram i n (fe , &i )-parameter space. I conclude 22  2  the analysis by investigating higher period orbits.  5.5.2  One-parameter bifurcation diagrams  Namkoong et al. [93] found that an interior period-2 attractor exists for & A n example is shown i n figure 5.1 for the value fe  22  5 .  22  > 0.526.  = 0.54. In this figure there is a  ,  .  ,  ,  N  0I 0  ,  p  I 1  Figure 5.1: D y n a m i c s i n the (p, 7V)-plane for m o d e l (5.2) w i t h a n = 2 . 1 , a i = 1.9, a 2  & n = 1.0, &12 = 0.904 a n d 6  2 2  2 2  = 1.1,  = 0.54. T h i s d i a g r a m was o b t a i n e d using D S T O O L .  period-1 source at (p,N) = (1.000,2.100) and period-1 saddles at (0.956,2.10007) and (0.000,2.037). T h e points (1.000,2.878) and (1.000,1.322) correspond to a period-2 saddle and the points (0.917,1.466) and (0.923,2.734) to a period-2 sink. Thus, i n this case both boundaries are repelling and there is an interior attracting period-2 orbit as expected.  Chapter 5. Population  Genetics Model I  120  C o m p a r i n g the carrying capacities K~u = 2.100, K\  = 2.102 and K 2 = 2.037 we see  2  2  that we are just w i t h i n the region of heterozygote superiority. We can now use A U T O to vary b . It is noted i n appendix B that for discrete systems 22  A U T O can detect period-doubling bifurcations but cannot continue the resulting period2 orbits, and hence cannot detect higher period orbits. This is clearly a disadvantage i n the present situation where we are specifically interested i n the period-2 orbits. A way of overcoming this problem is to study the second iterate of the model since period-2 orbits w i l l become equilibria i n this new model (see section A . 3 . 5 ) . In terms of the original model A U T O w i l l then be able to detect period-1 equilibria (since these are also period-2 equilibria), period-2 orbits and bifurcations to period-4 orbits. For a model as simple as the one under discussion, the second iterate is easy to determine. F r o m equations (5.2) we obtain  Pt+2  N  =  W  =w N t+1  t+2  -  Pt+i—TT x+L  t+1  Pt — w  w  1+l  ^nr  = w w*N t+1  (5.3)  t  where  u>i  +1  + {1  =  Pt+Mi  -  p ^exp(a  1  t  i  l  -  ~ Pt+i)w$i  bi^Nt)  + (1 -  p -~) t  exp(a  i2  -  b w Nt) t  i2  and w  t+1  =  p wl  +l  t+1  +  (l-p )w t+1  +l 2  Figure 5.2 shows the results obtained from varying 622 using the second iterate of the  Chapter 5.  Population  F i g u r e 5.2:  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m of m o d e l (5.3) w i t h a n = 2.1, ai2 = 1.9, a 2 =  121  Genetics Model I  2  11,  & i i = 1.0 a n d &12 = 0.904 o b t a i n e d u s i n g A U T O . ( B e h a v i o u r for s m a l l e r a n d larger values o f 622 t h a n i n d i c a t e d i n the figure can be found by e x t r a p o l a t i n g the curves and lines at the b o u n d a r i e s o f the T h e period-2 o r b i t s are i n d i c a t e d i n the  figure.  figure.  T h e p h e n o m e n a corresponding to b o u n d a r y values o f  p, n a m e l y p — 0 or p = 1, are l a b e l l e d . Branches m a r k e d w i t h a * correspond to i n t e r i o r values o f p , n a m e l y 0 < p < 1. H B stands for H o p f b i f u r c a t i o n .  T h i s is really a p e r i o d - d o u b l i n g b i f u r c a t i o n  but  A U T O m a r k s it w i t h the H B s y m b o l . )  m o d e l . I plotted N versus 622 instead of p versus 622 as the period-2 orbits can be seen 1  w i t h greater clarity this way. The p-values for the two points on these period-2 orbits are very similar and thus the continuation curves are difficult to distinguish. A l s o , a large number of bifurcations occur at p = 1. P l o t t i n g p versus 6 2 would result i n many 2  branches lying on top of one another. For 622 < 0.523 we can see that the boundary at p = 0 is attracting w i t h a single S o m e very c o m p l i c a t e d b i f u r c a t i o n d i a g r a m s can be generated w h e n s t u d y i n g this m o d e l , however not a l l c o n t i n u a t i o n branches are o f interest. T h o s e branches w h i c h have p < 0 , p > l o r i V < 0 are not p r a c t i c a l l y significant b u t it is g o o d practice to continue such branches w i t h i n the p a r a m e t e r range under s t u d y i n case a b i f u r c a t i o n occurs a n d they re-enter the ranges o f interest. 1  Chapter 5.  Population  Genetics Model I  e q u i l i b r i u m point. A t b  22  122  = 0.523 the genotype A A 2  2  loses superiority i n carrying ca-  pacity and we move into the region (b > 0.523) of heterozygote superiority. W h i l e the 22  difference i n carrying capacities is not too large we still have a unique e q u i l i b r i u m (an interior equilibrium this time) but as b  22  increases, causing the carrying capacity of  AA 2  2  to decline and the instability of A i A i to have a greater influence on the dynamics, this e q u i l i b r i u m bifurcates to become an attracting interior period-2 orbit. A s 622 increases further this period-2 polymorphism moves closer and closer to the p = 1 boundary where there is a repelling period-2 orbit as well as an unstable period-1 saddle. These results match those i n [93] and are in agreement with previous findings that heterozygote superiority i n equilibrium carrying capacity is a necessary condition for a stable polymorphic e q u i l i b r i u m when fitness is a decreasing function of population size. In addition, figure 5.2 shows clearly that heterozygote superiority is not a sufficient condition for a stable polymorphic equilibrium since no such phenomena occur for 622 > 0.526. Stable period-2 polymorphisms exist in this region. In [93] it was found that stable period-2 polymorphisms, unlike polymorphic equil i b r i a , can exist in the absence of overdominance. K\  2  Suppose we set b\ = 0.906, then 2  = 2.097 which means that we no longer have any regions of heterozygote superior-  ity. T h e one-parameter bifurcation diagram shown in figure 5.3 was obtained by using D S T O O L to calculate starting points and Interactive A U T O to vary b  22  once again.  In this case the dynamics are a little more complicated. We still have an equilibrium at p = 0 which is attracting for 622 < 0.525 (the region of heterozygote inferiority) and unstable otherwise. However, at p — 1 there is now a stable period-2 orbit for all values of b . For 0.520 < b 22  22  < 0.522 (2.116 > #22 > 2.109) there is also a stable interior period-2  orbit. Intuition could not have been used to guess the existence of these orbits since the bifurcation values do not coincide w i t h changes in the relative carrying capacities of the genotypes. It would also have been difficult to predict their existence analytically since  Chapter 5.  Population  Genetics Model I  123  P=l  2.9  2.5  -  1. l i m i t p o i n t 2. p e r i o d - d o u b l i n g P=0  N  2.1  p=l  -  -[  P=0  3. p e r i o d - d o u b l i n g 4. bifurcation  1.7 p=l  1.3 0.51  0.5  0.52  0.53  0.54  0.55  0.56  622 F i g u r e 5.3:  O n e - p a r a m e t e r bifurcation  d i a g r a m of m o d e l (5.3) w i t h a n = 2.1, a i  611 = 1.0 and 612 = 0.906 o b t a i n e d using A U T O . Ku  2  = 1.9, 022 =  1.1,  = 2.1 and K12 = 2.097. B r a n c h e s m a r k e d w i t h a  * correspond to interior values of p, n a m e l y 0 < p < 1.  the period-doubling bifurcation which initiates the interior period-2 orbit occurs when 622 — 0.519 but the orbit is initially unstable and only becomes stable after a further bifurcation. A n idea of the extent of the domain of attraction (see section A.2.5) of the period-2 p o l y m o r p h i s m in terms of population density can also be seen in figure 5.3 by noting the positions of the unstable period-2 orbit and the unstable equilibria since these phenomena separate the domains of attraction of the stable phenomena. Figure 5.4 shows the (p, AQ-plane corresponding to 6 2 = 0.521 and indicates the domains of attraction for 2  this particular value of 6 2- The region denoted by a is the domain of attraction for the 2  sink at p = 0, b is the domain of attraction for the period-2 p o l y m o r p h i s m , and c is the domain of attraction for the period-2 orbit at p = 1. The above-mentioned figures indicate the ranges of 622 values corresponding to stable  point  Chapter 5.  Population  Genetics Model I  124  5  N  0  F i g u r e 5.4:  T h e (p, 7V)-plane using m o d e l (5.3) for a  n  =  2.1, a  i2  =  1.9, a 2 2  — 1-1, 6 n  =  1.0,  &12 = 0.906 a n d 622 = 0.52 showing the d o m a i n s of a t t r a c t i o n for the stable p h e n o m e n a .  equilibria and period-2 orbits at the boundaries as well as in the interior.  Comparing  figures 5.2 and 5.3 we can see that the regions of occurrence of these phenomena vary w i t h 612 as well as 6 2- We can find out more about this dependence on 6 2  12  by generating  a two-parameter bifurcation diagram.  5.5.3  T w o - p a r a m e t e r bifurcation diagram  Using A U T O we can trace the paths of period-doubling bifurcations and l i m i t points in 2  two-parameter space. Points 2 i n figure 5.3 are period-doubling bifurcations and points 1 are l i m i t points. These points are only located when using the second iterate of the model since they are on the period-2 orbits. Point 3 is also a period-doubling bifurcation but from period-1 to period-2 orbits and thus is only labelled as such by A U T O when using the original model (5.2).  Hence, to continue this point in two parameters we need to  use the original model. The resulting two-parameter diagram can then be superimposed A U T O labels p e r i o d - d o u b l i n g bifurcations as H o p f bifurcations for discrete m o d e l s (see A . 3 . 5 ) . T h u s p o i n t s m a r k e d H B i n the figures are really p e r i o d - d o u b l i n g bifurcations. 2  section  Chapter 5. Population  Genetics Model I  125  0.909 0.908 ,  0.907  012  0.906 0.905 0.904 0.5  0.51  0.52  0.53  0.54  0.55  »22 F i g u r e 5.5: T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m o f m o d e l (5.3) w i t h a  u  = 2.1, a i 2 — 1.9, (Z22 = 1 . 1 a n d  611 = 1.0 o b t a i n e d using A U T O . C u r v e 1 is the l i m i t p o i n t c o n t i n u a t i o n a n d curve 2 the p e r i o d - d o u b l i n g bifurcation  continuation.  on the one obtained for the second iterate of the model by using the R E A D P c o m m a n d in Interactive A U T O . W e could choose any one of the parameters a i i , a i 2 , a 2 , 611 or 612 2  as the second parameter to vary.  Since the parameter 6  22  corresponds to one of the  homozygotes, it m a y be interesting to vary one of the heterozygote parameters.  The  parameter 612 was chosen for this illustration to compliment the results i n the previous section. Figure 5.5 shows the results obtained from continuing points 1 and 2 i n figure 5.3 i n two-parameter space. T h e two-parameter continuation of point 3 i n figure 5.3 can be obtained f r o m the original model as mentioned earlier. A U T O cannot continue transcritical bifurcation points such as point 4 i n figure 5.3. However, we can obtain a good approxi m a t i o n to the relevant line or curve by doing a number of one-parameter continuations as i n section 5.5.2 and noting the coordinates at which the bifurcation at p = 0 occurs. W e already have the points (622,012) = (0.523,0.904) and (622,612) = (0.525,0.906) from figures 5.2 and 5.3 respectively.  If the resulting approximate curve is combined w i t h  Chapter 5.  Population  Genetics Model I  126  0.909 0.908 0.907 A  C  H  F  0.906 0.905 0.904 0.5  0.51  0.52  0.53  0.54  0.55  '22 F i g u r e 5.6:  T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m of m o d e l (5.3) w i t h a n = 2.1, a\  2  = 1.9, 022 = 1.1 a n d  611 = 1.0 o b t a i n e d using A U T O . C u r v e 1 is the l i m i t p o i n t c o n t i n u a t i o n a n d curve 2 the bifurcation continuation.  period-doubling  C u r v e 3 is the p e r i o d - d o u b l i n g c o n t i n u a t i o n o b t a i n e d using the o r i g i n a l m o d e l  (5.2) a n d curve 4 is the curve where the bifurcation at p — 0 occurs.  figure 5.5 and the two-parameter continuation of point 3, then we obtain figure 5.6. There are many other bifurcation curves that could have been included i n figure 5.6 but only those of interest for the present discussion have been drawn. T h e dotted line has been included to demarcate, together w i t h line 4, the regions of heterozygote superiority and inferiority . In regions F and G there is heterozygote superiority and i n regions A , 3  B and D there is heterozygote inferiority. C, E and H are regions of partial dominance. These distinctions i n relative carrying capacities w i l l be referred to again shortly. To help us understand figure 5.6 we can refer back to the one-parameter bifurcation diagrams, figures 5.2 and 5.3, which are horizontal slices of figure 5.6 at 612 = 0.904 and bu — 0.906 respectively. Figure 5.6 indicates the types of phenomena corresponding to different parameter combinations and the one-parameter diagrams show the values of T h e s e lines can be o b t a i n e d a n a l y t i c a l l y using the definition of heterozygote superiority. T h e d o t t e d line c o u l d also have been o b t a i n e d f r o m one-parameter b i f u r c a t i o n d i a g r a m s where 612 is varied i n s t e a d of 622 as it is along this line t h a t the period-2 orbit at p = 1 changes its s t a b i l i t y properties. 3  Chapter 5.  Population  Genetics Model I  1  2.9  127  i  1  i p=i -  ——J^?i *,p=i  p=i NT :  HB  .•*  p=i  0.5  i  i  i  i  i  0.51  0.52  0.53  0.54  0.55  0.56  622 F i g u r e 5.7:  O n e - p a r a m e t e r bifurcation d i a g r a m of m o d e l (5.3) w i t h a n = 2.1, 012 = 1.9, a 2 = 2  1.1,  6 n = 1.0 a n d 612 = 0.908 o b t a i n e d using A U T O . Branches m a r k e d w i t h a * correspond to interior values of p, n a m e l y 0 < p < 1. H B m a r k s a p e r i o d - d o u b l i n g  bifurcation.  ./V at which these phenomena occur and, hence, their relative positions in state space. We can also use figure 5.6 to predict the behaviour corresponding to different parameter combinations. Consider fixing 6  12  at 0.908. F r o m figure 5.6 we expect a period-doubling  bifurcation to occur as 622 increases through line 3 but we do not expect any region of stable interior period-2 cycles as we are above the region where curve 1, the l i m i t point of such a phenomenon, occurs. Also from figure 5.6, we expect the e q u i l i b r i u m at p = 0 to change stability as we pass through line 4. Using A U T O to generate a one-parameter diagram w i t h 6  12  fixed at 0.908 gives figure 5.7 which is just as we predicted. T h e above  exercise can be repeated for other values of 6  12  as well.  A complementary way of obtaining insight into figure 5.6 is to choose points i n regions A to H and to display the dynamics i n the (p, AQ-plane. This is a straightforward exercise  Chapter 5.  Population  Genetics Model I  128  using D S T O O L . The results are shown i n figure 5.8. Strictly speaking dots should be used for the trajectories instead of continuous lines since the model is discrete. However, it is easier to denote the direction of flow and the qualitative behaviour when lines are used. These diagrams also give a better idea of the domains of attraction corresponding to the different stable phenomena. Using figures 5.6 and 5.8 we can make some important observations. In regions D, E, F and G stable polymorphisms occur. D lies in the area of heterozygote inferiority and E in a region of partial dominance, that is, of neither heterozygote inferiority nor superiority. W h i l e Asmussen [8] and Namkoong et al. [93] documented the occurrence of stable period-2 polymorphisms i n such regions, they did not investigate the extent of the regions corresponding to such phenomena. F r o m figure 5.6 we can see that regions D and E occupy a fairly small region i n the (622, 612) parameter space and thus may have l i m i t e d ecological significance. Another observation is that the region of stable polymorphic equilibria (region G) is very small. Thus, i n the region of heterozygote superiority, stable period-2 polymorphisms are much more likely than stable polymorphic equilibria. It appears that the instability of the A\A\ genotype (a result of choosing an > 2.0) has a significant effect on the dynamics of the model. In the next section it is shown how the preceding analysis can be extended to look for higher period polymorphisms. Asmussen [8] found cases of interior chaotic attractors. Since repeated period-doubling is a well-known path to chaos (see, for example, Seydel (1988) or Wiggins (1990)), we can expect to find higher period stable polymorphisms.  5.5.4  Orbits of p e r i o d four (and higher)  Consider first orbits of period 4. Since I did not find such orbits by varying 6  12  and 6 , 22  I chose b\2 and 622 to lie i n region E, a region where more complex behaviour already  Chapter 5.  Population  Genetics Model I  129  (A)  (B)  (C)  (D)  N  2.11*  (E)  (F)  (G)  (H)  13":  Figure 5.8: D y n a m i c s in the (p, 7V)-p]ane w i t h a n = 2.1, a\2 = 1.9, a 2 = 1 1 , & u = 1 0 and various 2  c o m b i n a t i o n s o f 612 and 622 w h i c h correspond to regions A to H in figure 5.6.  Chapter 5.  Population  Genetics Model I  130  4 -  1. l i m i t p o i n t 2. p e r i o d - d o u b l i n g 3. p e r i o d - d o u b l i n g  N  4. b i f u r c a t i o n p o i n t 5. p e r i o d - d o u b l i n g  2.7  F i g u r e 5.9:  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m of m o d e l (5.3) w i t h ai  2  = 1.9, a  22  = 1.1, bu  =  1.0,  &12 = 0.905 a n d 622 = 0.525 o b t a i n e d u s i n g A U T O . Branches m a r k e d w i t h a * correspond to i n t e r i o r values of p, n a m e l y 0 < p < 1.  occurs i n the form of period-2 cycles, and then varied the other parameter values i n turn. Consider &i = 0.905 and 6 2  22  = 0.525 and suppose a  n  is varied. This yields a period-  doubling bifurcation at a n = 2.526 (see figure 5.9). Since I used the second iterate of the m a p , this period-doubling is a bifurcation from period-2 to period-4 orbits at the boundary p — 1. This can be checked by using D S T O O L to generate diagrams in the (p, 7V)-plane for nearby values of a n . F r o m studies of other discrete systems (for example, [71]) it is likely that there w i l l be period-doublings to higher and higher order orbits as a n is increased. Unfortunately these cannot be detected using A U T O unless higher order iterates of the map (5.2) are determined analytically and then studied. However, using D S T O O L we can determine the dynamics i n the (p, AQ-plane for fixed values of a n .  Some examples are given i n  Chapter 5. Population  Genetics Model I  131  figure 5.10. Two-parameter continuations of the period-doubling bifurcation i n figure 5.9 show that the position of this bifurcation depends only on o n . ' None of the other parameters affects the value of a n at which it occurs. Thus we can conclude that it is the growth rate of the A\Ai genotype that determines the degree of instability of its dynamics. T h i s supports the comment i n [93] that alleles that affect seedling survival can increase carrying capacity and simultaneously destabilise population growth dynamics.  Interior p e r i o d - 4 orbits B u t what about the dynamics of the heterozygote? It seems likely that complex polymorphic behaviour would exist i n regions where both homozygotes exhibit unstable dynamics. In order to investigate this question I chose both a  n  the parameter values a  n  = 2.1, a  12  and a  22  to be greater than 2.0. Using  = 1.9, a 2 = 2.1, bu = 1-0, 612 = 0.908 and b 2  22  = 0.53,  I found a set of starting points using D S T O O L and then varied each parameter i n t u r n using A U T O . T h e only interior period-doubling bifurcation was found by increasing a i . Figure 5.11 2  shows the results. There are period-2 orbits at both p = 0 and p = 1 for a l l values of a i . 2  T h i s is expected since both a n and 022 are greater than 2.0. T h e period-2 orbit at p = 1 is attracting for a i < 1.907 (the region of heterozygote inferiority) and the orbit at p = 0 2  is attracting for a i < 3.598. A t this latter point the heterozygote becomes dominant, 2  a bifurcation occurs and a stable interior period-2 orbit is initiated. A s a  i2  further, a period-doubling bifurcation occurs at ai2 = 3.750.  is increased  A l t h o u g h the period-4  orbits are not shown by A U T O , we can verify that such orbits exist using D S T O O L . Figure 5.12 shows the dynamics i n the (p, iV)-plane for 012 = 3.8. F r o m the above results it appears that the relative carrying capacities of the genotypes determine whether the boundaries and/or interior are attracting but that the growth  Figure 5.10: D y n a m i c s i n the (p, 7V)-plane for a  i 2  = 1.9, a  2 2  = 1.1, & u = 1.0, 6  1 2  = 0.905, 6  2 2  = 0.525  a n d (a) a n = 2.68, (b) a n = 2.69 a n d (c) a n = 2.75. In (a) there is a period-8 a t t r a c t o r at p = 1. In (b) t h i s changes t o a period-16 attractor a n d i n (c) we have w h a t appears to be a c h a o t i c a t t r a c t o r .  Chapter 5. Population  Genetics Model I  133  1  1  *. • " '  -  HB  /  -  -  ••'  *  v  -  HB  -.  -  1  1.5  2.5  3.5  4.5  622  F i g u r e 5.11: 6  1 2  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m o f m o d e l (5.3) w i t h a n = 2.1, 022 = 2.1, bu =  1.0,  = 0.908 a n d 622 = 0.53 o b t a i n e d using A U T O . Branches m a r k e d w i t h a * correspond to i n t e r i o r  values o f p , n a m e l y 0 < p < 1. H B m a r k s a p e r i o d - d o u b l i n g b i f u r c a t i o n f r o m period-2 to p e r i o d - 4 orbits.  Figure 5.12: and 6  2 2  D y n a m i c s i n the (p, A^)-plane for a n = 2.1, a  = 0.53 s h o w i n g a stable period-4 p o l y m o r p h i s m .  i 2  = 3.8, a  2 2  = 2.1, & n = 1.0, 6 i = 0.908 2  Chapter 5. Population Genetics Model I  Figure 5.13: a n = 2.1, a  2 2  134  T w o - p a r a m e t e r c o n t i n u a t i o n of the p e r i o d - d o u b l i n g b i f u r c a t i o n ( H B ) i n figure 5.11 w i t h = 2.1, b  n  = 1.0 a n d 6  i 2  = 0.908.  rates, a ; j , determine the type of attractor, that is, whether the attractors are equilibria or periodic cycles of various orders. Since the only way to alter the carrying capacity of AiAj without affecting its growth rate is to vary bij, we can conclude that the bifs greatest influence is on stability whereas the a^'s determine the type or order of the behaviour. A l t h o u g h we have located a stable polymorphic period-4 orbit, the values of a  12  for  which it occurs are very high. We would like to know whether such a phenomenon is possible for lower values of a  1 2  but different values of some of the other parameters. It is  only i n the region of heterozygote superiority that polymorphic attractors exist i n figure 5.11 and since K  22  by decreasing A^  22  = 3.962 is large, this region is only entered when a  12  is large. Hence,  we may be able to find stable interior period-4 orbits for lower values of  a . Figure 5.13 shows a two-parameter continuation of the period-doubling bifurcation 1 2  labelled 5 i n figure 5.11. T h e parameter b  22  is varied i n addition to a\ - F r o m figure 5.13 2  it can be seen that the period-doubling bifurcation point is reduced to a\ = 2.831 if b 2  22  Chapter 5. Population Genetics Model I  135  is increased to 0.945. This is substantially lower than before. If we wanted to use A U T O to study orbits of period greater than 4, we would have to calculate higher order iterates of the map (5.2). This is rather tedious and is perhaps more of mathematical than of ecological interest.  However, using D S T O O L , we can  increase the a j values and observe the results for particular parameter combinations. 4  Some examples of more complex dynamics are shown in figure 5.14.  (a)  (b)  Figure 5.14:  E x a m p l e s of c o m p l e x d y n a m i c s , (a) A n interior period-8 o r b i t for a n = 2.3, a i  « 2 2 = 2.5, fen = a  i 2  = 3.1, a  2 2  1.0,  fci  2  = 0.908 and fc 2 = 0.95. 2  = 2.5, & n = 1.0, & i = 0.908 and 6 2  2 2  =  (b) A n interior chaotic a t t r a c t o r for a n 0.95.  2  = 2.9, =  2.6,  Chapter 5. Population Genetics Model I  5.6  136  Conclusion  In this chapter a partial analysis of a population genetics model w i t h monotone densitydependent fitness functions has been done. A l t h o u g h most of the theoretical results had been obtained by Asmussen [8] and Namkoong et al. [93], this study gave rise to a twoparameter bifurcation diagram which indicates the relative frequency of occurrence of the various types of dynamical behaviour in addition to proving their existence. U s i n g A U T O and D S T O O L , stable polymorphic period-2 and higher period orbits were located. Asmussen [8] found these phenomena particularly difficult to study by hand when using exponential fitness functions but they can be found without too much difficulty using the available software. In addition to locating equilibria and higher period orbits I concluded that the parameters 6,j (i,j = l,2) have the greatest influence on the stability of these phenomena whereas the a^'s determine the type or order of the behaviour. A number of opportunities for further research arise naturally from the results of this chapter. F i r s t , could the numerical results be used as a basis for arriving at analytical relationships between the parameters or at biological conditions that would determine the existence of polymorphisms? For the model that I have considered this would be particularly difficult to decide because of the exponential fitnesses, but a start could be made using linear fitness functions. It would also be informative to know the relationship between the behaviour of periodic attractors for the homozygotes and for the full genetic system. Specifically, if both homozygotes have period-2 dynamics when acting alone, is it only possible to obtain period-2 polymorphic behaviour or are polymorphic equilibria and higher period polymorphic attractors also possible? If one homozygote exhibits period-x dynamics and the other period-y dynamics when acting alone ( x < y ) , does the full system exhibit dynamics of period-x, period-y, or some combination of x and y?  Chapter 5. Population Genetics Model I  137  In the next chapter the same basic model is analysed but this t i m e non-monotone fitness functions are considered. T h e model has not been studied i n detail before.  Chapter 6  Population Genetics Model II  6.1  Introduction  In the last chapter a single locus, two allele population genetics model with monotone density-dependent fitness functions was analysed. Most studies assume such monotonically decreasing fitness functions because of the detrimental effects of population density on growth [110]. However, this assumption is not biologically realistic for a l l population densities [110]. A t low densities, increases in density m a y benefit both reproduction and survival. In this chapter I consider the same basic model as i n chapter 5 but w i t h nonmonotone density-dependent fitness functions which have a single h u m p . T h i s model is more difficult to study than that i n the previous chapter because of the increased complexity i n the fitness functions. Instead of a single carrying capacity each fitness function has two fixed points corresponding to the values of N where Wij  = 1. T h e terminology  described i n section 5.2 is used again in this chapter. A s i n chapter 5 the study focusses mainly on stable polymorphic  behaviour—both  equilibria and higher period orbits. Such phenomena correspond to the maintenance of genetic diversity i n a population. F r o m an ecological perspective we would like to know how c o m m o n or rare these phenomena are, that is, how likely they are to occur.  In  contrast to chapter 5 (see section 5.5.3) it is found that a large number of parameter combinations give rise to stable polymorphic equilibria but that stable periodic polymorphisms are not as common. T h e latter phenomena are only found to occur i n situations  138  Chapter 6.  Population  Genetics Model II  139  where the dynamics corresponding to one of the homozygotes are fairly unstable and where the fitnesses of the two homozygotes acting alone are very different. Other formulations for hump-shaped fitness functions may, however, lead to different results. A n interesting result is that there is always a possibility of one allele being excluded, even when a polymorphic attractor is present. Diagrams in the (p, A )-plane indicate the r  domains of attraction for the homomorphic and polymorphic attractors. T h e possibility of extinction is also high for most parameter sets. C r i t e r i a for determining the existence and stability of polymorphic equilibria are also given.  These combine theoretical and  numerical results. In the next section I outline some previous results which apply to the model. I then describe the functional forms of the non-monotone fitness functions. T h e model analysis begins by fixing the fitness function corresponding to one of the homozygotes and varying the two parameters corresponding to the other.  A two-parameter bifurcation diagram  shows which parameter combinations give rise to homomorphic attractors and which to polymorphic attractors.  Section 6.4.3 finds criteria for predicting the existence and  stability properties of polymorphic equilibria, and stable period-2 polymorphisms are the focus of section 6.4.4. A small modification to the fitness functions is required before the latter phenomena are located. The results are summarised i n a two-parameter bifurcation diagram. Further two-parameter bifurcation diagrams show that period-2 polymorphisms do not correspond to fitness functions where both homozygotes have similar properties but one of the homozygotes has a greater fitness than the other at every population density.  Chapter 6. Population Genetics Model II  6.2  140  Background  One of the few studies which considers the more general setting of non-monotone densitydependent fitness functions is by Selgrade and Namkoong [110]. T h e y prove a number of results concerning the existence and stability of polymorphic equilibria for both continuous and discrete two-dimensional models. In particular, a necessary condition for the existence of a polymorphism is that the heterozygote fitness be either superior or inferior to both homozygote fitnesses at the equilibrium. For the case of heterozygote inferiority the equilibrium is unstable. For the discrete model (which is the one considered i n this chapter), if the heterozygote is superior and - 2 < JVff < 0 (JV=population density, u)=mean fitness) at the polymorphic equilibrium then this e q u i l i b r i u m is stable. Higher order attractors are not studied i n [110] and it is noted that arguments concerning global dynamics are more difficult for these non-monotone fitness functions because of the complicated nature of the mean fitness curve.  6.3  Fitness functions  T h e fitness functions used i n this chapter are given by  j = 1,2  u>ij = kijNtexpln^l-kijNt)] and wlj = kijNtexpln^l - kijN )N ] t  t  i,j = 1,2.  (6.1) (6.2)  Expressions of the form (6.1) can be found i n [109] and also i n [110] for the analogous continuous model. These functions have a single hump (see figure 6.1) and are known as climax fitness functions i n some applications [109]. A p a r t f r o m the results for polymorphic equilibria obtained by Selgrade and N a m k o o n g [110], not much work has been done on discrete models such as (5.2) i n which both genotypes have climax fitnesses. In order to simplify the analysis I look at an additive model  Chapter 6. Population Genetics Model II  141  i n which the fitness parameters for the heterozygote are averages of the corresponding parameters for the homozygotes. T h a t is, hi +  k•12  and  r  k  22  2  ru + r  i 2  22  2 T h i s reduces the number of parameters for consideration.  6.4  M o d e l analysis  6.4.1  Approach  I begin by studying the existence and relative frequency of occurrence of stable equilibria for the additive model w i t h fitness functions given by (6.1). I then look for stable periodic polymorphisms using fitness functions given first by equations (6.1) and then by equations (6.2). A s i n the previous chapter bifurcation diagrams and plots of the (p, AQ-plane are the m a i n tools for communicating results. In some situations the fitness functions are plotted so that the relative fitnesses for various population densities can be seen. T h i s is analogous to calculating relative carrying capacities i n the previous chapter. In order to begin we need to choose a starting set of parameter values. There are four parameters in this m o d e l — A ; n , r n , & responding homozygote parameters). begin, I used M A P L E  1  22  and r  (h  22  2  and r i  2  are averages of the cor-  Since there were no prior results w i t h which to  [122] to plot the three fitness functions and chose parameter val-  ues which led to plausible-looking curves. Since there are four parameters and A U T O can vary at most two at a time, I fixed r W22)  22  and fc  22  (that is, I fixed the fitness function  and investigated the (hi, rn)-parameter space. Therefore, the results w i l l indicate  A n y other m a t h e m a t i c a l graphics package, such as M A T H E M A T I C A [126] or G N U P L O T c o u l d have been used. J  [125],  Chapter 6.  Population  Genetics Model II  142  1.2-  1— 0.80.60.40.20-  o  \  IN I  Figure 6.1: w 2 with r 2  22  I  To  = 0.8 and & = 0.6. 22  the effects of varying the relative positions and magnitudes of the fitness functions. should be kept in m i n d that altering ru and ku w i l l affect both wu and  6.4.2  It  w\ . 2  ( & n , r ) - p a r a m e t e r space n  After using A U T O to vary the parameters one by one to locate a stable polymorphism, I chose r  22  = 0.8 and k  22  = 0.6 as starting values, which gave the w  22  fitness function  shown i n figure 6.1. The a i m is to divide the (ku, r ) - p a r a m e t e r space into regions corren  sponding to stable fixed points at p = 0,p = 1 (homomorphic equilibria) or 0 < p < 1 (polymorphic equilibria). T h e first step is to generate a one-parameter bifurcation diagram. Choosing values for r n and k\i, I used D S T O O L to find fixed points as starting points for A U T O . Figure 6.2 was obtained by varying ku w i t h r  n  = 0.7. O n l y branches corresponding to 0 < p < 1  have been drawn and no distinction between sources and saddles has been made so as to keep the diagram as clear as possible. (Interactive A U T O uses magenta to represent sources and blue to represent saddles.)  Chapter 6. Population Genetics Model II  0 i 0  1  143  1  0.4  1  0.8  1.2  1  1.6  1  2  hi  Figure 6.2: A one-parameter bifurcation d i a g r a m o b t a i n e d by v a r y i n g k\\ w i t h r n fixed at 0.7 (r22 = 0.8 a n d fc 2 = 0.6). Branches m a r k e d w i t h a * correspond to interior values o f p , n a m e l y 2  0 < p < 1. L P m a r k s the l i m i t points a n d B P the t r a n s c r i t i c a l bifurcation p o i n t s .  There are many bifurcations i n figure 6.2. Most are transcritical bifurcations (see section A.2.25) but there are also two l i m i t points (see section A.2.13).  Diagrams of  the (p, AQ-plane for a number of different values of hi are shown i n figure 6.3 to help clarify the changes i n dynamical behaviour that occur as hi increases through these bifurcation points. Note that for this value of r n there are two ranges of fen-values where stable polymorphic equilibria occur. Another important observation from an ecological perspective is that the possibility of extinction is fairly high for all the situations shown in figure 6.3. A U T O can only continue l i m i t points and period-doubling bifurcations i n two parameters. A s can be seen from figure 6.2 we w i l l also need to know how the positions of the transcritical bifurcations vary w i t h r  n  if we want to delimit regions of stable behaviour  Chapter 6. Population Genetics Model II  O  1  P  0  P  144  1  0  P  1  Figure 6.3: D i a g r a m s of the (p, 7V)-plane (obtained using D S T O O L ) for a n u m b e r of different values of ku  w i t h r n fixed at  0.7.  in two-parameter space. In order to do this I chose a number of different r - v a l u e s (such n  as 0 . 3 , 0 . 5 , 0 . 7 , . . . ) . For each value I obtained starting points (that is, e q u i l i b r i u m points) for A U T O using D S T O O L . I then used A U T O to generate a one-parameter bifurcation 2  diagram by varying ku in both directions, and recorded the A; -values corresponding to xl  the various bifurcations—both transcritical and l i m i t point. Using the graphics package G N U P L O T [125] to plot the recorded points, I obtained figure 6.4(a). Figures 6.4(b), (c) I t is i m p o r t a n t t h a t a n u m b e r of different fixed points are located for each p a r a m e t e r set to ensure t h a t unconnected c o n t i n u a t i o n branches are not missed. 2  Chapter 6. Population Genetics Model II  145  and (d) show the same diagram with shaded regions corresponding to stable equilibria at 0 < p < l , p = 0 and p = 1 respectively. Although figure 6.4 only gives approximations to the various bifurcation curves, they are sufficient for a qualitative analysis. I am more interested in the different types of qualitative behaviour that can occur than the actual parameter values at which transitions take place.  Figure 6.4:  D i a g r a m s of the  {ku,  r n ) - p a r a m e t e r space, (a) T h e basic d i a g r a m s h o w i n g a n u m b e r of  bifurcation curves, (b) T h e regions corresponding to stable p o l y m o r p h i s m s are shaded, (c) T h e regions c o r r e s p o n d i n g to stable e q u i l i b r i a at p = 0, N > 0 are shaded, (d) T h e regions c o r r e s p o n d i n g to stable e q u i l i b r i a at p = l,N  > 0 are shaded.  Chapter 6. Population Genetics Model II  146  In order to clarify figure 6.4, diagrams of the (p, A )-plane corresponding to the rer  gions marked A to P are shown in figures 6.5 and 6.6. representations of diagrams obtained using D S T O O L .  These figures are schematic,  T h e dashed lines in these dia-  grams approximate the one-dimensional manifolds of the saddle points (see page 241) and indicate the boundaries of the domains of attraction of the sinks. In addition to revealing the extents of the regions corresponding to homomorphic and polymorphic equilibria, figure 6.4 shows where we can expect simultaneous homomorphic and polymorphic attractors. In fact, there are no regions where the only attractor is i n the interior. Thus, there is always the possibility that one of the alleles w i l l be excluded. T h e relative sizes of the domains of attraction of the homomorphic and polymorphic equilibria are shown in figures 6.5 and 6.6. These diagrams also highlight the ever-present possibility of extinction.  6.4.3  C r i t e r i a for polymorphisms  It would be helpful to be able to predict when a stable polymorphic e q u i l i b r i u m is likely to occur.  One possibility is to investigate the fitness functions corresponding to the  different regions i n figure 6.4(a). Computer packages such as G N U P L O T [125], M A P L E [122] and M A T H E M A T I C A [126] are convenient for such investigations as the fitness functions K J , W\2 and W22 can be plotted on the same pair of axes. F i x e d points at p = 0 U  and p = 1 occur at those values of N where w  22  = 1 and u>  n  = 1 respectively.  The  stability of these points depends on the relative fitnesses of the homozygotes and the heterozygote. In each of the regions corresponding to a stable polymorphism it was found that the heterozygote is superior at the point where the downward slope of the W12 fitness function crosses the line w\2 = 1 (see figure 6.7(a)).  Similarly, interior saddle points  correspond to heterozygote inferiority at this point (see figure 6.7(b)).  However, not  Chapter 6. Population Genetics Model II  0  P  1  0  P  1  Figure 6.5: Diagrams of the (p, A )-plane corresponding to the regions A to H infigure6.4( r  Chapter 6. Population Genetics Model II  Figure 6.6:  D i a g r a m s of the (p, A ) - p l a n e corresponding to the regions N to P i n figure f  6  Chapter 6.  Population Genetics Model II  149  6.7: Examples of the fitness functions for parameter values corresponding to (a) a stable polymorphism (rn = 0.7, ku = 2.0, r i = 0.8, £22 = 0.6) and (b) an unstable polymorphism (rn = 0.4, jb = 2.0, r = 0.8, k = 0.6). Figure  2  n  22  22  Chapter 6. Population Genetics Model II  150  all parameter combinations which gave similar configurations of the fitness functions correspond to the existence of polymorphisms. This supports the conclusion i n [110] that heterozygote inferiority or superiority is necessary but not sufficient for a p o l y m o r p h i s m to occur. Hence, it appears that this method of looking at the fitness functions is not too informative when it comes to the question of existence of polymorphisms. A n alternative is to investigate the mean fitness functions:  w[ = ptw^ + (1  -pt)w\ , 2  w\ = ptw{ + (1 -pt)w* , 2  and w  t  22  = p w\ + (1 - p )wl, t  t  where w\ is the marginal fitness of allele A{ and w is the mean population fitness at time l  t. A n interior equilibrium (0 < p < 1, N > 0) requires  Pt+i = Pt and  N +i t  =  N. t  F r o m equations (5.2) i n the previous chapter we can see that the above equations w i l l be satisfied if  w* = and  1  w\ — w*.  (6.3) (6-4)  For 0 < p < 1 the latter condition can only be satisfied if  w[ = w . 2  (6.5)  W e need to solve conditions (6.3) and (6.5) simultaneously for p and N but an explicit mathematical solution is not possible because of the exponential terms i n the fitness functions. W e can use a computer package such as G N U P L O T [125] to plot these equations numerically. A n example is shown i n figure 6.8.  Chapter 6. Population Genetics Model II  151  3  0  1  1  0  Figure 6.8:  C u r v e s given by  1  0.5 P  w\ — w  2  ( t h i n dotted line) and  1  w  = 1 (thick solid line) for p a r a m e t e r  values c o r r e s p o n d i n g to a stable p o l y m o r p h i s m ( r n = 0.7, k\\ = 2.0, r  2 2  = 0.8, fc  22  =  0.6).  F r o m the above mathematical analysis we know that the curves corresponding to the two equations (6.3) and (6.5) always intersect when a polymorphism is present and do not intersect in other regions.  Each intersection corresponds to a unique polymorphic  equilibrium. Thus, we can predict the existence of polymorphic equilibria. However, it is not possible to distinguish between stable polymorphisms and interior saddle points on the basis of these diagrams. Using both plots of the mean fitness function configurations and the results in [110] mentioned in section 6.2 for determining the stability of interior equilibria, we can predict the existence and the stability properties of polymorphic equilibria. T h e relative positions of the fitness functions, such as in figure 6.7, give most of the required stability information but for a stable polymorphism the quantity i V | ^ also needs to be checked (see [110]). Although it is satisfying to have neat mathematical criteria, for a given situation it is probably easier and quicker to use D S T O O L to find and classify the fixed  Chapter 6. Population  152  Genetics Model II  points. In the next section I turn to higher order dynamics.  As i n the previous chapter  we would like to know whether stable period-2 (and higher period) polymorphisms are possible w i t h this additive model since the maintenance of genetic diversity need not be restricted to the existence of polymorphic equilibria.  6.4.4  Stable p e r i o d - 2 p o l y m o r p h i s m s  A t t e m p t s to find a stable period-2 polymorphism using D S T O O L and A U T O  proved  to be time-consuming. However, I was finally successful in locating a period-2 sink for r n = 1.3, &n = 0.5, r 2 = 7.5 and k 2  = 4.57. The fitness functions and (p, AQ-plane  22  corresponding to these parameter values are shown in figure 6.9. As can be seen from figure 6.9(b), the domain of attraction for this period-2 orbit is very small. A l s o , figure 6.9(a) shows that the fitness function w  22  has an unrealistically high m a x i m u m . Neither  situation is particularly satisfying. T h e rij values determine the heights of the fitness function m a x i m a . We would like to know whether altering one of the other parameter values would allow r  to be lowered  22  while still maintaining the interior period-2 attractor. two-parameter diagrams for this purpose. bifurcation diagram by varying r . 22  A U T O can be used to generate  The first step is to create a one-parameter  The relevant curves are shown i n figure 6.10. T h e  period-doubling bifurcation HB* is the point at which stable period-2 orbits are initiated. Using A U T O we can see how the position of this period-doubling bifurcation changes as a second parameter is varied. The diagrams obtained using r n , kn and k  22  parameter are shown in figures 6.11(a), (b) and (c) respectively. these diagrams, r  22  as the second  As can be seen from  needs to be greater than 6.4 for a stable period-2 polymorphism.  T h i s is still rather large and not particularly satisfactory.  Chapter 6. Population Genetics Model II  153  (a)  W2  2-  2  1—  T N  (b)  Figure 6.9:  (a) F i t n e s s functions and  (b)  (p, 7V)-plane for  rn =  1.3, k  n  = 0.5, r  22  = 7.5 a n d k  22  =  4.57.  There is no reason why we should be confined to the fitness functions given i n equations (6.1).  T h e only requirements for this study were that the fitness functions be  density-dependent and have a single hump, and that the model be additive (that is, the heterozygote parameters must be linear combinations of the homozygote parameters). T h e steeper the slopes of the fitness functions (the right-hand slope i n particular), the less stable the dynamics corresponding to that homozygote. T h i s is similar to the previous chapter where higher growth rates, which caused exponential fitnesses w i t h steeper  Chapter 6. Population Genetics Model II  154  0.75  Figure 6.10: A p a r t i a l one-parameter bifurcation d i a g r a m o b t a i n e d by v a r y i n g r  2 2  using Interactive  A U T O . O n l y branches satisfying 0 < p < 1 are shown. T h e second iterate of the m o d e l was used so t h a t the period-2 o r b i t s c o u l d be c o n t i n u e d . However, using this m o d e l the p e r i o d - d o u b l i n g b i f u r c a t i o n H B * is l a b e l l e d as a b i f u r c a t i o n p o i n t by A U T O w h i c h means t h a t it cannot be continued i n two p a r a m e t e r s . T h e o r i g i n a l m o d e l needs to be used for such a  continuation.  slopes, resulted i n more complex dynamics. We expect higher period interior orbits to occur i n regions of less stable behaviour at the boundaries p = 0 or p = 1 and, thus, one-humped fitness functions whose slopes are steeper than those given by equations (6.1) may give more reasonable results.  Fitness functions of the form (6.2) have the  required property. W i t h these fitness functions I found a stable period-2 p o l y m o r p h i s m for m = 0.2, fe = 5.0, r n  22  = 0.3 and fe = 0.4. B o t h D S T O O L and A U T O were used in 22  the search. The fitness functions and (p, AQ-plane corresponding to these values can be found i n figure 6.12. A s can be seen from figure 6.12(a), the fitness functions are much more reasonable than before. A g a i n we can get some idea of the size of the region i n parameter space for which  Chapter 6. Population Genetics Model II  Figure 6.11:  155  T w o - p a r a m e t e r c o n t i n u a t i o n s o f the p e r i o d - d o u b l i n g b i f u r c a t i o n H B * i n figure 6.10 o b -  t a i n e d b y v a r y i n g (a) ru,  (b) & n a n d (c) k  22  i n a d d i t i o n to r - T h e shaded regions i n d i c a t e w h i c h side 2 2  of the b i f u r c a t i o n c o n t i n u a t i o n s corresponds to p e r i o d i c b e h a v i o u r .  Chapter 6. Population Genetics Model II  Figure 6.12:  (a) F i t n e s s functions a n d  (b) (p, i V ) - p l a n e  156  for r  u  = 0.2, ku = 5.0, r 2 = 0.3 a n d 2  k 2 = 0.4. 2  Chapter 6. Population Genetics Model II  (a)  2  157  (b)  2  i  ?  1  H B  B P I •  N  1.5 -  N  1.5  _  1 4  ^ ^ H B  2  i  ' • •.  5  e  Figure 6.13: One-parameter diagrams obtained by varying ku using (a) the original model and (b) the second iterate of the model. Only branches satisfying 0 < p < 1 are shown.  these higher order stable polymorphisms occur. I decided to fix W22 (that is, fix r-22 and & ) and examine the (ku, r n ) - p a r a m e t e r space as was done earlier. T h e first step is 22  to use A U T O to create a one-parameter bifurcation diagram by varying ku-  DSTOOL  was used to generate starting points for A U T O . Using the first iterate of the model equations resulted i n figure 6.13(a). O n l y the period-doubling bifurcation indicating the change from a stable equilibrium to a stable period-2 orbit is shown. In order to plot the period-2 orbits the second iterate of the model needs to be used. T h i s gives figure 6.13(b). Notice that the period-doubling at ku = 4.226 is now labelled as a bifurcation point (transcritical) by A U T O instead of as a period-doubling bifurcation. T h i s second bifurcation diagram shows a further period-doubling at ku  — 5.358 from period-2 to  period-4 orbits but the stable period-4 orbits are not continued by A U T O .  However,  their existence can be verified using D S T O O L . Using the first iterate of the model we can see how the position of the first perioddoubling (at ku = 4.226) changes as ru is varied i n addition to ku-  In order to obtain  a rough idea of the extent of the region of interior higher order stable behaviour, we can vary ru using the second iterate of the model and trace out the path of the second period-doubling (at ku = 5.358 i n figure 6.13(b)). B o t h curves are plotted on the same  Chapter 6. Population Genetics Model II  158  0.1 -  0  Figure 6.14:  I  0  1  2  1  4  1  hi  1  6  8  10  A bifurcation d i a g r a m showing the two-parameter c o n t i n u a t i o n of the  at & n = 4.226 and the subsequent p e r i o d - d o u b l i n g  at ku  = 5.358 i n figure 6.13.  period-doubling  T h e shaded region  a p p r o x i m a t e s the region of stable p o l y m o r p h i s m s of p e r i o d greater t h a n 1.  set of axes i n figure 6.14. The shaded area in this figure approximates the region of parameter space corresponding to higher order stable interior behaviour. In general, the further the parameters are from the solid line (first period-doubling) w i t h i n this shaded region, the more complex the dynamics. For example, for r n = 0.18 and k-u = 5.0 there is an interior chaotic attractor as shown in figure 6.15. In figure 6.12 the fitness functions corresponding to the two homozygotes are fairly different i n terms of magnitude and the steepness of their slopes. This is true throughout the shaded region in figure 6.14 and can be deduced from the diagrams in the (p, 7V)-plane by noting the contrast between the complex dynamics near the p = 1 boundary and the m u c h simpler behaviour near the p = 0 boundary. A n important question is whether a fitness function configuration of the form shown i n figure 6.16 could result i n period-2 (or  Chapter 6. Population Genetics Model II  159  3.5  Figure 6.15: A n e x a m p l e of an interior chaotic a t t r a c t o r o b t a i n e d for r n = 0.18, ku = 5.0,r22 = 0.3 a n d &22 = 0.4.  higher period) stable polymorphisms. In such a situation both homozygotes have similar fitness properties but w i t h one of the homozygotes slightly out-competing the other at each population density. The region in ( & , r ) - p a r a m e t e r space corresponding to such n  n  fitness configurations is plotted together w i t h figure 6.14 to produce figure 6.17. Clearly, i n this case the two regions do not overlap. Thus in this range it is not possible to have a d d i t i v i t y (in the sense just described for the fitness functions) and stable polymorphic behaviour. So far w 2 has been fixed to have the shape shown i n figure 6.12(a). T h e slopes of 2  this function are fairly gentle. Suppose we replace W22 by the function shown i n figure 6.18.  W e expect such an alteration to reduce the stability of the dynamics near the  p — 0 boundary and hope that this might reduce the differences between the homozygote fitnesses that were previously required to obtain a stable period-2 p o l y m o r p h i s m . Figure 6.19 was obtained using the same procedure as for figure 6.17. A g a i n the two regions do not overlap but their relative positions have now changed. A n obvious question  Chapter 6. Population Genetics Model II  160  Figure 6.17: T h e (ku, rn)-parameter space showing the region of higher order stable polymorphic behaviour corresponding to figure 6.14 (vertical lines) and the region corresponding to fitness function configurations of the type shown in figure 6.16 (horizontal lines).  Chapter 6. Population Genetics Model II  F i g u r e 6.19'.  161  T h e new two-parameter space s h o w i n g the region o f higher order stable p o l y m o r p h i c  b e h a v i o u r ( v e r t i c a l lines) a n d the region corresponding to fitness f u n c t i o n configurations o f the t y p e s h o w n i n figure 6.18 ( h o r i z o n t a l lines).  Chapter 6. Population Genetics Model II  162  would be to ask whether there are intermediate parameter values where the two regions do overlap. Numerous investigations using both D S T O O L and A U T O did not reveal any situations of this type.  For the additive model with fitness functions given by (6.2) it  appears that the fitness properties of the homozygotes need to be fairly different before higher order stable polymorphisms are found. This result is comparable w i t h results in the previous chapter for period-4 orbits.  6.5  Conclusion  T h e population genetics model studied i n this chapter is more complicated than that of the previous chapter due to the form of the fitness functions. A s a result any m a t h e m a t i c a l analysis using pencil and paper is very difficult, if not impossible, since many of the fixed points do not have closed algebraic forms. However, A U T O and D S T O O L proved invaluable for investigating certain aspects of the behaviour of the model and led to some important conclusions. In particular, it was found that for the additive model w i t h one-humped fitness functions, period-1 stable polymorphisms (interior equilibria) are much more probable than period-2 (and higher period) stable polymorphisms. Homozygote fitnesses need to differ greatly in magnitude and slope properties for the latter to occur.  A method for pre-  dicting the occurrence of interior equilibria from mean fitnesses was demonstrated and the relationships between interior and boundary stable equilibria were shown using twoparameter bifurcation diagrams and corresponding diagrams of the (p, A )-plane. r  The  latter diagrams also highlight the high possibility of extinction for most parameter combinations.  Other one-humped fitness functions may lead to different conclusions than  those obtained i n this study. The techniques outlined i n this chapter could be used for such investigations.  Chapter 7  Spruce B u d w o r m M o d e l  7.1  Introduction  In this chapter I concentrate on a discrete model of a defoliating insect system. T h e insect is the spruce budworm and its preferred host trees are balsam fir and white spruce. The model that I have chosen to study was developed by Clark and L u d w i g [22]. In it the b u d w o r m , the branch surface area of the trees and their foliage are all state variables. In the next section I give some background to the budworm-forest system as well as to a few of the models which have been formulated to describe it.  T h e model by  Clark and L u d w i g [22] is fairly complicated and includes a number of processes such as dispersal, predation, food l i m i t a t i o n , and parasitism. Section 7.3 gives a description of the equations. Discrete models of this complexity have not been analysed i n detail before. Section 7.4 contains the model analysis. D S T O O L is the m a i n package used. Because the system is discrete and because of its complexity, continuation packages such as A U T O are of l i m i t e d value. This is discussed in more detail i n section 7.4.1. For the analysis I chose to focus on one aspect of the model rather than attempt an exhaustive parameter study. T h e process I chose is dispersal as it is thought to have a significant effect on the budworm dynamics (Clark [19]). Two parameters that affect small larval dispersal and female adult dispersal, respectively, are allowed to vary. Regions of this two-dimensional parameter space which correspond to budworm extinction, m u l t i p l e stable states and  163  Chapter  7. Spruce Budworm  164  Model  periodic outbreak behaviour are identified. W h i l e C l a r k and L u d w i g [22] found parameter values which give rise to some of this behaviour, a few additional possibilities are found in this study. A n important result is that insect outbreaks are possible for a large number of realistic parameter combinations. It would be interesting to know which of the many component processes i n the model is responsible for the observed behaviour and which ones have a lesser effect. K n o w i n g how each process affects the system behaviour can greatly help in understanding and managing the budworm-forest system. Using a variety of techniques I show that the m a i n process responsible for outbreak cycles is small larval dispersal. T h i s agrees w i t h the findings of C l a r k [19].  After studying the effects of small larval dispersal on the  behaviour of the model in more detail, I use bifurcation analyses once again to compare the effects of predation and small larval dispersal. The m a i n influence of predation is at fairly low budworm densities which means that it affects the time period between outbreaks.  7.2  Background  T h e eastern spruce budworm, Choristoneura  fumiferana  C l e m . (Lepidoptera:  Tortrici-  dae) is found throughout the Canadian Maritimes and northern New E n g l a n d as well as westward and northward through middle Canada up to the boreal forest ( M c N a m e e [89]). In some regions budworm densities remain low as a result of predators, inadequate resources and weather (Clark [19]). However, these controls break down periodically, particularly i n the eastern regions, resulting i n budworm outbreaks of epidemic proportions. Damage to the preferred host trees, balsam fir (Abies balsamea) and white spruce (Picea glauca), is extensive and can approach 100% i n dense, mature stands [19]. These  Chapter 7. Spruce Budworm Model  165  outbreaks are documented as far back as the 1700's w i t h some of the worst ones occurring i n the Canadian province of New Brunswick. Intensive insecticide spraying began i n this area i n 1952 in an attempt to protect the foliage and, thus, l i m i t tree m o r t a l i t y [19]. Contrary to expectations, this led to high endemic populations of budworm which began to k i l l significant portions of the forest (Baskerville [13]). T h e budworm itself is a univoltine insect which means that there is one budworm generation per year. Its life cycle can be divided into egg, larval, pupal and adult stages. The large larvae have the most effect on the dynamics of the budworm-forest system (Jones [65]). T h i s stage causes the most defoliation and large larval feeding levels influence both fecundity and adult dispersal. T h e large larvae are also subject to b i r d predation and parasitism and are the target for insecticide spraying. A s mentioned earlier, dispersal also affects the dynamics. The small larvae spin silk threads and are transported aerially by w i n d . If the adults disperse, they may fly from 10 up to 100 kilometres [89]. The system has received a substantial amount of research attention, both empirical and theoretical, in the past few decades (for example, [19, 32, 42, 65, 74, 89, 91, 105, 106]). In the 1970's Jones [65] developed a process-oriented simulation model which takes into account the annual dynamics of the insect and the forest in which it resides. T h i s model has been used as a research tool by forest managers and scientists i n New Brunswick (Clark and H o l l i n g [21]). However, a full understanding of the behaviour exhibited by the model has been hindered by the large number of component processes involved and the complexity of the equations. A t the other end of the scale, L u d w i g et al. [74] developed a simplified model consisting of a system of three ordinary differential equations which they studied qualitatively. A l t h o u g h they obtained some interesting results and demonstrated the potential of the model to exhibit outbreak behaviour, Clark and L u d w i g [22] note that many processes are ignored or aggregated when simplifying the situation to such an extent.  Chapter 7. Spruce Budworm Model  166  Another approach is to combine the two approaches and apply qualitative methods to a fairly complicated model. W i t h this a i m i n m i n d , Clark and L u d w i g [22] developed !  a condensed version of Jones's model by aggregating new and old foliage into a single variable and ignoring age structure in trees. The result is a discrete system describing the annual dynamics of three basic state variables: budworm density, foliage density and branch surface area density. Their model is more manageable than the one i n [65] but still includes important biological components. This model is described in the next section.  7.3  M o d e l equations  7.3.1  Foliage  The foliage variable, F , is the density of green needles found i n a unit of branch surface area. It is an average value representing conditions on the whole site and is measured i n 'foliage units' where one foliage unit (fu) is the quantity of new foliage produced per unit of branch surface area in the absence of budworm-induced defoliation. In addition to being consumed by budworm larvae, the foliage provides oviposition sites for adult moths [19]. A l t h o u g h balsam fir retains its foliage for about eight years, it is sufficient to consider only two classes ('new' or present year foliage and ' o l d ' foliage which includes all remaining foliage) since this is the only distinction made by the budworm. If budworm density is low, the ratio of new to old foliage is 1:2.8 [65]. For simplicity it is assumed that this ratio is fixed. Total foliage density can then be scaled to have a m a x i m u m of K  F  = 3.8 fu. If Ft, is the i n i t i a l foliage density then, following Jones, the effect of larval constimption  on new foliage can be described by: Fb  Remaining new foliage = e~ —— A  KF  (7.1)  Chapter  7. Spruce Budworm  Model  167  where A =  d L —^. 0  b  Here Lb represents the i n i t i a l budworm larval density and d  0  is the m a x i m u m foliage  consumption rate for an individual larva during the feeding season. E q u a t i o n (7.1) is a standard competition function (see appendix D for an explanation) and it is used to represent the competition between budworm that results from high population densities [65]. If the budworm's food requirements are not met by new foliage alone then old foliage is consumed. Analogues of the above equations are: Remaining old foliage = e~ —^——-Ft,  (7-2)  B  Kp where B = c {A - 1 + e~ ). A  x  (ci = 0.357 is a constant.) T h e total amount of foliage, JPI, remaining after consumption by budworm is obtained by combining equations (7.1) and (7.2) to give: Fr = -^[e-  A  + (K  (7.3)  - l)e~ ]F . B  F  K  b  F  Density-dependent growth of foliage also needs to be taken into account. If we let rp represent the average growth rate of foliage at low densities and remember that F  e  (the  foliage density after one year) cannot exceed Kp, then we obtain  F  e  =  ?\  ,  where the denominator introduces density-dependence. namics.  (7.4) This completes the foliage dy-  Chapter  7.3.2  7.  Spruce Budworm  Model  168  B r a n c h surface area  Another feature of trees that is important to budworm is the surface area of branches. T h i s serves as the budworm habitat. In this model the branch surface area density, 5", is an average value for the whole site and is measured in units of ten square feet (tsf) per acre.  The original model of Jones used 75 age classes. Since outbreaks tend to  synchronise tree development, there is some justification for the simplification to a single age class. Severe defoliation by budworm may k i l l branches. This is modelled by setting  S  1  (7-5)  = [ l - d (l s  where Sb is the i n i t i a l branch surface area density and d$ is an average death rate. S\ represents the living branch surface area which remains after defoliation. Because of the quadratic t e r m (1 —  ) , the difference between Sb and S is only significant if there is 2  ±  substantial defoliation, that is, if F\ is very different from Ff>. Subsequent density-dependent growth of surface area is taken into account by setting  Se =  7\  ,  (7.6)  where r$ is the average growth rate and Ks is the m a x i m u m branch surface area density.  7.3.3  Budworm  T h e preceding equations are only slightly more complicated than the ones in L u d w i g et al. [74]. However, those for the budworm dynamics are much more complex. M o s t of the following equations are based on Jones's model [65]. Following his approach, an i n i t i a l large larval density of Lb larvae per unit of branch surface area (per tsf) is assumed. Parasitism of these larvae is considered first.  Chapter 7.  Spruce Budworm Model  169  Parasitoids are not treated as a dynamic variable in the model since under normal conditions the parasitoids' numerical response is too slow to raise parasitism rates significantly before the outbreak collapse has begun [19]. According to Jones, the rate of parasitism is a decreasing function of larval density, with a m a x i m u m of 4 0 % at low b u d w o r m densities that decreases exponentially. The number (density) of larvae surviving parasitism is given by  Li = (1 - qmaxe- )L ,  (7.7)  c  b  where C = 0.003.L& and qmax = 0.4. Large larval survival is also influenced by the amount of food consumed. Following Jones it is assumed that survival is proportional to the average amount of food consumed:  > = (^fer)L  <-  L  7 8)  where kr, — 0.425 is a proportionality constant. Predation by birds is limited to the large larvae. A s i n L u d w i g et al. [74] this process is modelled by a H o l l i n g type III functional response. Thus,  (7.9)  L = e~ L , D  3  2  where  " S( F b  PseLt  2 b  + Lj)'  and pmax is the m a x i m u m predation rate and p ^ is a half-saturation rate. T h e exsa  pression for D requires some explanation. If it is assumed that the number of predators (birds) per acre is fixed, then the number per branch is proportional to pmax/S . T h e b  predators search foliage and they switch to alternate prey if the ratio of larvae to foliage, L /F , 2  b  is too small. If pmax/S is small, then predator consumption is b  Chapter  7.  Spruce Budworm  L  Model  - L  2  3  170  (l-e- )L  =  D  «  2  DL  2  =  P  m  a  x  "  L  (7 10)  which indicates that p ^ is the half-saturation value of the ratio L\jF . 2  sa  C l a r k and  L u d w i g [22] chose the more complicated exponential form (7.9) over (7.10) i n order to take into account competition among predators. T h e survival of pupae is correlated w i t h the survival of large larvae [65] and is given by L  4  where A  = 0.473 and B  (7.11)  3  P  = 0.828 are regression constants obtained by Jones [65]. T h i s  P  v  + BAL  = (A  expression gives the density of adult moths. A c c o r d i n g to Jones, female fecundity depends on their weight.  He calculates this  weight, W, as W = A  K F1  where AFI  = 34.1, Ap  2  A  ' + A (K F2  = 24.9 and BF  - 1)  {  F  A  ]  + B  F  (7.12)  = — 3 . 4 . This formula expresses the differing  nutritional values of new and old foliage. Fecundity is proportional to the cube root of W [65] which results i n the following equation for egg density:  L where E\ = 165.64 and E  2  and A  sr  = {ExW ' - E )A„L 1  5  3  2  4  (7.13)  = 328.52 are regression constants obtained by Jones [65]  is an adult sex ratio which gives the average proportion of females. There is an  additional constraint that fecundity be at least 40 eggs per female. If n u t r i t i o n were so poor as to produce fewer than this, the pupae would not have survived [65].  Chapter 7.  Spruce Budworm  Model  171  Dispersal is another process which is thought to have an important effect on local budworm dynamics. It is convenient to think of eggs dispersing rather than adults, as a female m o t h w i l l deposit some of her eggs on the site under consideration and w i l l remove some to other locations [65]. Following Clark [19], it is assumed that dispersal always leads to death. Thus, i n this model, adult dispersal serves to increase egg mortality. U n l i k e normal mortality, however, this removal of eggs from the population depends on female adult density i n the following way:  A  fP  m  Le = (l~ jfj^)L.  (7.14)  where  A L\ & = —. ST  A-thr is the ratio of female adult density to a threshold density, Athr- If the parameter ra is large then E  m  w i l l be large if E > 1 and small if E < 1. This means that a fraction Adi  sp  of adult females (and hence eggs) disperse if E > 1 and no dispersal occurs if E < 1. The steepness of the transition between no dispersal and dispersal is controlled by the size of m . Foliage density and branch surface area are both important factors i n determining the survival of small larvae. T h e reason is that small larvae disperse twice (using silken threads to give them buoyancy i n the wind) and the success of their dispersal depends upon landing on suitable foliage. It is assumed that the scaled probability of successful dispersal is given by G = H{2-H)  (7.15)  where H =  K-  n F  -  The parameter n is analogous to ra i n equation (7.14). If n is small, H w i l l only vary slightly as F\ decreases below its m a x i m u m , Kp. B u t if n is large, changes i n Fi w i l l  Chapter  7.  Spruce Budworm  Model  172  greatly affect H. Thus n determines the extent to which foliage density affects the success of small larval dispersal. A graph of G versus H is given i n figure 7.1. Notice that G is  0  0.25  0.5  0.75  1  H  Figure 7.1:  G r a p h of G versus H (equation (7.15)).  close to 1 unless H differs substantially from 1. The effect of both dispersals is included by taking the above factors into account twice. This gives L  = d ^G L ft 5 2  e  SL  6  (7.16)  where d$L is an average survival rate for small larvae and Sb/Ks expresses the dependence of dispersal success on branch surface area density. The lifecycle is now complete and L  e  represents the new large larval density. T h i s completes the description of the three-dimensional model developed by Clark and L u d w i g [22].  A summary of the equations can be found in appendix D.  7.1 gives the standard parameter values.  Table  As with all models there are a number of  simplifying assumptions. It may already have been noted that adult dispersal is not accurately represented in the model since the model only deals with a single region and has no spatial component. Thus the validity and accuracy of the model predictions need  Chapter  7. Spruce Budworm  Model  173  to be evaluated, but this is outside the scope of the present study.  Parameter  Description  Value  maximum foliage density maximum foliage consumption rate/larva foliage growth rate  3.8 0.0074 1.5  branch surface area death rate branch surface area growth rate maximum branch surface area density  0.75 1.15 24 000  maximum parasitism rate maximum predation rate half-saturation value (predation) adult sex ratio (females/total) fraction of females dispersing steepness of dispersal transition female threshold density (dispersal) effect of foliage density on larval dispersal small larval survival rate  0.4 23 000 0.085 0.46 0.5 4.0 5.0 1.0 0.28  Foliage KF  do r Branch surface area d r F  s  s  K  S  Budworm <lmax Pmax Psat A\ ST  m A hr t  n dsL  Table 7.1: Table of standard parameter values.  7.4 7.4.1  M o d e l analysis Preliminaries  Traditional methods of qualitative analysis, such as isocline analyses, are not possible i n this case as the equations are complex and there are too many component processes. For models of this nature, numerical solution of the equations for a given set of parameter values has been the principal tool for investigating the dynamics. However, numerous solutions for various parameter combinations are required for a meaningful study. T h i s is time-consuming and often misleading as the solutions depend on the i n i t i a l conditions and on complex joint distributions of parameters [19]. Since many ecological models of  Chapter  7.  Spruce Budworm  174  Model  practical interest are similar i n complexity to the one under discussion, it is desirable that methods be found for understanding the behaviour of these models. W i t h o u t an understanding of the range of behaviour that a model can exhibit, it may be difficult to explain results obtained from computer simulations or time series analyses of the model [82]. A l t h o u g h A U T O can perform continuations of equilibrium points for simple systems of difference equations, the complexity of this system causes the numerical algorithms to encounter problems. First of a l l , for discrete systems A U T O can only detect a perioddoubling from an equilibrium point to a 2 point (period-2) cycle (refer to section A.3.5) but this model exhibits 9 point through to 70 point (and multiples thereof) cycles. A n analysis using A U T O would clearly be l i m i t e d as none of these higher period cycles would be detected. A l s o , as certain parameters are varied the periods of the cycles change. In this model changes occur over small parameter ranges.  Since each change in period  corresponds to a bifurcation, the stepsize would have to be made very small if these were to be detected using a numerical continuation method. However, very small stepsizes lead to increased computer round-off error and decreased accuracy. Despite these limitations of continuation software i n the context of complex discrete models, it is still possible to obtain the desired qualitative information. However, bifurcation diagrams have to be obtained 'manually' rather than through automated continuation programs. T h i s can be done using D S T O O L as w i l l be described shortly. T h e types of numerical routines involved in using D S T O O L are generally more robust than those for A U T O when it comes to these complicated discrete models since they involve solving the system for a fixed set of parameter values. N o further discretisation or approximation is required as i n the case of differential equations. As i n the previous chapters the first step is to decide which parameters to investigate. Those whose values are uncertain or which are thought to have a significant influence on  7. Spruce Budworm  Chapter  Model  175  the dynamics are obvious choices. I chose to study the effects of dispersal on the budworm dynamics. A n important parameter affecting female adult dispersal is Athr, the threshold value below which no dispersal occurs. The effects of small larval dispersal are controlled by n, which determines the extent to which foliage density affects the success of small larval dispersal, or dsL, which is the average survival rate for small larvae (see equation (7.16)). If we set n = 1 then the effects of small larval dispersal can be determined by varying dsL which is more biologically meaningful. (A large value for dsL means that the average survival rate is high and also, that foliage condition has a significant effect on survival after dispersal.) In what follows I look first at d$L and then at Athr- T h i s ordering is arbitrary.  The  aim is to analyse the effects of varying the values of these parameters and to divide the (dsL,  A / )-parameter space into regions of different qualitative behaviour. Following this i  ir  I study the effects of different processes on model behaviour to determine which one(s) is(are) responsible for the outbreak cycles. Finally, I study the two processes predation and small larval dispersal in more detail. Because there are three state variables i n the model, a decision had to be made as to the two-dimensional space into which the dynamics should be projected. Since branch surface area varies at a much slower rate than the budworm and foliage dynamics, I decided to look at the budworm versus foliage plane. It should be noted that i n all the D S T O O L calculations, branch surface area, S, is still allowed to v a r y — t h e results are just projected into the (Foliage,Budworm)-plane. This differs from C l a r k and Ludwig's study where S was constant in their analysis and means that the following analysis applies to the full three-dimensional model whereas that in [22] is restricted to a simpler two-dimensional system of equations.  Chapter  7.4.2  7. Spruce Budworm  Model  176  The effects of small larval dispersal  M y first objective was to determine the effects of varying the parameter dsL on model behaviour. Since d$L is a survival rate it must lie between 0 and 1. To begin I set dsL = 0.05 i n the S E T T I N G S - S E L E C T E D window and fixed all the other parameters at their respective values given i n table 7.1. Using D S T O O L I found and recorded the equilibrium points corresponding to this parameter set as well as their local stabilities.  T h i s is  done using icons in the F I X E D P O I N T window. I also used D S T O O L to calculate the corresponding dynamics in the (Foliage,Budworm)-plane. This is done using the mouse to click on different i n i t i a l points, or entering i n i t i a l points manually in the S E T T I N G S S E L E C T E D window and using F O R W A R D S and C O N T I N U E i n the O R B I T S window to calculate trajectories. The resulting diagrams show whether cyclical behaviour occurs and the amplitudes of the cycles as well as the approximate domains of attraction (see section A.2.5) i n the two-dimensional plane. After recording the results I increased dsL to 0.10 and repeated the above procedure. I continued i n this way (incrementing dsL by 0.05 each time) until I reached dsL = 1-0. In regions where qualitative changes occurred (such as an equilibrium point changing stability or a change from stable to unstable oscillatory behaviour) I decreased this increment to 0.01. Greater accuracy would have been easy to obtain but it is time-consuming and I did not think it necessary for a qualitative study. T h e results of the above parameter study are summarised in the one-parameter b i furcation diagram i n figure 7.2. (Similar diagrams for F and S can also be drawn.) T h i s diagram was obtained by plotting the budworm densities of the e q u i l i b r i u m points for each value of dsh and then connecting these p o i n t s — s o l i d lines for stable equilibria and dotted lines for unstable equilibria. M a x i m a and m i n i m a of the cycles that were found were then p l o t t e d — s o l i d dots for stable cycles and open circles for unstable cycles.  Chapter  7. Spruce Budworm  250  1  200 Budworm larval density (larvae/tsf)  177  Model  •  •  •  1  i  i  —outbreak maxima  ^gg  100  -  50 1outbreak  0  0  0.2  • *  minima  I . 0.6  0.4  .  • •  .  • • •  ol 0  .  o o  —  .  •o  0  0.8  dsL  F i g u r e 7.2:  O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m of b u d w o r m l a r v a l density versus  dsL-  We can see from figure 7.2 that for low values of d$L there is a single stable e q u i l i b r i u m value.  A diagram of the (Foliage,Budworm)-plane for the particular value d$L = 0.2  is shown in figure 7.3(a). A l l orbits spiral in to the equilibrium point (denoted by the triangle). A s dsL increases a bifurcation occurs, and for dsL > 0.28 there is the possibility of outbreak cycles. Figure 7.3(b) shows the (Foliage,Budworm)-plane for d$L = 0.35. In this case, depending on the i n i t i a l values of the state variables, either stable equilibrium or outbreak behaviour can occur. This is further exemplified i n figure 7.4 by the plots of the temporal variation of budworm, foliage and branch surface area for dsL = 0.35. As can be seen f r o m these plots, the m a x i m u m of the outbreak cycle varies but the period remains fixed at 15 years. (Strictly speaking the period may be some larger multiple of 15 but from a biological viewpoint, we are most interested in the fact that peaks (outbreaks) occur every 15 years, even if the cycle m a x i m u m varies slightly i n consecutive outbreaks. Cycles of very long period or totally aperiodic motion w i l l both appear to be almost periodic (or chaotic) in a practical biological setting [84].) Since both the equilibrium and the outbreak cycle are stable phenomena, there must be some k i n d of boundary  Chapter  7.  Spruce Budworm  178  Model  (a)  350  (b)  350  (c)  350  (d)  350  / /  •i  Figure 7.3:  ] \  '  D i a g r a m s o f b u d w o r m l a r v a l density versus foliage for (a) dsL = 0.2, (b) dsL — 0.35, (c)  d$h — 0.8 a n d (d) dsL = 0.9. T h e dots i n d i c a t e densities i n consecutive years.  Chapter  7. Spruce Budworm (a)  179  Model 350  B  1/4" u  Time  100  (b)  Time  (c)  25000  Time  Figure  7.4: Time  plots of (a) b u d w o r m l a r v a l density, (b) foliage density and (c) b r a n c h surface area  density versus t i m e for dsL — 0.35. In each case two trajectories are shown for a t i m e p e r i o d of 100 years.  As can be seen, i n i t i a l values of the three state variables affect the r e s u l t i n g b e h a v i o u r of the  s y s t e m . The two s t a r t i n g points, A and B, are the same in all three graphs.  Chapter  7.  Spruce Budworm  Model  180  delimiting their domains of attraction (see section A.2.5). This is not easy to locate i n three dimensions and may not be a smooth surface. The dashed line in figure 7.3(b) gives a rough approximation to a two-dimensional projection of part of this boundary (the boundary also varies with S) and is only included to indicate that the domain of attraction for the outbreak cycle is much larger than that for the equilibrium point. For 0.28 < d$L < 0.68 the behaviour remains the same qualitatively. Near dsL — 0.68 the stable equilibrium undergoes a bifurcation resulting in an unstable e q u i l i b r i u m (denoted by a plus sign i n figure 7.3(c)) surrounded by stable cycles of small amplitude. These cycles have periods of 7 or 8 years. Figure 7.3(c) gives the (Foliage,Budworm)plane for dsL = 0.8. Note that the outbreak cycle now has a much larger amplitude than i n figure 7.3(b) and that the points corresponding to these outbreaks appear to fill a defined region rather than being confined to a curve as in figure 7.3(b).  T i m e series  corresponding to these outbreaks are similar td those in figure 7.4 but the amplitudes of the cycles vary more and there is more variation i n the magnitudes of consecutive points in the cycles. However, the period is fixed for a given value of dsLReturning to figure 7.2 again we can see that the two cycles become unstable as d$L is increased further, as indicated by the open circles. The (Foliage,Budworm)-plane for dsL = 0.9 is shown i n figure 7.3(d).  For any i n i t i a l values the system still oscillates  but the amplitudes of these oscillations get larger and larger until the budworm finally becomes extinct. H a v i n g classified the dynamical behaviour of the system for different values of dsL, I w i l l now vary A h , the threshold value for female adult dispersal. t  7.4.3  r  T h e effects of adult dispersal  To begin I chose three values of dst (0.2, 0.45 and 0.7) which correspond to regions of different qualitative behaviour in figure 7.2. For each of these values of <7sx I varied Athr  Chapter  7.  Spruce Budworm  Model  181  in the same manner as described i n the previous section i n order to obtain one-parameter bifurcation diagrams. In this case Athr is not restricted to lie between 0 and 1 since A hr t  is the threshold density of female moths above which dispersal occurs.  Female adult  outbreak densities are around 30 females/tsf [19]. However, dispersal is not l i m i t e d to outbreaks so we would like to investigate values of A h t  r  between 0 and, say, 20. I used  an increment of 0.5 for this study, decreasing this to 0.1 i n regions of qualitative change. T h e resulting bifurcation diagram for dsi = 0.45 is shown i n figure 7.5. In this case  120 Budworm larval density (larvae/tsf)  80  40 h  F i g u r e 7.5: O n e - p a r a m e t e r bifurcation d i a g r a m o f b u d w o r m l a r v a l density versus A h t  r  for dsL = 0.45.  there are two Hopf bifurcations resulting i n two regions, L\ and L , where periodic orbits 2  of small amplitude (when compared with the outbreak cycles) occur. These orbits have periods of 9 or 10 years. For this value of dsL, outbreak cycles are possible for a l l Athr values. For clarity of the smaller amplitude orbits, only the m i n i m a of the outbreaks are shown i n figure 7.5 (these are just above zero). T h e m a x i m a vary between 355 larvae/tsf at Athr = 0.01 and 480 larvae/tsf at A h t  corresponds to that at A h t  r  r  = 14. For larger Athr values the behaviour  = 14 but with greater outbreak amplitudes. T h e diagrams  for the other values of dsL are qualitatively s i m i l a r — o n l y the values of A hr at which the t  Chapter  7. Spruce Budworm  0  182  Model  0.2  0.4  0.6  0.8  1  dsL Figure 7.6:  T w o - p a r a m e t e r bifurcation d i a g r a m o f A hr t  versus dsL-  bifurcations occur and the amplitudes of the cycles are different. For d$L = 0.2 outbreaks only occur for A h t  r  > 10.5.  H a v i n g obtained an idea of the qualitative behaviour which corresponds to varying Athr, we can begin constructing a two-parameter bifurcation diagram i n the (dsL, A hr)t  parameter space. The features we can expect to locate as a result of the above studies are two curves denoting where the Hopf bifurcations occur and a curve dividing the parameter space into regions where outbreak behaviour is or is not possible. Other curves of interest include those indicating the extent of stable cycling behaviour. T h e values of Athr at which each of the above phenomena occur can be found through incrementing dsh by 0.1 (or 0.05 i n regions where significant changes occur). T h i s results i n the two-parameter bifurcation diagram shown i n figure 7.6. T h e solid lines indicate where Hopf bifurcations occur and the small dotted lines indicate the outer limits for  Chapter  7. Spruce Budworm  Model  183  stable cycling behaviour corresponding to these bifurcations. The thick dotted line separating regions A and B from C and D indicates the boundary for outbreak cycles, that is, to the left of this curve no periodic outbreak behaviour occurs. T h e other thick dotted lines indicate the boundaries of the regions F, G and H where budworm extinction occurs and region I where two equilibrium states are possible. T h i s single diagram summarises nine qualitatively different types of behaviour that can be obtained by varying dsL and A hr- The nine regions are marked A - I . Diagrams t  of the (Foliage,Budworm)-plane corresponding to each region are shown i n figure 7.7. Strictly speaking dots should be used for the trajectories instead of continuous lines since the model is discrete. However, it is easier to denote the direction of flow and the qualitative behaviour when lines are used. In region A there is one spiral sink (indicated by the triangle) corresponding to positive budworm densities, and an unstable saddle point (indicated by a plus sign) at (F, S, L) (KF,KS,0)  =  (L represents budworm larval density). A l l trajectories with positive i n i t i a l  values spiral i n towards the sink (see section A.2.21). In region B this sink has become an unstable saddle (see section A.2.20). (In (Foliage,Budworm)-space the behaviour near this saddle resembles that near an unstable spiral (see section A.2.23).) starting near this point spiral out towards a stable periodic orbit.  Trajectories  Trajectories from  other i n i t i a l points still spiral inwards but in this case they approach the periodic orbit instead of the equilibrium point. Region C is similar to region A in that the equilibrium points are again a spiral sink and a saddle. However, i n this region outbreaks are also possible. Since both the sink and the outbreak cycle are attracting, there must be a basin boundary d i v i d i n g their domains of attraction. A rough approximation to part of this boundary is denoted by the dashed line. A g a i n , the position of the boundary w i l l vary w i t h S. It appears that in most cases the domain of attraction for the outbreak cycle is much larger than that for  Chapter  7.  Figure 7.7:  Spruce Budworm  184  Model  Foliage  Foliage  Foliage  Foliage  Foliage  Foliage  Foliage  Foliage  Foliage  D i a g r a m s o f b u d w o r m l a r v a l density versus foliage for the regions m a r k e d  A-I  7.6. T r i a n g l e s denote sinks (usually s p i r a l sinks i n this m o d e l ) and plus signs denote u n s t a b l e  i n figure saddles  ( e q u i l i b r i u m points h a v i n g at least one stable and one unstable eigenvalue—the t h i r d eigenvalue m a y be either stable or unstable).  Chapter  7.  Spruce Budworm  Model  185  the sink. Region D is similar to region C for most of the state space. T h e only difference is that the spiral sink has been replaced by a stable periodic orbit. A s Athr is varied (and we move from region D to E ) , the periodic orbit becomes unstable and we are left w i t h all the trajectories approaching the outbreak cycle. A s we move into region F even the outbreak cycle becomes unstable. The system may oscillate a few times exhibiting cycles of very large amplitude (in terms of budworm density) but then the budworm population crashes and becomes extinct. Region G also has unstable outbreaks but there is a small region where trajectories spiral in towards a stable equilibrium. T h e remaining two regions are H and I. These correspond to low dsL values. In region H the point (F,S,L) = (Kp, Ks,0)  is a spiral sink. A l l trajectories approach this point  and thus the budworm becomes extinct without any outbreak occurring since small larval survival is too low for the budworm population to survive even in the most favourable conditions.  In region I there are two spiral s i n k s — o n e corresponding to no b u d w o r m  and the other to a positive budworm density. The i n i t i a l values of F, S and L (budworm larval density) determine which sink is approached.  7.4.4  Biological interpretation  T h e first point to note is that, from an experimental viewpoint, it may be difficult to distinguish between the equilibrium behaviour associated with regions A and C and the cycling behaviour of regions B and D, respectively, due to random variation and measurement errors.  This is true even i n the simplest cases. Near a spiral sink the  values of the state variables oscillate with the amplitude of oscillation getting smaller as the sink is approached.  W h e n a periodic orbit is present the behaviour is similar  but the oscillations approach the stable cycle. Because of statistical variation in nature and measurement errors, it is almost impossible to determine whether the system is  Chapter  7.  Spruce Budworm  Model  186  approaching a sink or a periodic orbit of small amplitude. Therefore the locations of the solid lines (representing Hopf bifurcations) are not as important as the positions of the small dotted lines. Beyond the latter (in regions E , F and G) no 'desirable' stable behaviour is possible—only periodic outbreaks are found i n these regions.  In regions  F and G , the amplitudes of these outbreaks are so large that the budworm population crashes to zero after a few cycles and becomes extinct. T h e actual position of the boundary line of these latter two regions is fairly arbitrary as even before the thick dotted curve is reached the outbreak cycle leads to very small budworm densities for certain parts of the cycle. Such small densities may be equivalent to extinction due to statistical variation. Essentially we have to decide how low budworm densities can drop before extinction occurs. T h e other region of extinction is region H. This region corresponds to very low (average and dispersal) survival rates for small larvae. In this region survival is too low to allow the budworm population to be self-sustaining, even if conditions are favourable. contrast regions F and G correspond to high survival rates.  In  These have the effect of  destabilising the system and causing wild oscillations. This destabilisation as a survival rate is increased is a phenomenon characteristic of many ecological models [84]. Region I corresponds to two stable e q u i l i b r i a — o n e at very low budworm densities and the other at higher population densities. Although the possibility of m u l t i p l e stable states is of interest to ecologists, this region exists for such a small range of dsL values that it is probably not of practical importance as a small perturbation would move the system into regions A or H. However, C and D are regions of multiple stable states with significant area. F r o m the above discussion figure 7.6 can be simplified to include only the most important phenomena. A simplified diagram is given in figure 7.8. T h e above results show that outbreak behaviour is possible for a large number of parameter combinations. In the region of two stable states an alternative, more desirable  Chapter  7. Spruce Budworm  187  Model  I  i  1  ,  .1  d  ' •,  -  i  e  -  a I)  -  :  •  ;  -  ' ' • ' '•: \  c  . •• "  0  -  R e g i o n of m u l t i p l e stable states \  .. . • i • •  0.2  d i  i  0.4  0.6  f '.  e  i  0.8  1  dsL a. b. c. d. e. f.  Figure 7.8:  Extinction S p i r a l s i n k / s t a b l e l i m i t cycle b e h a v i o u r (endemic p o p u l a t i o n ) S p i r a l s i n k / s t a b l e l i m i t cycle + o u t b r e a k b e h a v i o u r O n l y outbreak b e h a v i o u r W i l d oscillations l e a d i n g to e x t i n c t i o n S m a l l region of stable b e h a v i o u r + same as region e  S i m p l i f i e d two-parameter bifurcation d i a g r a m of A hr t  versus dsL-  stable state is possible but for the rest of the outbreak region no alternative is possible. A l t h o u g h figures 7.6 and 7.8 provide a concise summary of results, the one-parameter diagrams (figures 7.2 and 7.5) are also useful. They show the budworm densities corresponding to the equilibria and cycling behaviour as well as the amplitudes of the cycles. T h e corresponding one-parameter bifurcation diagrams for foliage and branch surface area density show that the equilibrium values for these quantities decrease as dsL and/or Athr are increased. Thus they show the severity of the defoliation corresponding to different dispersal rates. A l t h o u g h the state space diagrams i n the preceding analysis were only for the (Foliage, Budworm)-plane, they could also have been projected into the (Surface A r e a , B u d w o r m ) -  Chapter  7. Spruce Budworm  Model  188  or (Foliage,Surface Area)-plane, since the branch surface area, S, varies in all the situations studied. T h e dynamics are similar i n all the planes—just the shapes of the cycles and the positions of the equilibrium points are different.  7.4.5  W h a t causes outbreak cycles?  Approach In the preceding sections parameter values corresponding to low (0-2 larvae/tsf) and m e d i u m (30-50 larvae/tsf) equilibrium budworm densities as well as values giving rise to outbreaks were obtained. A l l three types of behaviour (namely, very low budworm densities, endemic equilibria and outbreaks) have been observed i n the field [90]. However, the model under discussion is a complex one w i t h many component processes. W e would like to know just which processes are responsible for the observed behaviour and which ones are of lesser importance. In particular we would like to know which processes cause the outbreak cycles. One way to investigate the effects of the different processes i n the current model is to start w i t h a basic model and add i n d i v i d u a l processes one at a time (such as parasitism or the effect of larval weight on fecundity) to see what effect this has on model behaviour. Clark and L u d w i g [22] included a number of switches in their model so that the different processes can be turned on and off. In order to remove parasitism, equation (7.7) can be replaced by L\ — Lb-  (7.17)  T h e effect of food on large larval survival (equation (7.8)) can be replaced by a constant survival rate, namely L  2  -  k Li. L  (7.18)  Chapter  7.  Spruce Budworm  Model  189  As for parasitism, predation (equation (7.9)) can be removed by setting L  3  = L. 2  (7.19)  Instead of correlating pupal survival with that of large larvae, equation (7.11) can be replaced by  L = BL. 4  P  3  (7.20)  T h e effect of pupal weight (and hence feeding history) on female fecundity (equation (7.13)) can be removed by using a constant average fecundity. This gives  L =BL 5  where Bj  e  Se  A  (7.21)  = 96 was the chosen average fecundity per female m o t h [22]. Female adult  dispersal is removed by setting Ansp  = 0 in equation (7.14) and small larval dispersal  can be precluded by replacing equation (7.16) with L = d L. e  SL  &  (7.22)  In order to obtain an idea of the effects of these processes on model behaviour, I turned off all the switches initially and then added each process to the model i n turn. For each process I varied its associated parameters to .see what range of behaviour could be obtained, and more specifically, whether outbreak cycles could occur.  Results W i t h all the switches off, the budworm density increases exponentially causing foliage density (and hence branch surface area) to decline to zero. T h e only process that affects this behaviour significantly when added to the model is small larval dispersal. In fact, the inclusion of this process leads to outbreak cycles for certain values of dsL- (In section 7.4.1 it was explained how varying the small larval survival rate, dsL, effectively varies  Chapter  7.  Spruce Budworm  Model  190  the success of small larval dispersal. Higher small larval survival rates also mean that the larvae have a greater chance of dispersing successfully,, that is, of surviving dispersal, because of the formulation of equation (7.16). Thus, the two processes are referred to interchangeably.) E v e n with very high predation rates, predation alone cannot produce cyclical behaviour. T h e other processes in the model, namely parasitism, predation, large larval and pupal survival, fecundity and adult dispersal, do not alter the qualitative behaviour significantly when operating alone but do affect the rates at which the budworm population grows and at which foliage and branch surface area vary.  In other words, their effects are  quantitative rather than qualitative. To demonstrate this more conclusively, I obtained a diagram similar to figure 7.6 using a simplified model containing only a few processes. S m a l l larval dispersal was included in this simplified model as the dynamics depend on it. In order to obtain a diagram similar to figure 7.6, adult dispersal also needs to be included so that Athr can be varied. The switches for all the other processes mentioned above were turned off, that is, equations (7.17).. .(7.21) replaced the corresponding equations i n the original model. (For brevity this simplified model w i l l be referred to as the dispersal model.) The two-parameter bifurcation diagram i n (dsL, A - ) - p a r a m e t e r space shown i n t  ir  figure 7.9 was generated in the same way that figure 7.6 was obtained.  Discussion If we compare figures 7.6 and 7.9 we can see that all the m a i n phenomena are still present, which supports the c l a i m that dispersal is responsible for the qualitative behaviour of the model. T h e only feature that is missing from figure 7.9 is the region where the outbreaks lead to extinction (regions F and G in figure 7.6). However, by generating diagrams of the (Foliage,Budworm)-plane or time plots for the simplified model (as is done i n the process of obtaining figure 7.9), it is easily seen that the outbreaks for this model have  Chapter  7. Spruce Budworm  0  I—  L  0  Model  L-J  0.2  •  191  1  1  1  1  0.4  0.6  0.8  1  dsL  Figure 7.9:  T w o - p a r a m e t e r bifurcation d i a g r a m of A hr t  versus dsL for the s i m p l i f i e d m o d e l w h i c h o n l y  includes dispersal. R e g i o n s are m a r k e d according to figure 7.6.  very large amplitudes.  Even when dsL = 0.4 budworm densities vary between 1 0  - 9  and 820 larvae/tsf within an outbreak cycle. This outbreak amplitude increases as dsL increases w i t h m a x i m a around 2200 larvae/tsf and extremely small m i n i m a around 1 0 ° -5  larvae/tsf for d$L — 0.9. These values are clearly unrealistic and equivalent to extinction from a biological viewpoint. If we compare the above observations with the results from the original m o d e l , then we can make another important deduction.  The processes which have been left out  of this simpler dispersal model are important for biological realism. A s stated above, their effects are quantitative rather than qualitative. T h e y have a moderating effect on the system dynamics and prevent densities becoming too high or too low for reasonable parameter values. E a c h process may have a relatively small influence but together they exert considerable control over the system.  Chapter  7.  Spruce Budworm  192  Model  T h e observation that small larval dispersal is responsible for the outbreaks contrasts w i t h M c N a m e e ' s [89] explanation that cycling is the result of movement between b u d worm equilibria at low and high budworm densities. However, the result agrees w i t h the findings of Clark [20] who did a detailed study of the effects of dispersal on budworm dynamics. Royama's results [105] also indicate that larval survival, which is linked to dispersal, is the determining factor in the occurrence of outbreaks. In their study of the larch b u d m o t h , Baltensweiler and Fischlin [12] suggest that the cycles i n their system appear to stem from regional migration rather than long range migration of adults.  Why  does small larval dispersal regulate the outbreak cycles? Clark [20] suggests  the following argument. The survival of dispersing larvae depends on the quality and quantity of the foliage on which they land (see equations (7.15) and (7.16)).  If foliage  density is high then more larvae survive. For an outbreak to occur the forest must be i n good condition with high branch surface area and foliage densities. T h e  budworm  population then grows rapidly and escapes from the control of parasitoids and predators [90, 119]. However, high budworm densities lead to forest defoliation which induces rapid branch mortality. This lowers the success of small larval dispersers [20] resulting i n an epidemic collapse. Fischlin and Baltensweiler [37] come to similar conclusions i n their study of the larch/larch budmoth system. They also note that their model is sensitive to the recovery rate of the trees after defoliation but that not much field data is available on this. M c N a m e e et al. [90] recognise the importance of forest biomass on outbreak behaviour but m a i n t a i n that the outbreaks are movements between high and low e q u i l i b r i u m budworm densities. Their analysis is based on isorecruitment curves in which certain variables are held fixed. However, the analysis of this chapter leads to different conclusions. T h e state space and bifurcation analyses show that the equilibria and the outbreak cycles are different phenomena and that small larval dispersal is responsible for the cycles. A l l  Chapter  7.  Spruce Budworm  Model  193  the processes were allowed to vary simultaneously in this analysis and the results agree w i t h Clark's extensive analysis of dispersal [20]. Clark notes that the epidemic-collapse behaviour is the hardest to explain because of the strong dynamic feedbacks between the forest and budworm. He also states that direct application of equilibrium manifolds, the method used by M c N a m e e et al. [90], is not particularly informative (cf. chapter 4).  7.4.6  T h e effects of the other processes  I stated above that the effects of the processes other than small larval dispersal are quantitative rather than qualitative. However, a better understanding of these effects would be helpful. It would be informative to know the relative effects of each process on the dynamics of the system and the budworm densities over which their effects are greatest.  Method A technique suggested by Clark and L u d w i g [22] involves beginning with all the processes turned off (as described earlier) and then turning them on one at a time and determining budworm recruitment values over one year for a wide range of i n i t i a l budworm larval densities. The density range over which each process has the greatest influence can then be determined. This can be done a number of times w i t h the processes being turned on i n different orders so that the results can be checked. A n example is shown in figure 7.10.  Discussion F r o m figure 7.10 we can draw a number of conclusions. First of a l l , the effect of food on large larval survival is only noticeable at high (65 larvae/tsf) budworm densities and even  Chapter  7. Spruce Budworm  Model  194  31  2  Figure 7.10:  I  u p  Isoclines of r e c r u i t m e n t versus b u d w o r m density w h e n 1) a l l processes are s w i t c h e d off,  2) the dependence of large l a r v a l s u r v i v a l on food is added, 3) the effect of feeding h i s t o r y o n f e c u n d i t y is also a d d e d , 4) s m a l l l a r v a l dispersal is added, 5) p a r a s i t i s m is added, 6) p r e d a t i o n is a d d e d , a n d 7) a d u l t dispersal is added g i v i n g rise to the full m o d e l .  then the effect is not very pronounced. Feeding history has a small effect on fecundity at all b u d w o r m densities and has the most effect at high densities. P a r a s i t i s m substantially reduces recruitment for densities below about 45 larvae/tsf (log(budworm)  < 3.8) but has  less influence at higher densities (the parasitism curve approaches the preceding curve as budworm density increases). The effect of predation is clearly noticeable for fairly low budworm densities between 0.4 and 20 larvae/tsf ( — 1 < \og(budworm) < 3). However, control by predation declines as budworm densities increase. A d u l t dispersal exerts most control for budworm densities between 20 and 90 larvae/tsf (3 < \og{budworm) < 4.5) while the influence of small larval dispersal is clearly noticeable for densities greater than 90 larvae/tsf (\og(budworm) > 4.5). These results reiterate that small larval dispersal  Chapter  7. Spruce Budworm  Model  195  is the most important process at high larval densities and hence is the process which is most responsible for outbreak collapses. In summary, the processes having the greatest effects on budworm dynamics at high larval densities are small larval dispersal and adult dispersal. A t low to m e d i u m densities we have predation and parasitism. The latter two processes are responsible for controlling budworm densities between outbreaks. It is only when the budworm escapes their control that outbreaks occur. T h e extent of their influence w i l l therefore affect the length of time between outbreaks.  7.4.7  T h e effects of predation  C o m p a r i n g p r e d a t i o n and dispersal In order to further substantiate some of the above claims, a bifurcation analysis can be employed once again. The effect of predation is very pronounced at lower b u d w o r m densities (as can be seen from figure 7.10). I thus decided to add predation to the simpler model used earlier, which only included small larval and adult dispersal, to see what effect this would have. This new model w i l l be referred to as the predation model for simplicity (although it also includes dispersal). T h e two parameters affecting predation are p , max  the m a x i m u m predation rate, and  p , a half-saturation value. Increasing p x or decreasing p ma  sat  sat  both lead to increased  predation (see equation (7.9)). W i t h these two parameters at their n o m i n a l values given in table 7.1, the new two-parameter bifurcation diagram is shown i n figure 7.11. Some observations can be made by comparing this diagram with figures 7.6 and 7.9. First of a l l , predation does not affect the position of the lower Hopf bifurcation curve (see figures 7.9 and 7.11). predation.  However, the upper Hopf bifurcation curve is lowered by  Thus predation decreases the size of the region ( A , B , C , D and G) where a  Chapter  7. Spruce Budworm  196  Model  14 r 12 • 10 • 8• 6• 4 • 2• 0• 0  Figure 7.11:  T w o - p a r a m e t e r bifurcation  d i a g r a m of A hr t  versus dsL for the p r e d a t i o n m o d e l w h i c h  includes dispersal as well as p r e d a t i o n . Regions are m a r k e d according to figure 7.6.  stable endemic state is possible. T h e other noticeable effect of predation is on the parameter values for which outbreak behaviour occurs. Outbreaks occur for lower dsL values in figure 7.11 than i n figure 7.9. Thus predation seems to have a destabilising effect from this point of view. T h e m a x i m a of the outbreaks are similar to those for the dispersal model at each value of dsL but the budworm population crashes to zero for values of dsL greater than about 0.48 instead of continuing to oscillate (with ever-increasing amplitude as d$L increases) as occurred i n the dispersal model. This is due to the appearance of a sink corresponding to zero budworm density i n the predation model. However, this technicality is not of biological consequence since the outbreaks in the dispersal model attained such large amplitudes that they became unrealistic and equivalent to extinction. F r o m figures 7.9 and 7.11 it appears that the m a i n effect of predation is to destabilise  Chapter  7. Spruce Budworm  Model  197  the system as outbreaks occur for lower dsL values and the region of m u l t i p l e stable states is smaller (due to the shift i n the upper Hopf bifurcation curve).  However, the  qualitative dynamics have not been significantly altered. C o m p a r i n g figures 7.6 and 7.11 we can deduce the effects of adding the remaining processes (other than dispersal and predation) which are included i n the full model. In figure 7.6, the region of endemic stable states ( A , B , C , D and G ) has been shifted to lower dispersal thresholds (that is, lower  A h t  r  values) by adding these processes since both  Hopf bifurcation curves are lower i n figure 7.6 than i n figure 7.11. In other words, i n the full model endemic equilibria occur for higher rates of female adult dispersal than i n the predation model since lower dispersal thresholds i m p l y that there is more dispersal (see equation (7.14)). T h e region of extinction for low dsL values (region H ) is larger i n figure 7.6 but the amplitudes of the outbreak cycles are greatly reduced so that the region of extinction for high dsL values (regions F and G ) is much smaller. For example, outbreak m i n i m a and m a x i m a for dsL — 0.3 and A  t  h  r  = 5 are respectively 1 0  - 2 1  and 680 larvae/tsf for the  predation model as opposed to 1 and 216 larvae/tsf for the full model. For A hr t  < 12,  the onset of outbreaks occurs for higher dsL values i n the full model than i n the predation model. T h i s emphasises the above observation that the additional processes i n the original model have a stabilising effect on the system provided that adult dispersal is sufficiently high, that is, A hr t  is sufficiently low (below 12 larvae/tsf i n this case).  T h e role of predation In order to study the effects of predation i n more detail, one-parameter bifurcation diagrams can be generated by varying p  max  or p t. sa  I used the original model which includes  all the processes since its output is more biologically meaningful. T h e method used to obtain figures 7.2 and 7.5 was used again and the resulting one-parameter bifurcation  Chapter 7. Spruce Budworm Model  diagram for p ,  198  corresponding to d  max  = 0.4 and A  SL  simplify the scale p  max  = 5, is shown i n figure 7.12. To  thr  is given i n multiples of 23 000 which is the nominal value for  p  max  given i n table 7.1.  100 80 Budworm larval density  gg  (larvae/tsf) 40 20 0 0  1  2  3  ~F~  a Figure 7.12: One-parameter bifurcation diagram for  4 c  p  m a x  with  dsL  Pmax . = 0.4 and  Athr  = 5.  For Pmax less than half its nominal value there is a single e q u i l i b r i u m state, corresponding to endemic budworm densities (see figure 7.12). N o outbreaks occur i n this range although the system oscillates as it approaches the equilibrium. Outbreaks are possible i n the range of p x values denoted by a. M a x i m a for these cycles are around 270 ma  larvae/tsf for p  max  — 0.6 x 23 000 and increase as p x increases. A t p ma  max  = 2.7 x 23 000  a m a x i m u m of 360 larvae/tsf is attained. These values are much more realistic than the m a x i m u m of 820 larvae/tsf produced by the dispersal model for the same values of dsL and AthrAround p  max  = 2.3 x 23 000 the endemic equilibrium bifurcates to produce a nine  year periodic orbit of small amplitude . This periodic orbit is attracting for the range 1  To be mathematically correct I should say that a number of bifurcations occur leading to orbits of higher and higher period. However, these bifurcations occur over such a small range of parameter values that it is difficult to detect them using the present techniques. Also, because they occur over 1  Chapter  7.  Spruce Budworm  Model  199  of parameter values denoted by 6. In region a, i n i t i a l values for the budworm and forest variable determine whether the outbreak cycle or an endemic e q u i l i b r i u m state (stable e q u i l i b r i u m point or small amplitude periodic orbit) is attained. For p ax > 3.3 x 23 000 m  (region c) there is a single equilibrium corresponding to low budworm densities. T h i s suggests that if predation were to control the budworm population and keep it at low levels, the amount of predation would have to be much higher than has been observed i n the field. (Recall that the nominal or standard value for p x ma  is 1.0 x 23 000.) T h e  recruitment curves generated earlier (figure 7.10) using the standard parameter values i n table 7.1 show that predation has its most significant effect at budworm densities around 5 to 7 larvae/tsf.  However, during outbreaks larval densities increase to much  higher values very rapidly. This supports the above observation that predation does not have a significant influence on outbreak behaviour, except when predation is so high that b u d w o r m densities cannot escape from the low numbers where predation is prevalent. A diagram similar to figure 7.12 can be obtained by decreasing p . sat  F r o m the d i -  agrams of the (Foliage,Budworm)-plane and the time plots generated i n doing these parameter studies, an important observation can be m a d e — v a r y i n g the predation parameters has a significant influence on the periods of the outbreaks. For example, when Pmax — 0.6 x 23 000 the outbreak cycle has a period of 13 years. This increases to 50 years for p  max  = 3.4 x 23 000. These increased periods do not have much effect on the  t i m e span of the actual outbreak which is usually around 7 or 8 years.  Instead they  increase the number of years for which budworm densities remain below 1 larva/tsf. F i g ure 7.13 illustrates the above comments. These results again support the conclusion that predation only affects the behaviour at low budworm densities. Once the budworm have escaped the control exerted by predation, an outbreak occurs and the attributes of this such a s m a l l range, they are not of p r a c t i c a l i m p o r t a n c e i n themselves. W e are m o r e interested i n the q u a l i t a t i v e change f r o m an e q u i l i b r i u m to cycles.  Chapter 7. Spruce Budworm Model  (a)  200  350  Time  (b)  100  350  Time  Figure 7.13: p  m a x  T i m e plots o f b u d w o r m l a r v a l density for (a)  = 3.4 x 23 000.  Pmax  — 0.6 x 23 000 a n d  (b)  O u t b r e a k s last 7 or 8 years i n b o t h cases but the t i m e between o u t b r e a k s is  longer i n (b). B o t h figures are plots o f the b e h a v i o u r after the i n i t i a l transients have d i e d away.  Chapter 7.  201  Spruce Budworm Model  outbreak (such as its time span) are independent of the rate of predation. However, for higher rates of predation it takes much longer for the budworm to escape this control, hence the longer time periods between outbreaks. A s a final test of the effects of predation relative to small larval dispersal, I constructed a two-parameter bifurcation diagram of p x ma  versus dsi using the full model (which  includes a l l processes). T h e results are shown i n figure 7.14. Clearly, the higher dgL (and  •4  3  Pmax  (x23000)  2  1  0  0  0.2  0.4  0.6  0.8  1  dsL A . O n l y lower sink ( b u d w o r m : 0-2 l a r v a e / t s f ) B . T w o sinks (lower a n d interior) C . Interior s i n k ( b u d w o r m : 30-50 l a r v a e / t s f ) D . Interior E . Interior F . Interior G. Interior  s i n k + outbreaks saddle + outbreaks saddle + stable l i m i t cycle saddle + stable l i m i t cycle + o u t b r e a k s  F i g u r e 7.14: T w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m of p  m a x  versus dsL-  hence the greater small larval dispersal success) the greater the chance of outbreaks. O n l y for low dsL and high p x ma  values does predation exert sufficient control over the  budworm population to prevent outbreaks occurring. Note that the nominal value for Pmax is 1 x 23 000 and at this value the lower budworm equilibrium only exists for  Chapter 7.  d  SL  Spruce Budworm Model  202  < 0.12. T h i s section emphasises once again the overwhelming importance of small larval dis-  persal success, and hence the importance of forest condition, on the budworm dynamics.  7.5  Conclusion  This chapter has focussed on analysing the budworm-forest model developed by Jones [65] and Clark and L u d w i g [22] using the techniques of dynamical systems theory. Procedures for obtaining one- and two-parameter bifurcation diagrams using diagrams i n the (Foliage,Budworm)-plane and time plots from D S T O O L were explained. In the first part of the analysis a classification of the (dsL, A / )-parameter space was obtained. f  ir  While  Clark and L u d w i g [22] found parameter combinations corresponding to behaviour i n a number of these regions, namely A , B , C, D and H, this study found some additional possibilities for model behaviour and is much more comprehensive as the resulting bifurcation diagrams summarise the behaviour for all possible combinations of dsL and AthrT h e diagrams also show how the system behaviour changes as the two parameters, dst and A^r, are varied. Following this the various processes in the model were investigated i n more detail using a variety of techniques. It was found that small larval dispersal, and hence forest condition, has the most effect on outbreak cycles i n the model and that predation has an added destabilising effect. The m a i n influence of predation is at fairly low budworm densities which means that it affects the time period between outbreaks but not the length or amplitude of the outbreak. Again the use of bifurcation diagrams, state space diagrams and t i m e plots was crucial in the analysis.  Chapter 8  Conclusion  8.1  M a i n results  A variety of models, both continuous and discrete, theoretical and practical, have been analysed i n the preceding chapters. The same basic techniques have been used i n each example. Conclusions which are specific to a particular model have already been noted at the end of each chapter. There are also a number of general results that I wish to highlight. F i r s t , the dynamical systems techniques can lead to greater insight into the behaviour of a model and the interactions between various processes in a system than is possible w i t h traditional techniques.  For the sheep-hyrax-lynx model i n chapter 3 bifurcation  diagrams were found to give more information than a traditional sensitivity analysis, and for the ratio-dependent model i n chapter 4 the techniques proved more accurate and informative than an isocline analysis.  In both these cases the additional information  resulted in improvements in the formulations of the models. Thus d y n a m i c a l systems techniques can be helpful in constructing more plausible models. T h e techniques can also identify which parameters or processes are crucial for determining the behaviour of the model. This is illustrated i n chapters 3 and 7. T h e available computer packages allowed us to obtain the results fairly easily without the prerequisite of an extensive mathematical knowledge of dynamical systems theory and without intensive mathematical manipulations. This was highlighted i n chapters 2  203  Chapter 8. Conclusion  204  and 5 where the numerical results were compared w i t h previously obtained theoretical results. In chapter 2 the numerical results were i n fact more accurate than B a z y k i n ' s approximate analytic results [14]. T h e computer packages also allow more complicated models to be studied than is possible by hand. A l l the models illustrate this point. A s a result, previously unobtainable insights can be discovered. In addition, bifurcation diagrams provide a concise way of summarising results and two-parameter diagrams give an idea of the relative frequency of occurrence of the various phenomena. A l t h o u g h the dynamical systems techniques can be applied i n a variety of situations, they are obviously not suitable for a l l types of ecological models.  8.2  Limitations  Systems of difference equations or ordinary differential equations can be studied but not systems of partial differential equations.  A l s o , the models must not be t i m e or space  dependent. These limitations are serious, however it is generally the case that models which are fundamentally different i n structure require different methods of solution. For certain types of partial differential equations and time-dependent models it is possible to overcome these limitations by transforming the equations so that they fall into the required categories, but the mathematics required to do this is not t r i v i a l . A l t h o u g h the computer packages allow the dynamical systems techniques to be applied to large models, it is usually more difficult to interpret the results when many state variables and interactions are involved. There is also a greater risk of encountering software restrictions with these large models and the dynamics can become extremely complex because of the higher dimension of the system. Hence, the ease w i t h which the computer packages can be used should not be taken as an argument for building  Chapter 8.  Conclusion  205  complicated models. Simple models are still most likely to give us insight into system behaviour because of our own limits i n understanding. A s anyone who has used a computer w i l l know, computers and software hardly ever work as smoothly as one might wish. Some of the problems that I encountered have been discussed i n previous chapters and others are recorded i n appendix B . It is usually a good idea to check the results using an alternative technique or software package. A l t h o u g h a variety of models have been studied i n this thesis, there are many other possibilities for the application of dynamical systems techniques.  8.3  Future possibilities  I did not study any examples of models which include seasonal variation but it is possible to do so (see, for example, [48, 102]). 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Introduction Row  to Theoretical Ecology, New York: Harper and  Appendix A  Dynamical systems theory  A.l  Introduction  T h i s appendix explains the dynamical systems terminology used i n m y thesis. Section A . 2 is a glossary of the basic concepts such as equilibrium point, domain of attraction and bifurcation point. Extensive use is made of diagrams i n order to introduce the concepts as simply and intuitively as possible.  Section A . 3 describes some of the more formal  mathematical details associated with these concepts. However,the mathematics is kept to a m i n i m u m since the appendix is intended for biologists. Further details can be found i n any introductory text on dynamical systems theory. A few examples which I found particularly readable include [57, 111, 128].  A.2  Basic concepts  T h i s section is ordered alphabetically. A l l the examples given are for continuous systems of equations. Discrete systems are discussed i n section A . 3 . 5 . W i t h i n each subsection in this glossary, italics is used to highlight terms which are explained i n a separate subsection.  A.2.1  Bifurcation diagram  A one-parameter bifurcation diagram summarises the qualitative behaviour corresponding to different values of a parameter. A state variable (or combination of state variables) is  216  Appendix A. Dynamical systems theory  217  plotted on the y-axis and the parameter on the x-axis. T h e positions and local stabilities of equilibrium points as well as limit cycles are indicated using different line types. Solid curves are used to represent locally stable equilibria and dotted curves are used for locally unstable equilibria. M a x i m a and m i n i m a of l i m i t cycles are indicated using c i r c l e s — s o l i d ones for stable cycles and open circles for unstable cycles. See, for example, the figures i n sections A.2.10, A . 2 . 1 3 , A.2.16, A.2.18, A.2.25. It is important to note that bifurcation diagrams summarise the behaviour associated w i t h a range of parameter values. They do not represent the dynamics corresponding to a continually varying parameter [124]. In order to read a bifurcation diagram, fix the parameter at a particular value and mentally draw a vertical line at that value.  Each  crossing of this line with a curve i n the diagram corresponds to an equilibrium point or a periodic orbit (limit cycle). T h e local stability properties of a particular phenomenon are given by the type of curve, that is, solid, dotted, or open or closed circles. For example, the phase portraits'mfigure  A.17(b)(i) and (ii) were obtained by mentally drawing vertical  lines at the parameter values p = pi and p = p respectively i n figure A . 17(a). 2  A two-parameter bifurcation diagram shows how the positions of bifurcation points change as two parameters are varied. For example, if a bifurcation point is encountered i n a one-parameter bifurcation diagram, a second parameter may be varied to see how it affects the position of the bifurcation point. A n example involving limit points is shown i n figure A . 1 9 . A and p are parameters and X\ is a state variable. Part (a) of this figure shows a two-parameter bifurcation diagram and part (b) shows one-parameter bifurcation diagrams corresponding to different fixed values of the parameter A.  A.2.2  Bifurcation point  A bifurcation point is a point i n parameter space at which the qualitative behaviour of the system changes. A stable equilibrium may become unstable at this point or there may  Appendix  A. Dynamical  systems theory  218  be a change from a stable equilibrium to oscillatory behaviour. Examples can be found in sections A . 2 . 1 0 , A . 2 . 1 3 , A.2.16, A.2.18, A.2.25.  A.2.3  Chaos  Chaos is difficult to define but intuitively it refers to the (apparently) irregular and u n predictable behaviour which many nonlinear mathematical models (systems of equations) exhibit [11]. If a system is chaotic then i n i t i a l values which are very close together m a y lead to vastly different behaviour as time progresses.  However, this behaviour is still  bounded by a region i n space. Chapters 5 and 6 contain examples of chaotic behaviour.  A.2.4  Continuation branch  A solution or continuation  branch is a curve of equilibrium points (or limit cycles or  bifurcation points) that indicates how the position and properties of the e q u i l i b r i u m point (or l i m i t cycle or bifurcation point) change as a Together a number of these branches make up a  A.2.5  parameter (or parameters) is altered.  bifurcation diagram.  Domain of attraction  Suppose the system i n which we are interested has a stable A.2.14).  T h e n the collection of all i n i t i a l  equilibrium point (see section  state variable values from which the system  tends towards this equilibrium as time progresses is the domain (or basin) of attraction of the equilibrium point.  T h e equilibrium point is called an 'attractor'.  phenomenon, such as a stable to as an attractor.  A n y stable  limit cycle, also has a domain of attraction and is referred  Appendix A. Dynamical systems theory  219  For example, in figure A . l a population density of 10 is locally stable. For initial population values which lie between 5 and 20 the population tends towards a density of 10 as time progresses. For initial values below 5 the population tends to extinction and for i n i t i a l values greater than 20 it increases steadily. The range of values between 5 and 20 is the domain of attraction for the equilibrium point at 10. Part (a) of figure A . l shows time plots corresponding to various initial points and part (b) is a one-dimensional phase portrait of the situation.  (a) Population density  20-  Time  (b)  A—• q • — * 0  5  1  0  •  —2  0  Population density  Figure A . l :  (a) T i m e plots showing the d o m a i n of a t t r a c t i o n of an e q u i l i b r i u m p o i n t . A p o p u l a t i o n  density of 10 is stable a n d the range of i n i t i a l p o p u l a t i o n values between 5 and 20 constitutes its d o m a i n of a t t r a c t i o n . T h e values 5 and 20 are unstable e q u i l i b r i u m points, (b) A o n e - d i m e n s i o n a l phase p o r t r a i t of the s i t u a t i o n in (a).  T h e arrows indicate the direction of change corresponding to i n i t i a l p o i n t s in  each range of values.  Suppose our system consists of two competing populations. We can represent their dynamics using a phase portrait such as in figure A . 2 . In this case the points A and C are stable equilibria. The domain of attraction of A is the shaded region and the remaining  Appendix A. Dynamical systems theory  220  region is the domain of attraction of C . T h e curve separating these regions is called a separatrix.  Curves with arrows indicate how the population densities vary over time  beginning at various i n i t i a l points.  Population 2  A  Population 1  Figure A . 2 :  P h a s e p o r t r a i t showing the d o m a i n s of a t t r a c t i o n o f two e q u i l i b r i u m p o i n t s i n two d i m e n -  sions. A a n d C are stable e q u i l i b r i a . T h e arrows indicate how the p o p u l a t i o n densities vary over t i m e b e g i n n i n g at various i n i t i a l points.  A.2.6  E q u i l i b r i u m point  If the values of the state variables representing an ecological system do not change as time progresses then we say that the system is at an equilibrium point. Other commonly used terminology is singular point or fixed point. See also section A.2.14.  A.2.7  H a r d loss of stability  In the case of a Hopf bifurcation this occurs when there is a sudden change from stable equilibrium behaviour to stable limit cycles of large amplitude. A n example is shown in figure A . 3 . W h e n the parameter fi is increased beyond the Hopf bifurcation at //*, the system suddenly jumps to l i m i t cycles of large amplitude instead of starting off with  Appendix A. Dynamical systems theory  221  (population density)  \  HB  o oo e o  Mi P-*  Figure  A.3:  H a r d loss o f s t a b i l i t y (adapted from  [111],  P-74).  T h i s p h e n o m e n o n gives rise t o sudden  changes between stable e q u i l i b r i u m b e h a v i o u r a n d limit cycles of large a m p l i t u d e .  small l i m i t cycles which grow i n size as p increases. T h e latter phenomenon is shown i n figure A.6(a) and is called soft loss of stability. Also, as p is decreased, there is a j u m p from large amplitude cycles to a zero amplitude equilibrium point but this takes place at pi which is less than p.*. For pi < p < p* there are two stable a t t r a c t o r s — a n equilibrium point and a limit cycle. This is a kind of hysteresis phenomenon. Examples of hard loss of stability arise i n the analysis of the ratio-dependent model i n chapter 4. E q u i l i b r i u m states can also undergo a hard loss of stability (for example, in the vicinity of a pitchfork bifurcation). However, such phenomena are not encountered i n the m a i n body of the thesis.  A.2.8  Heteroclinic orbit  Consider a system having both predator and prey populations and suppose there are two equilibrium points one of which is unstable (a saddle or a source) and where the other is either a saddle or a sink. If an unstable manifold (see page 241 for an explanation of this term) of the unstable equilibrium point intersects a stable manifold of the other  Appendix A.  Dynamical systems theory  222  e q u i l i b r i u m point then the system is said to have a heteroclinic orbit.  A n example is  shown in figure A . 4 . For any initial point on this orbit the system w i l l tend towards the  predator  prey . Figure A . 4 : Example of a heteroclinic orbit. An unstable manifold of saddle point 1 intersects a stable manifold of saddle point 2.  equilibrium point 2. The parameter value at which the heteroclinic orbit occurs is called a heteroclinic bifurcation point. For a more detailed explanation of this phenomenon see section A.3.3.  A.2.9  Homoclinic orbit  This is similar to a heteroclinic orbit except that in this case the unstable and stable manifolds of the same equilibrium point (which must be a saddle) intersect. A n example is shown in figure A.5. Further details are given in section A.3.3 and an example arises in chapter 2. T h e unique parameter value giving rise to the homoclinic orbit is a homoclinic bifurcation point.  Appendix A.  Dynamical systems theory  223  predator  -homoclinic orbit  saddle point prey  Figure A . 5 : E x a m p l e of a h o m o c l i n i c o r b i t . A n unstable a n d a stable m a n i f o l d of the saddle p o i n t intersect.  A.2.10  H o p f bifurcation  A Hopf bifurcation  1  (HB)  is a bifurcation point at which an equilibrium point alters  stability and a limit cycle (period orbit) is initiated. A n example is given in figure A . 6 . Part (a) of the figure is a bifurcation diagram. The large dots denote the m a x i m a and m i n i m a of the l i m i t cycles. The phase portrait, in figure A.6(b)(i) shows the dynamics in the (xi, x2)-phase space for p = p\.  A is a stable equilibrium point or sink. After the  Hopf bifurcation is encountered at p = p*, A becomes a source and a stable l i m i t cycle is initiated. The corresponding dynamics at p = p  2  are shown in figure A.6(b)(ii). Notice  that the amplitude of the l i m i t cycle increases as p increases (see figure A.6(a)). T h i s is called soft loss of stability. In figure A . 6 a stable l i m i t cycle surrounds an unstable equilibrium point. It is also possible for an unstable l i m i t cycle to encircle a locally stable equilibrium point (see figure A . 7 ) . Unstable periodic orbits are indicated by open circles instead of solid ones. ' A l t h o u g h the n a m e H o p f bifurcation is usually used, A r n o l d [7] points out t h a t this is i n a c c u r a t e . B o t h P o i n c a r e and A n d r o n o v studied this bifurcation prior to Hopf. W i g g i n s [124] refers to the PoincareA n d r o n o v - H o p f bifurcation.  Appendix A. Dynamical systems theory  Figure A . 6 :  224  (a) A bifurcation d i a g r a m of a H o p f bifurcation ( H B ) , (b)(i) a phase p o r t r a i t c o r r e s p o n d i n g  to fi = fii at w h i c h there is a stable e q u i l i b r i u m p o i n t and (b)(ii) a phase p o r t r a i t c o r r e s p o n d i n g to n = /J at w h i c h there is an unstable e q u i l i b r i u m p o i n t surrounded by a stable l i m i t cycle.  2  Appendix A. Dynamical systems theory  225  (population density)  o°  Figure A . 7 :  B i f u r c a t i o n d i a g r a m o f a H o p f bifurcation where unstable p e r i o d i c o r b i t s s u r r o u n d a stable  equilibrium point.  T h e first examples of Hopf bifurcations occur i n chapter 2. See also sections A . 2 . 7 , A.2.22.  A.2.11  Hysteresis  A hysteresis phenomenon occurs when two limit points are connected as shown i n figure A . 8 . This results i n multiple equilibria corresponding to a single parameter value. For [i < u*^ and [i > u.* there is a single equilibrium point, which is stable i n this example. 2  For parameter values n* < fj, < pi* there are three equilibrium p o i n t s — t w o stable A  2  and one unstable. T h e unstable equilibrium point divides the domains of attraction of the stable equilibria. For initial population densities above the dotted line i n the range //*! < \i < u* the population tends towards C . For i n i t i a l values below the dotted line 2  the population is attracted towards A . T h e way i n which the behaviour of the system changes differs depending on whether p is increased or decreased. Suppose we are at equilibrium point C at \i =  If fi is  increased then solution trajectories will continue to tend towards C for all p such that  Appendix A. Dynamical systems theory  226  (population density)  •LP  LPC  Mi  M*  M2  M2  p  F i g u r e A . 8 : B i f u r c a t i o n d i a g r a m of hysteresis.  p < p* since our initial point is above the boundary B of the domain of attraction of 2  C. However, as p increases beyond p* a catastrophe or sudden change occurs and the 2  system tends towards A instead. W h a t is more, if-// is now decreased we do not return to the equilibrium at C. T h e system continues to tend towards A as we are below the dividing point B . This occurs until p \ is passed. Then the system jumps up towards C again. T h e situation that has been described is known as hysteresis. Occurrences of this phenomenon arise i n chapter 4. In nature there are many unpredictable influences on a system which means that there w i l l be fluctuations around any equilibrium. Notice that the domain of attraction of C is smaller the closer p is to p* . This makes the system much more susceptible to 2  crashing towards A as p* is approached [128]. 2  A.2.12  L i m i t cycle  L i m i t cycle behaviour occurs when a state variable (such as a population density) oscillates in a regular repetitive manner. Temporal behaviour for a single population is shown in figure A.9(a) and a phase portrait for two interdependent cycling populations  Appendix A. Dynamical systems theory  227  is shown in part (b). These diagrams show the eventual or l i m i t i n g behaviour once the i n i t i a l transients have died away. A l i m i t cycle is also called a periodic orbit and is often associated w i t h a Hopf bifurcation.  L i m i t cycles may be locally stable or unstable (see  section A . 3 . 4 for more details).  (b)  (a) Population density  predator  Time  Figure A . 9 :  A.2.13  prey  (a) T i m e plot and (b) phase p o r t r a i t of stable l i m i t cycle b e h a v i o u r .  Limit point  A l i m i t point or saddle-node bifurcation occurs when there are two equilibrium  points  on one side of the bifurcation point but none on the other side. Figure A.10(a) shows an example of a bifurcation  diagram of a l i m i t point ( L P ) . For p < p* there are no  equilibrium points at which both populations are nonzero, p* is thus the l i m i t i n g value of p for which equilibrium points exist, hence the name limit point.  A possible phase  portrait in two dimensions for p > p* is shown in figure A.10(b) for the particular value p — p\.  A is a locally stable equilibrium point and B is a saddle point. The i n i t i a l values  of xi and x  2  determine the subsequent behaviour of the system. If the initial point is  in the domain of attraction  of A (to the right of point B in figure A.10(b)), then the  system w i l l approach A . If the initial point lies on the other side of B , however, it will be repelled away from B in the opposite direction to A . Notice how the size of the domain  Appendix A. Dynamical systems theory  (a)  228  (b)  . 0-2-  x  (population  2  (predator)  density)  •B  LI  /i!  Ax? 1  0-2 (prey)  Figure A . 1 0 :  ( a ) B i f u r c a t i o n d i a g r a m showing a l i m i t p o i n t ( L P ) a n d ( b ) a phase p o r t r a i t c o r r e s p o n d i n g  to p = H\. F o r fi < fi* there are no e q u i l i b r i u m points at w h i c h b o t h p o p u l a t i o n s are nonzero.  of attraction of A , in terms of the state variable x\, decreases as p decreases towards p* (see figure A.10(a)). T h e first example of this type of bifurcation in the main body of the thesis occurs in chapter 2.  A.2.14  Local stability  Suppose the system in which we are interested is disturbed slightly from its  equilibrium  point. For example, a week of warmer weather may cause an insect's growth rate to increase slightly. If after the disturbance is removed the system returns to its original e q u i l i b r i u m , then the equilibrium point is said to be locally stable and is called an 'attractor'. Otherwise it is said to be unstable and is a 'repeller'. Locally stable equilibrium points are called sinks and locally unstable ones are called saddle points or sources.  A.2.15  Parameter  A parameter is a quantity such as a fecundity rate or predation rate which is used in describing the dynamics of a state variable. Whereas a state variable evolves with time,  Appendix A. Dynamical systems theory  a parameter is kept constant as time progresses.  229  In this thesis parameter values are  varied across ranges of values to see how their values affect the qualitative behaviour of the state variables. For example, increasing the fecundity rate of a population which is at e q u i l i b r i u m may cause the population to start cycling.  A.2.16  Period-doubling bifurcation  A period-doubling bifurcation occurs when a limit cycle undergoes a bifurcation and there is an exchange of stability to cycles having double the period. T h e situation is depicted graphically i n figure A . l l . Part (a) shows a bifurcation diagram w i t h period-doubling bifurcations at A i and A , and part (b) shows the behaviour over time for different values 2  of the parameter A. Chapter 4 contains the first examples of this phenomenon i n the m a i n body of the thesis.  A.2.17  Phase portrait  Suppose our system has two state variables, say a prey ( x i ) and a predator ( £ 2 ) . W e can represent the behaviour of both populations i n a single diagram called a phase portrait. A n example is shown on the square base of the diagram i n figure A . 12. In this example both x\ and x  2  exhibit oscillations of decreasing amplitude as they approach the stable  equilibrium point. This translates into an inward spiral i n the (x , x )-phase space. x  2  In this thesis sinks are represented by triangles, saddles by plus signs, and sources by squares. In most cases solid lines are used to denote solution trajectories and dashed lines denote boundaries of domains of attraction.  Appendix A. Dynamical systems theory  230  (a) •oooo • • •  * •  o  # O O O O  , l o o o o •o O O h O O O 6 O O 0 0 » • a  (b) i)  iii)  ii)  Xl  A < AH  Figure A . 1 1 :  A/^ <  A < Ax  T  i  m  e  Ai  < A <  A  2  (a) P e r i o d - d o u b l i n g bifurcations at \ \ and A (adapted from [111], p.259). (b) 2  over t i m e for the state variable  Time  Behaviour  for (i) X < \JJ (stable e q u i l i b r i u m ) , (ii) A # < A < A i ( l i m i t cycle)  and (iii) Xi < A < A (period-2 cycle—each cycle consists of a b i g h u m p and a s m a l l h u m p ) . 2  A.2.18  P i t c h f o r k bifurcation  A pitchfork bifurcation occurs when there is a unique equilibrium  point for  parameter  values on one side of the bifurcation point but there are three equilibrium points on the other side. A n example is shown in figure A . 13. Part (a) shows a bifurcation and part (b) gives phase portraitsior  diagram  parameter values on either side of the bifurcation  point. In the example shown in figure A . 1 3 , A is a stable equilibrium point for p < p*. For n > fi* A is a source and the other two equilibrium points, B and C, are locally stable. These stability properties vary from situation to situation but the symmetry is  Appendix  A. Dynamical  Figure A. 12:  systems theory  231  D e r i v a t i o n of a phase plane showing the time-dependent b e h a v i o u r of two variables,  xi  and X2 (adapted from [61], p-3). D a m p e d oscillations over t i m e give rise to an i n w a r d s p i r a l i n the phase plane.  always maintained, that is, B and C always have the same stability assignment and this assignment is the same as that for A on the opposite side of the bifurcation point, u*• For p > p* A is a k i n d of threshold point as it separates the domains  of attraction of B  and C. T h e i n i t i a l values of Xi and x determine whether the system tends towards B or 2  C (see figure A.13(b)(ii)). No examples of this type of bifurcation occur in this thesis but the above description is included for completeness.  A.2.19  Q u a l i t a t i v e behaviour  W h e n we refer to the qualitative behaviour or dynamics of a system we are interested in the long-term general behaviour of the system rather than exact (quantitative) population densities for each instant in time.  For example, different types of qualitative  Appendix A.  Dynamical systems theory  232  (a)  oi  «2  (prey)  Figure A . 1 3 :  (a)Bifurcation  a  2  CL3  (prey)  d i a g r a m of a pitchfork bifurcation, (b)(i) a phase p o r t r a i t c o r r e s p o n d i n g  to ix = Hi and (b)(ii) a phase p o r t r a i t corresponding to LI — p - In (b)(ii) the unstable m a n i f o l d s from 2  A d i v i d e up the d o m a i n s of a t t r a c t i o n for B and  C.  behaviour include a population declining to extinction, a population tending towards a stable equilibrium point, or a population undergoing limit cycle oscillations.  Thus,  qualitative behaviour is determined by the presence and nature of attractors (see section A . 2 . 5 ) .  Appendix  A.  Dynamical  A.2.20  Saddle point  systems  theory  233  A saddle point is an equilibrium point which attracts in certain directions and repels in others. In figure A . 1 4 the equilibrium point has an unstable manifold (see page 241) and a stable manifold as indicated by the dashed lines. Initial points lying on these manifolds are repelled from or attracted towards the equilibrium point respectively. Other i n i t i a l points m a y first be attracted and then repelled as shown by the solid lines.  predator  prey  Figure A . 14: Example of a saddle point (plus sign) and the associated dynamics. Stable and unstable manifolds are indicated by the dashed lines and solution trajectories from different initial points by the solid lines.  A.2.21  Sink  A sink is a locally stable equilibrium point. A sink may be either a stable node or a spiral attractor. Phase portraits and time plots corresponding to these two possibilities are shown i n figures A.15(a) and A.15(b) respectively. T h e time plots begin at the point marked with a * in the phase portraits.  Appendix  A. Dynamical  systems theory  234  (a)  f  i)  H)  Xi  0  Time  (prey)  Figure A . 1 5 :  (a)(i) Phase p o r t r a i t of a stable node and (a)(ii) t i m e plots  starting  at p o i n t * i n (a)(i).  (b)(i) P h a s e p o r t r a i t of a spiral a t t r a c t o r and (b)(ii) t i m e plots starting at p o i n t * i n ( b ) ( i ) .  A.2.22  Soft loss of stability  For a Hopf bifurcation this occurs when there is a continuous change from stable equilibrium behaviour to limit cycles of small amplitude. The amplitude of these cycles increases gradually for parameter values further from the Hopf bifurcation. Figure A.6(a) gives an example of soft loss of stability.  See also section A . 2 . 7 .  The first example of this  phenomenon in the main body of the thesis occurs in chapter 2.  Appendix  A. Dynamical  A.2.23  Source  systems  A source is an equilibrium  235  theory  point which is locally unstable. A n y disturbance to the system  will cause the state variables to move away from this point.  A n unstable node and a  spiral repeller are shown in figures A.16(a) and A.16(b) respectively. B o t h are examples of sources.  (a) i)  |  ")  a  .  X  i  0  Time  (prey) F i g u r e A.16:  (a)(i) Phase p o r t r a i t of an unstable node and (a)(ii) t i m e plots s t a r t i n g at p o i n t * i n  (a)(i). (b)(i) Phase p o r t r a i t of a spiral repeller and (b)(ii) t i m e plots s t a r t i n g at p o i n t * i n b ( i ) .  A.2.24  State variable  Suppose we are interested in a system consisting of plants, herbivores and predators. T h e n the 'state' of the system can be described by the relative biomasses or densities of  Appendix A. Dynamical systems theory  236  these populations. T h e variables that are used in a mathematical model of the system to represent these biomasses or densities are called state variables.  A.2.25  Transcritical bifurcation  A t a transcritical bifurcation point two equilibrium points coincide and exchange stabilities. A n example is shown in figure A . 1 7 . There are two equilibrium points, A and B ,  (a) (population density)  a>- A -  BfJ-i  (b) i)  ii) X2  X2  (predator)  (predator)  Xi  (prey)  Figure A . 1 7 :  (a) A bifurcation d i a g r a m of a t r a n s c r i t i c a l b i f u r c a t i o n , (b)(i) a phase p o r t r a i t corre-  s p o n d i n g t o fi = fi\ a n d (b)(ii) a phase p o r t r a i t corresponding to / i = [in. T h e s t a b i l i t i e s o f the t w o e q u i l i b r i u m p o i n t s interchange for parameter values on either side of the b i f u r c a t i o n p o i n t .  at each value of the parameter fi. A is stable for \i < u* and a saddle point for u > fi"  Appendix A. Dynamical systems theory  237  T h e situation is reversed for B . Figures A.17(b)(i) and A.17(b)(ii) show possible phase portraits i n two dimensions for p = pi and p = p respectively. Note that the bifurcation 2  diagram i n figure A.17(a) only indicates the positions of the equilibrium points i n terms of one of the state variables, x\. Chapter 2 contains an example of this type of bifurcation.  A.3  Some mathematical details  A.3.1  Introduction  T h i s section gives a brief introduction to some of the mathematical details of dynamical systems theory. Texts such as [49, 57, 111, 124, 128] give more complete expositions. For most of the section I w i l l assume that the model under study consists of a system of m ordinary differential equations of the form:  x = f (x)  (A.l)  where x is a vector of m state variables and the dot denotes differentiation w i t h respect to t i m e , that is, x =  For example, if we.were studying a predator-prey model then  we would have m = 2 and equation ( A . l ) i n expanded form would be  X\  =  fi{xi,x )  X  =  f2{xi,X )  2  where / i and f  2  say) and x  2  2  2  are the components of f representing the dynamics of xi (prey density,  (predator density), respectively.  T h e results for systems of difference equations (discrete models) of the form: X I—>  f(x)  (A.2)  t = 0,l,2,...  (A.3)  or x  t+1  = f(xO,  Appendix A. Dynamical systems theory  238  are very similar. Section A . 3 . 5 highlights some of the differences.  A.3.2  E q u i l i b r i u m points and local stability  A n e q u i l i b r i u m point (fixed point) x* of the system of equations ( A . l ) satisfies f(x*) = 0. If system ( A . l ) were to start at x* at time zero, it would remain there for a l l time. However, i n nature it is very unlikely that a system w i l l remain exactly at an e q u i l i b r i u m point since numerous factors perturb systems continually.  So we would like to know  whether solutions of the system of equations ( A . l ) starting near x* move towards or away f r o m x* as time progresses. T h a t is, we would like to determine the local stability behaviour near x*. We can sometimes do this by using a linearised analysis . W e begin by perturbing 2  the system slightly from x*. T h a t is, we replace x by x* + u i n equation ( A . l ) where u is a small perturbation. (We use a local  small  perturbation since we are investigating the  behaviour near the equilibrium point.) O u r new vector of state variables is u since  x* is fixed. E x p a n d i n g f i n a Taylor series about x* and neglecting nonlinear terms i n u (since u is small) we obtain the linearised system  ii  - Au  where A is the m a t r i x of first order partial derivatives of f evaluated at the e q u i l i b r i u m point x = x*. B y solving the characteristic equation of A we obtain rn numbers, A = ( A i , . . . , A ) , m  known as the eigenvalues of A . Eigenvalues may be real numbers or complex numbers. It is these eigenvalues which determine the local stability properties of x*. 2  M o r e d e t a i l e d i n t r o d u c t i o n s t o linear analysis c a n be found i n [34, 128].  If a l l the  Appendix A. Dynamical systems theory  239  eigenvalues of A have non-zero real parts then x* is said to be a hyperbolic e q u i l i b r i u m 3  point of ( A . l ) . If any eigenvalue has a zero real part then x* is said to be a e q u i l i b r i u m point.  nonhyperbolic  T h e local stability behaviour near hyperbolic e q u i l i b r i u m points is  relatively easy to determine.  Nonhyperbolic equilibrium points are more difficult to  classify but it is at these points that interesting bifurcations (see section A.2.2) occur. Let us consider hyperbolic equilibrium points first.  Hyperbolic equilibrium points If the real part of A is negative (that is, 9£A < 0) for all eigenvalues A of A , then x* is an asymptotically stable equilibrium point of ( A . l ) (that is, trajectories starting near x* move towards x* as time progresses).  If 3^A > 0 for any eigenvalue A of A , then x* is  said to be unstable. In the special case of a two-dimensional system ( m = 2 i n ( A . l ) ) even more information can be obtained about the behaviour near  In this case there are two (since m — 2) eigenvalues, A i and A , of A . T h e y satisfy 2  the equation A  2  where Tr A = trace of A = the sum of the diagonal elements of A and Det A = determinant of A . In two-dimensional cases we can represent the behaviour near x* using a phase portrait (see section A.2.17). T h e various possibilities are summarised i n figure A . 18. Alternatively, the results can be summarised as follows:  A c o m p l e x n u m b e r , A, has a real p a r t , 9£A, a n d a c o m p l e x or i m a g i n a r y p a r t , 9 A . F o r a b r i e f i n t r o d u c t i o n t o the relevant theory o f c o m p l e x n u m b e r s see the a p p e n d i x i n [128]. 3  Appendix A. Dynamical systems theory  240  Tr  2  = 4Det  Tr  3 Saddle point  Figure A . 1 8 :  ^ Saddle point  A s u m m a r y of the l o c a l s t a b i l i t y b e h a v i o u r near an e q u i l i b r i u m p o i n t , x*, of ( A . l ) w h e n  m = 2 (adapted f r o m [34], p.190).  A i , A real 2  • Ai < 0, A < 0 => x* is a stable node (region 1 in figure (A.18)) 2  • Ai > 0, A > 0 => x* is an unstable node (region 2 in figure (A.18)) 2  • Ai > 0 , A < 0 (or vice versa) 2  Ai, A  2  x* is a saddle point (region 3).  complex  • 9£Ai < 0, 9?A < 0 => stable spiral or focus (region 4) 2  • SftAi > 0,!RA > 0 => unstable spiral or focus (region 5). 2  Appendix  A. Dynamical  systems theory  241  Saddle points and sources are both unstable equilibrium points (see section A.2.14) but a saddle point differs from a source i n that solutions may be attracted towards it for a while before being repelled, depending on the i n i t i a l values of the variables (see region 3 i n figure A.18). In the case m = 2 a saddle point has one unstable eigenvalue while a source has two. This means that (for m = 2) a saddle point has one stable  manifold  and one unstable manifold associated with it. These manifolds are curves i n phase space such that i n i t i a l points on these curves are attracted towards the saddle point (for i n i t i a l points on the stable manifold) or repelled away from the saddle point (for points on the unstable manifold) (see region 3 i n figure A . 18).  N o n h y p e r b o l i c equilibrium points As mentioned earlier it is the nonhyperbolic equilibrium points that are associated w i t h bifurcations, that is, with changes i n the qualitative behaviour of the system of equations. There w i l l be a threshold value at which the change i n behaviour o c c u r s — t h e bifurcation value.  This value corresponds to (at least) one eigenvalue (or its real part) passing  through zero as it changes sign from negative to positive or vice versa.  Examples of  bifurcations can be found i n sections A.2.10, A . 2 . 1 3 , A.2.16, A.2.18, A.2.25. For nonhyperbolic equilibrium points the principle of linearised stability used above does not apply and other methods need to be used. T w o of these are centre manifold theory and normal form theory. Centre manifold theory reduces or simplifies the system of equations so that only those parts which affect the local dynamics near the bifurcation point remain. N o r m a l form theory uses systematic coordinate changes to transform this reduced system of equations into a 'normal form'. T h e behaviour corresponding to a number of normal forms has already been classified by various mathematicians and can be found i n most dynamical systems texts. T h e abovementioned examples have a l l been  Appendix  A. Dynamical  systems theory  242  classified using normal form theory. Another method due to Liapunov is described i n Wiggins [124].  One-parameter local bifurcations  Suppose that the system of equations ( A . l ) has  the form (A.4)  x = f(x,/x)  where fj, is a parameter and suppose that the equilibrium point x* undergoes a bifurcation 4  at pL — (j,*. (We assume initially that there is only one zero eigenvalue or one pair of complex conjugate eigenvalues w i t h zero real p a r t s — t h e greater the number of zero eigenvalues associated with a bifurcation point the more degenerate it is and the more complicated the dynamics associated with it.) Such a bifurcation point is called a oneparameter local bifurcation. Examples of bifurcation points having one zero eigenvalue include l i m i t point (see section A.2.13), pitchfork (see section A.2.18), and transcritical (see section A.2.25) bifurcations. A Hopf bifurcation (see section A.2.10) is also a oneparameter bifurcation but it has one pair of complex conjugate eigenvalues whose real parts are zero.  Two-parameter local bifurcations  Suppose we allow two parameters i n our model  to vary, that i s , (A.5)  x = f(x,//,A)  where fj, and A are parameters. W i t h two parameters more complex behavioural patterns such as hysteresis, which is described i n section A.2.11, are possible. The extent of the region of overlap i n figure A . 8 (that is, the difference  fi*  2  — fi\)  may  vary w i t h a second parameter, A, as shown i n the fwo-parameter bifurcation diagram T y p i c a l l y a s y s t e m of equations has more t h a n one parameter b u t we o n l y need t o consider one of these e x p l i c i t l y at the present t i m e . W e assume t h a t the values of any other parameters are fixed. 4  Appendix  A. Dynamical  in figure A.19(a).  systems theory  243  Part (b) of this figure shows one-parameter bifurcation diagrams  (a)  Ml  fJ-2 '_P3 M  M  M4  iii)  M  —i—i— Pi  M3  M  Mi  M4  M  Figure A . 19: ( a ) T w o - p a r a m e t e r bifurcation d i a g r a m s h o w i n g a cusp p o i n t and the p o s i t i o n s of the two l i m i t points associated w i t h the hysteresis as b o t h fi a n d A are varied. ( b ) O n e - p a r a m e t e r b i f u r c a t i o n d i a g r a m s c o r r e s p o n d i n g to different, fixed values of A i n part (a) and w i t h fi as the b i f u r c a t i o n p a r a m e t e r , (i) A = A i , (ii) A = A  2  and (iii) A = A 3 . These one-parameter bifurcation d i a g r a m s correspond to the  h o r i z o n t a l dashed lines in p a r t (a).  corresponding to different values of A (that is, corresponding to the horizontal dashed lines i n part (a)). A t A = Ai the equilibrium point does not undergo any bifurcations in behaviour.  For A = A the two l i m i t points are close together and for A = A 3 they are 2  further apart. The point (p, A) = (M,A*) is called a cusp point. A t this point the two l i m i t points coincide. The curves in figure A.19(a) thus show how the positions of the l i m i t points (bifurcation  points) vary with p and A. Compare this w i t h one-parameter  Appendix A. Dynamical systems theory  244  bifurcation diagrams which show how the positions of equilibrium points vary as a single parameter changes.  A.3.3  Global bifurcations  So far we have been looking at the local dynamics associated with bifurcations of equil i b r i u m points and l i m i t cycles. However, some dynamical properties cannot be deduced from local information [49]. These are called global properties.  The simplest situation  involves homoclinic and heteroclinic orbits. Suppose we have two equilibrium points and let fi be the bifurcation parameter. Phase portraits of two possible degenerate situations that can arise are shown in figure A . 2 0 . In  (b)  x  2  Xi  Figure A . 2 0 : (a)Phase p o r t r a i t saddle loop or homoclinic o r b i t .  of a saddle connection or  Xi  heteroclinic  orbit.  ( b ) P h a s e p o r t r a i t of a  part (a) of this figure a heteroclinic orbit joins two saddle points. T h a t is, the unstable manifold of one saddle point coincides with the stable manifold of the other saddle point. In (b) the stable and unstable manifolds of the same saddle point coincide and encircle the other equilibrium point. The dynamics associated with the second e q u i l i b r i u m point vary depending on the model equations. T h e situations in figure A.20 are degenerate. T h a t is, they only exist for a particular value of f i . A l m o s t any small perturbation will disrupt the coincidence of the stable and  Appendix A. Dynamical systems theory  245  (a) i)  iii)  x  X  X  2  2  2  7  P< P  P— P  Xl  (b) i)  T  Xl  Xl  p>  Xl  iii)  ii)  ^2  X2  p < n  Figure A . 2 1 :  P> P  Xl  p -  p  Xl  p  (a)Phase p o r t r a i t s for parameter values near a saddle connection or h e t e r o c l i n i c o r b i t .  ( b ) P h a s e p o r t r a i t s for p a r a m e t e r values near a saddle loop or h o m o c l i n i c o r b i t , (i) p < p*, (ii) /j, = fi* and (iii) p > p*. (p*\s the p o i n t at w h i c h the heteroclinic or h o m o c l i n i c o r b i t occurs.)  unstable manifolds. In figure A.21 we see what happens to the stable and unstable m a n ifolds when p is perturbed from the bifurcation point, p*. T h e reader may be wondering what happens near the second equilibrium point in part .(b) of this figure. Some examples of possible phase portraits are shown i nfigureA . 2 2 . A n important point to note is that the time period required to get from one saddle point to the other along a heteroclinic orbit, or to return to the same saddle point along a homoclinic orbit, is infinite. This has important consequences for practical studies as can be seen i n chapter 2 (section 2.4).  Appendix  A. Dynamical  Figure A.22:  systems theory  246  Phase p o r t r a i t s near a saddle loop or h o m o c l i n i c o r b i t s h o w i n g possible b e h a v i o u r near  the second e q u i l i b r i u m p o i n t .  Appendix  A.3.4  A. Dynamical  systems theory  247  P e r i o d i c orbits  Local stability So far we have discussed equilibrium points and the stability behaviour associated w i t h them. The concept of a Hopf bifurcation introduced the idea of periodic orbits or l i m i t cycles. Cycles have been studied in many biological settings (for example, the spruce budworm [74], nerve action potentials [58], glycolysis [47], cellular slime m o l d [108], predator-prey interactions [14]).  A n important aspect is whether the periodic orbits  exhibited by a system of equations are locally stable or unstable. For this purpose I introduce the concept of Poincare maps. However, only the m a i n results are presented here. M o r e detailed discussions can be found in [49] and [124]. In general , a Poincare section S is an (m — l)-dimensional hypersurface chosen so 5  that all trajectories of ( A . l ) a) intersect the hypersurface transversally, and b) cross the hypersurface i n the same direction. In particular, the l i m i t cycle passes through the hypersurface transversally at a particular point, q*. Figure A . 2 3 shows a periodic orbit i n three dimensions and a two-dimensional Poincare section, S. The periodic orbit intersects S at the point q*. If T is the period of the limit cycle and ip(t;z) is a solution of ( A . l ) starting at z (that is, satisfying the initial condition x(0) = z), then  q* = <p(T;q*). Let q be a point on S and let T (q) v  be the time taken for a trajectory ,x(t; q) to first  return to S. T h e n the Poincare map or first return map P(q) is defined by  P(q) = <p(T (q);q). v  T h i s is illustrated in figure A . 2 3 . Note that P(q*) = q* and T (q*) = T. v  5  T h e f o l l o w i n g is s u m m a r i s e d f r o m  [111].  Appendix  A. Dynamical  systems theory  248  Figure A . 2 3 : S c h e m a t i c representation o f a P o i n c a r e section a n d a l i m i t cycle in three d i m e n s i o n s (from [111], p.244). q*is a p o i n t on S t h a t lies on the l i m i t cycle, q is another p o i n t o n S a n d P ( q ) is its p o i n t of first r e t u r n to S.  In order to determine the stability of the periodic orbit, we need only investigate the behaviour of P near its equilibrium point q*. That is, we need to determine whether this equilibrium point is attracting or repelling. A s i n section A . 3 . 2 we linearise P about the e q u i l i b r i u m point q*. In this case A =  a  P  J ^ \  Stability is again related to the  eigenvalues of A but the conditions are slightly different as we are dealing w i t h a map (discrete system) here and not a continuous system such as ( A . l ) . We have the following result (see [111]): a) If the moduli of all the eigenvalues are smaller than 1, then q* is stable; b) If the modulus of at least one eigenvalue is larger than 1, then q* is unstable. It turns out that the eigenvalues of A =  a  P  g ^ ^ can be found from the eigenvalues  of the m a t r i x M  =  d<p{T;q*) dz  The m a t r i x M always has an eigenvalue equal to 1. It can be shown that the remaining (m — 1) eigenvalues are the eigenvalues of A Floquet  multipliers.  . T h e eigenvalues of M are called  Analogous to the discussion on equilibrium points of ( A . l ) , the  Appendix  A. Dynamical  systems theory  249  stability of a periodic orbit can be determined by calculating the Floquet multipliers. These are easier to find than the eigenvalues of A . In general, i n the vicinity of a Hopf bifurcation, unstable periodic orbits encircle stable equilibrium points and stable periodic orbits encircle unstable equilibrium points however there are some other possibilities. A few cases are shown i n figure A . 2 4 . B o t h the m i n i m a and m a x i m a of the periodic orbits  Figure A . 2 4 :  E x a m p l e s of H o p f bifurcations h a v i n g soft loss of s t a b i l i t y (adapted from [111], p-72).  are shown. Figure A . 2 4 gives examples of soft loss of stability or soft generation of l i m i t cycles (see section A.2.22).  Section A . 2 . 7 gives an example of hard loss of stability or hard  generation of l i m i t cycles.  Bifurcations of periodic orbits T h e above discussion has focussed on local stability near a Hopf bifurcation. It is also possible for the periodic orbits themselves to undergo stability changes.  These occur  when the Poincare map undergoes bifurcations. If the Poincare map undergoes a simple  Appendix  A. Dynamical  systems theory  250  bifurcation, then there may be an exchange of stability of the periodic orbit from stable to unstable or vice versa. If the Poincare map P has an eigenvalue of — 1 at the bifurcation point p* then the second iterate of the map, P  2  = P(P), undergoes a bifurcation. We call  this a period-doubling or flip bifurcation. A t this point there is an exchange of stability of the periodic orbits to orbits having double the period. The situation is depicted i n figure A . 1 1 . Period-doubling occurs in many situations such as chemical reactions, nerve models, the Navier-Stokes equations and ecological models involving three trophic levels. (Perioddoubling does not occur i n fewer than three dimensions for continuous systems, that is, m > 3 is required.)  In some cases a sequence of period-doublings may occur and this  may lead to chaotic behaviour (see section A.2.3). Another way in which a periodic orbit may exchange stability is through bifurcation into a torus (Hopf bifurcation of the corresponding Poincare map). This occurs when a complex pair of Floquet multipliers moves into or out of the unit circle. A g a i n m > 3 is required. Details of this can be found i n [111, 124]. F r o m a practical point of view it is probably more informative to return to generating numerical solutions of a model over t i m e when these complicated phenomena are encountered i n a bifurcation analysis so that the behaviour of the system near these points can be seen explicitly. I w i l l not present the mathematical details of these bifurcations here.  Chaos A brief description of chaos was given i n section A.2.3.  Further mathematical details  can be found i n [11, 16, 124, 128]. Currently there is wide debate as to the practical application of chaos. In ecological systems it is very difficult (probably impossible) to distinguish between stochastic noise and chaotic behaviour (see the preface i n [72]). In Stone [114] it is shown that the addition of a single small term to a logistic type model  Appendix  A. Dynamical  systems theory  removes the chaotic behaviour.  251  However, in [82, 84] it is demonstrated that chaos is  prevalent i n many discrete ecological models and that higher order systems display chaotic behaviour more readily than one-dimensional systems. A good overview o f t h e current debate is given in the collection of papers in [72].  A.3.5  M a p s (systems of difference equations)  T h e discussion so far has been restricted to models consisting of systems of ordinary differential equations. A very similar theory can be developed for maps given by (A.2) or (A.3). I w i l l briefly mention some of the differences that occur. More detailed discussions can be found in [124, 128]. Referring to equations (A.2) and ( A . 3 ) , an equilibrium point occurs when  f (x) = x or  x i = x«. i +  A linear stability analysis can again be done for hyperbolic equilibrium points. For maps a hyperbolic equilibrium point is one for which none of the eigenvalues of the m a t r i x of partial derivatives has unit modulus (that is, no eigenvalues have a magnitude of 1). A g a i n it is the nonhyperbolic equilibrium points which result i n interesting bifurcations. If the linearised matrix has a single eigenvalue equal to 1 then a l i m i t point, transcritical or pitchfork bifurcation may occur. The bifurcation diagrams are the same as for continuous models. However, it must be remembered that the 'phase portraits' or diagrams i n state space consist of discrete points rather than continuous curves. Examples corresponding to a spiral sink are shown i n figure A . 2 5 . Consecutive points are labelled i n the discrete case.  Appendix A. Dynamical systems theory  252  Figure A . 2 5 : State space diagrams of a spiral sink for (a) a continuous model and (b) a discrete model. For maps the special case of a single eigenvalue equal to -1 introduces another type of bifurcation called a period-doubling  bifurcation.  Figure A.26(a) shows a stable equilibrium  point undergoing a, period-doubling bifurcation to become a stable period-2 orbit as fx is increased. T h i s period-2 orbit undergoes a further period-doubling to produce a stable period-4 orbit. Figure A.26(b) gives examples of diagrams i n state space corresponding to these situations.  Notice that this situation is analogous to the period-doublings of  periodic orbits for continuous models. This is not surprising since that theory is based on Poincare maps which are discrete. It is important to note that periodic orbits for discrete systems are different from those for continuous systems.  In particular they have integral periods.  second iterate, f , of the map (A.3): 2  f (x ) = f ( f ( x 0 ) = f ( x , 2  In general, the k  th  f  iterate of the map is given by f (x ) = x fc  t  t+Jfe  Suppose there exists a value of x , x , such that f (x) = X fc  .  + 1  )-x  t + 2  .  Consider the  Appendix  A. Dynamical  systems theory  253  (a) Xi - «  •  O O  0  c  oo  ••••••  ••BOOooo  -f  i—I—I—I  Mi  (b)  9,  s.A 3«  2  *•• \  lll)  1.  X2  and p = p*  M  ii)  1•  Figure A . 2 6 :  Mi* M2 M2 M3  «4  ^2  z  2* 4».  2  1 4 • . 10 • ,14 2» 3' 12 V 6  1  5* .9 +7' .13 11*  (a)One-parameter bifurcation d i a g r a m showing p e r i o d - d o u b l i n g bifurcations at // =  p\  for a discrete s y s t e m . (b)State space d i a g r a m s s h o w i n g the d y n a m i c s at (i) \x — \x\ (stable  e q u i l i b r i u m ) , (ii) p = \in (stable period-2 o r b i t ) and (iii) u = fi  3  (stable period-4 o r b i t ) .  but f ( x ) ^ x f o r j = l,2,...,k-l. j  T h e n x is called an equilibrium or fixed point of period k. This means that the system has a cycle or periodic orbit whose period is equal to k time units. Suppose k = 2 and we have a one-dimensional system. Then / (x) = /(/(*)) = x 2  but f(x)  +  X.  Appendix  A. Dynamical  systems theory  254  Hence, if we start at x then after the first time unit we are at f(x) but after the second time unit we have returned to x. This is shown in figure A . 2 7 . A state space diagram  0  i  i  i  \  i  \—i—-  1  2  3  4  5  6  7  Time  Figure A . 2 7 : T i m e plot of a period-2 orbit for a discrete s y s t e m .  for such a situation in two dimensions (that is, for two state variables) is shown in figure A.26(b)(ii). In this example both xi and x will have time plots resembling figure A . 2 7 . 2  A n analogue of the Hopf bifurcation also exists for maps. It is sometimes referred to as the Naimark-Sacker  bifurcation  [124] but is also simply called the Hopf  bifurcation  for maps. T h i s bifurcation corresponds to a pair of eigenvalues of modulus 1. Instead of a periodic orbit, however, an invariant circle is initiated at this bifurcation point. W h i l e geometrically similar to a periodic orbit, the dynamics are different. A state space diagram of an invariant circle is shown in figure A.28 together with a time plot.  The  stability of this circle is intuitively similar to a periodic orbit but the methods of analysis are quite different [124]. There are two possibilities for an invariant circle. Either point 10 coincides w i t h point 1 in figure A.28(a), point 11 coincides with point 2 and so on, or subsequent points are distinct from all the earlier ones but still lie on the circle. If the map is allowed to iterate for a long time in the latter case, then what is eventually observed in state space will  Appendix  (a)  A. Dynamical  255  systems theory  (b)  X2  1»  X l  »9  6' 5«  5«  «9  •4  2• 4t  .8  .3  • 6  Time  Figure A . 2 8 :  (a) State space d i a g r a m showing an i n v a r i a n t circle,  (a) i n t e r m s of  x\.  (b)  T i m e p l o t of the s i t u a t i o n i n  appear to be a continuous circle. In general, the behaviour associated with discrete models is more complicated than that for continuous models because of the b u i l t - i n time delays in the feedback relationships [128]. Even one-dimensional maps can exhibit chaotic behaviour. M a y [82] shows how the behaviour of the discrete analogue of the logistic equation changes from stable e q u i l i b r i u m behaviour, to periodic behaviour, and finally to chaos as the growth rate is increased. The reader is referred to the literature that has been cited for further details on the dynamics of discrete models.  A.3.6  S t a b i l i t y of bifurcations under p e r t u r b a t i o n s  T h e question of robustness or structural stability of a model is an important one.  In  order to determine how robust a model is, we need to see whether or not perturbing the model alters its qualitative structure. It turns out (see [124]) that l i m i t point bifurcations, Hopf bifurcations and hysteresis phenomena are stable under small perturbations but transcritical and pitchfork bifurcations are not unless constraints or symmetries are preserved by perturbations.  That is, small perturbations of the model do not affect  whether or not the former three bifurcations occur (they only affect properties such as  Appendix  A. Dynamical  systems theory  256  the parameter value at which the bifurcations occur and the positions of the e q u i l i b r i u m points and periodic orbits) but can affect the occurrence of the latter two. Figure A . 2 9 illustrates the destruction of transcritical and pitchfork bifurcations graphically. It turns out that all bifurcations of one parameter families of equations which have an equilibrium point with a single zero eigenvalue (or single eigenvalue of modulus 1 for maps) can be perturbed to l i m i t point bifurcations [49].  Figure A . 2 9 :  Possible results of p e r t u r b i n g t r a n s c r i t i c a l a n d pitchfork bifurcations (adapted  from  [111], p.83).  However, we cannot make such rapid conclusions when the model contains more than one free parameter.  In these circumstances the idea of codimension of a bifurcation  becomes important.  However, for m > 3 the analysis is very complicated. A n intro-  duction to this theory can be found in [124]. T h e only point I would like to make here is that, because the models in this dissertation have more than three parameters, none of the bifurcations that have been described in this appendix can be dismissed as being unimportant.  Appendix  A.3.7  A. Dynamical  257  systems theory  M u l t i p l e degeneracy  A l l the cases discussed so far have assumed simple zero eigenvalues or a single complex conjugate pair of eigenvalues with zero real part (or the analogous situations for maps). Higher order singularities, or multiple degeneracies do, of course, occur but^the behavioural dynamics associated w i t h them can be difficult to interpret and there is still much research being done i n this area. Thus, at this stage, it is probably best to either solve the system of equations numerically or to generate phase portraits for parameter combinations i n a region surrounding these complex points rather than struggling w i t h the details of the bifurcation structure. For those interested some of these higher order degeneracies are investigated i n [124].  A.4  Conclusion  (  T h e introduction to dynamical systems theory given i n this appendix has been m a i n l y intuitive and not mathematically rigorous.  Although the concepts are fairly simple,  the mathematics involved i n studying a particular system of equations can be quite formidable. A p p e n d i x B describes how computers can be useful i n this regard.  Appendix B Numerical details  B.l  Introduction  T h i s appendix describes some of the computer software that is available for analysing systems of equations. Packages such as A U T 0 8 6 [28], A U T 0 9 4 [31], Interactive A U T O [117], X P P A U T [35] and some others that will be mentioned later, enable particular solutions to be 'continued' as a parameter is varied in order to produce a bifurcation diagram. In other words, these continuation programs trace out the location of equilibrium points and periodic orbits as a parameter is varied.  Bifurcation points that are encountered  along the way are also detected and classified. In this way a whole range of different modes of model behaviour can be obtained with much greater ease than if the mathematics had to be done by hand. The process can be repeated for different parameters i n order to obtain a more comprehensive picture of the qualitative behaviour associated w i t h different regions in the parameter space. Section B.2 introduces some of the techniques used by continuation programs for continuing equilibrium points and determining the stability of the solution branches (see section A . 2 . 4 ) . Methods for detecting bifurcations along these branches are also described w i t h particular reference to A U T O . The section is intended as a brief overview. 1  More  H w i l l use the a b b r e v i a t i o n A U T O to refer to A U T 0 8 6 , A T J T 0 9 4 a n d Interactive A U T O as the l a t t e r is i n essence j u s t a g r a p h i c a l interface for A U T 0 8 6 and A U T 0 9 4 is an u p d a t e d version of A U T 0 8 6 together w i t h a g r a p h i c a l interface. E v e n when using X P P A U T I w i l l refer to A U T O w h e n t a l k i n g a b o u t b i f u r c a t i o n d i a g r a m s since X P P A U T generates these d i a g r a m s t h r o u g h a g r a p h i c a l interface w i t h A U T 0 8 6 . I n cases where I need to d i s t i n g u i s h between the packages, I w i l l use their f u l l n a m e s .  258  Appendix B. Numerical details  259  detailed discussions as well as a comprehensive list of references can be found i n Allgower and Georg [4] and Seydel [111]. Following this theoretical introduction section B.3 describes the capabilities and l i m itations of the packages that I used and section B.4 describes how to obtain t i m e plots, phase portraits and bifurcation diagrams. Finally, section B.5 gives a few pointers and warnings regarding the use of some of the packages. These are based on m y experiences gained through analysing the models in the m a i n body of the thesis.  B.2 B.2.1  Theory Continuation methods  A s has already been mentioned, continuation or path-following methods generate a chain of solutions (equilibrium points, periodic orbits or bifurcation points) as a parameter is varied. A typical path-following method is the predictor-corrector method. T h i s involves the repetition of two different steps. The predictor step approximates the next point on the curve, often by using the direction of the tangent to the curve (Euler predictor) [4]. A number of iterative steps (called corrector steps) then a i m to improve this approximation and bring it back to the actual curve [4]. Typically Newton or gradient type methods are used i n this step [4]. Some f o r m of parameterisation of the curve is required for these steps. T h e obvious choice is the control parameter (that is, the parameter being varied) as it has physical significance. However, this leads to difficulties at l i m i t points (see section A.2.13) [111]. A n alternative is to choose another variable which involves adding another equation to the system. T h i s extended system can then be solved using the predictor-corrector methods mentioned above [111]. Popular choices for this alternative parameter are arclength or a pseudo-arclength parameter proposed by Keller [67]. A U T O uses the latter choice and  Appendix  B. Numerical  details  260  details are given i n [30, 67]. T h e accuracy of the predictor-corrector method depends on the choice of steplength. In general, shorter steplengths lead to greater accuracy (provided that they are not so small that computer round-off errors become large) but they are more costly in terms of time.  In some cases the objective is jusf to follow the curve as rapidly and safely  as possible u n t i l a critical point (such as a bifurcation point) is reached [4] and then a smaller steplength is needed for greater accuracy. Thus, for an efficient algorithm, the steplength needs to be adaptive and not fixed [4, 111]. Ideally a continuation method should also allow the user to have some control over the choice of steplength.  AUTO  fulfills both criteria. Stepsizes are changed automatically i n the program depending on the speed of convergence of Newton's method (that is, depending on the number of iterations required to fulfill the stopping criteria).  M a x i m u m and m i n i m u m stepsizes  and convergence criteria are given by the user. There are many different ways i n which choices of predictor, corrector, parameterisation and step control can be combined to produce a continuation method. Because of this no numerical comparison of different path-following methods has so far been done and, hence, no particular method can be recommended exclusively [111]. Simple E u ler predictors together with Newton-type correctors have been found to be satisfactory in many circumstances [4]. Because of the stability of Newton correctors, more stable higher order predictors based on polynomial interpolation (instead of on the direction of the tangent to the curve as w i t h an Euler predictor) are often advantageous [4]. N o continuation method can guarantee that all possible solutions w i l l be found i n a given example [111].  Isolated branches (branches which are not attached to other  branches v i a bifurcation points) are very difficult to detect. It is suggested i n the A U T 0 8 6 manual [30] and by Seydel [111] that time integration of the governing system of equations using random i n i t i a l data may be worthwhile for generating a starting point on an as yet  Appendix B. Numerical details  261  undetected solution branch. D S T O O L [10] can also be useful for locating e q u i l i b r i u m points which lie on these isolated branches.  B.2.2  Detection of bifurcations  A s a continuation method traces out a path, it needs to be able to detect bifurcation points. Techniques for doing this can be divided into direct and indirect methods [111]. Direct methods involve enlarging the original system of equations by including additional equations which characterise the bifurcation point. Indirect methods on the other hand utilise data obtained during a continuation together with a test function.  T h e latter  are generally recommended i n practical computations (and are used i n A U T O ) as direct methods involve solving much larger systems of equations and have higher storage requirements. Indirect methods do have more difficulty achieving high accuracy but when discretisation errors are present and when bifurcation points are unstable to perturbations, this greater accuracy is not needed [111]. If higher accuracy is required then a direct method can be applied once an indirect method has obtained an approximate result. A n indirect method detects bifurcation points using 'test functions' which are evaluated during branch tracing [111]. A bifurcation point is indicated by a zero of the test function, r.  (In practice, an algorithm checks for a change of sign of r.)  For simple  bifurcation points a natural choice for r is the m a x i m u m of all real parts a ; of the eigenvalues of the Jacobian matrix A (see section A.3.2). However, this choice m a y not be smooth and it does not signal bifurcations in which only unstable branches coalesce [111]. T h i s problem can be overcome by choosing r =  w i t h |afc| =  m i n . . . ,  |a |}. m  T h i s choice also detects Hopf bifurcations. In general, the accuracy of T depends on the accuracy w i t h which A is evaluated. There are many possibilities for the choice of test function.  However, not much  Appendix B. Numerical details  262  attention has been paid to which drawbacks are significant or which test function is best for which purpose [111]. Some authors have studied classes of test functions for various types of singular points (see [4] for references). Once a bifurcation or branch point has been detected, a method needs to be found for switching branches. A l l that is needed is a single point on the new branch as then the continuation method can be used to trace out the rest of the branch.  Predictors  and correctors can again be used for switching branches. Seydel [111] and Allgower and Georg [4] describe a few different approaches and give further references. A U T O uses a method suggested by Keller [67]. Doedel [30] notes that this method performs well i n most applications although difficulties can occur if the angle of intersection of the branches is very small. A t a Hopf bifurcation point periodic orbits are introduced. Once a solution point on the branch of periodic solutions has been located, a continuation procedure can be used to trace out the branch. After imposing an integral condition i n order to fix the phase of the orbit, the continuation procedure becomes a special case of the path-following techniques that have already been discussed. Details of the method as well as further references can be found i n the A U T 0 8 6 manual [30].  B.2.3  Stability  Stationary branches T h e stability of stationary branches (that is, of continuations of equilibrium points) is determined from the real parts of the eigenvalues of the Jacobian m a t r i x A . T h e calculation of eigenvalues (for example, v i a Q R factorisation or L U decomposition methods [4, 111]) can be very time-consuming especially when accurate approximations are required, such as near bifurcation points, and when the system of equations is large.  However, the  Appendix B. Numerical details  263  fact that the eigenvalues play an important role in the detection of bifurcations, as well as i n determining stability, provides justification for using more accurate techniques. A U T O uses the I M S L subroutine E I G R F for computing the eigenvalues of a general real m a t r i x [30]. Interactive A U T O [117] assigns different colours to continuation branches depending on the number of unstable eigenvalues while X P P A U T [35] uses thick lines to indicate stable branches and thin lines to indicate branches having one or more unstable eigenvalues.  Periodic branches It is the values of the Floquet multipliers (see page 248) that determine the stability of periodic orbits.  These multipliers are eigenvalues of a particular m a t r i x and, thus,  an eigenvalue solver is again required. Since Floquet multipliers can be very large or very small i n value, there may be a loss of accuracy when evaluating them numerically. T h i s is especially true for unstable orbits [111] and near orbits of infinite period [30]. In the A U T 0 8 6 manual [30] it is noted that orbits normally retain their accuracy even when the computation of Floquet multipliers (and hence stability determination) breaks down. In X P P A U T and A U T 0 9 4 the routine for calculating Floquet multipliers has been improved and is more accurate than that in A U T 0 8 6 , and hence i n Interactive A U T O . Floquet multipliers are also used for detecting higher order periodic bifurcations such as period-doubling bifurcations or bifurcations to tori.  B.3  Available computer packages  In this section I list the capabilities of a number of computer packages and also list some advantages and disadvantages related to choosing one package over another.  Appendix B. Numerical details  B.3.1  264  AUTO86  Since Interactive A U T O , X P P A U T and A U T 0 9 4 are all based on A U T 0 8 6 [28], their basic capabilities are very similar. Although I have not used A U T 0 8 6 directly (it runs as a batch process), I w i l l describe its capabilities and limitations since most of t h e m apply to the abovementioned packages. I highlight the differences between the packages in subsequent sections.  Capabilities (These are taken directly from the A U T 0 8 6 manual [30].) A U T O can do a l i m i t e d bifurcation analysis of algebraic systems of the f o r m f(x,A)=0,  x,fe7e  (B.6)  m  where A denotes one or more free parameters, and of systems of ordinary differential equations of the form x = f(x,A),  x,fe7t  (B.7)  m  It can also do certain continuation and evolution computations for the diffusive system x  t  = Dx  u u  + f(x,A),  x,fe7l  m  where x = x ( u , £ ) and D denotes a diagonal m a t r i x of diffusion constants.  (B.8)  For the  algebraic system (B.6) A U T O can: • trace out branches of solutions. • locate bifurcation points and compute bifurcation branches. • locate l i m i t points (saddle-node bifurcations) and continue these i n two parameters.  Appendix B. Numerical details  265  • do all of the above for fixed points of the discrete dynamical system x  < + 1  = f(x,,A).  (B.9)  • optimisation: find extrema of an objective function along the solution branches and successively continue such extrema i n more parameters. For the ordinary differential equations (B.7) A U T O can: • compute branches of stable or unstable periodic solutions and Floquet multipliers. Starting data for the computation of periodic orbits are generated automatically at Hopf bifurcation points. • locate l i m i t points, transcritical and pitchfork bifurcations, period-doubling bifurcations and bifurcations to tori along branches of periodic solutions. B r a n c h switching is possible at transcritical, pitchfork and period-doubling bifurcations. • continue Hopf bifurcation points, l i m i t points and orbits of fixed period i n two parameters. • compute curves of solutions to (B.7) on the interval [0,1] subject to general nonlinear boundary or integral conditions. • locate l i m i t points and bifurcation points for such boundary value problems. B r a n c h switching is possible at bifurcation points. Curves of l i m i t points can be computed i n two parameters. For the parabolic system (B.8) A U T O can: • detect bifurcations to wave train solutions of given wave speed from spatially homogeneous solutions.  These are detected as Hopf bifurcations along fixed point  branches of a related system of ordinary differential equations.  Appendix B. Numerical details  266  • trace out the branches of wave solutions to (B.8) and detect bifurcations. T h e wave speed c is fixed but the wave length L w i l l normally vary. • trace out branches of waves of fixed wave length L i n two parameters. If L is large, then one gets a branch of approximate solitary wave solutions. • do time evolution calculations for (B.8) with periodic boundary conditions on [0,L]. In this thesis systems of the form (B.7) and (B.9) are analysed. T h e discretisation used in A U T O to approximate ordinary differential equations and for calculating periodic solutions is orthogonal collocation with 2 , . . . , 7 Gauss collocation points per mesh interval.  The mesh automatically adapts to the solution so that a  measure of the local discretisation error is equi-distributed.  A l s o , the adaptive mesh  guards to some extent against computing spurious solutions. W h e n spurious solutions do occur they are often easy to recognise by the jagged appearance of the solution branch. Some general limitations of A U T 0 8 6 are listed below.  Limitations T h e following difficulties are noted in the manual [30]: • degenerate (multiple) bifurcations cannot be detected i n general. A l s o , bifurcations that are close together may not be noticed when the pseudo-arclength step size is not sufficiently small. • Hopf bifurcation points may go unnoticed if no clear crossing of the imaginary axis takes place. T h i s may happen when other real or complex eigenvalues are near the imaginary axis and when the pseudo-arclength step is large compared to the rate of change of the critical eigenvalue pair. A n often occurring case is a Hopf bifurcation close to a l i m i t point (saddle-node bifurcation).  Appendix  B. Numerical  details  267  • similarly, Hopf bifurcations may go undetected if switching from real to complex conjugate, followed by crossing of the imaginary axis, occurs rapidly w i t h respect to the pseudo-arclength step size. • secondary periodic bifurcations may not be. detected for very similar reasons. • for periodic orbits the numerical output should be checked to make sure that points labelled as bifurcation points, l i m i t points, period-doubling bifurcations or bifurcations to tori have been classified correctly. Some of the above problems may be solved by decreasing the m i n i m u m step size, dsmin, to allow A U T O to take smaller steps. This is particularly helpful when two continuation branches or two bifurcation points are very close together. In the former situation A U T O needs to take small steps to ensure that it does not switch branches during the calculation. Decreasing dsmax may also help as this prevents A U T O from taking large steps and missing important bifurcations. A s noted above A U T O may have some difficulty w i t h identifying bifurcations on periodic orbits.  Problems arise when Floquet multipliers are close to the unit circle.  A l s o , unstable orbits can be difficult to locate, especially when they are close to a stable orbit or equilibrium. Accuracy may be increased by decreasing dsmin or increasing ntst, the number of mesh points used i n the discretisation of the periodic orbits. Some other limitations of A U T O which have resulted from doing bifurcation analyses on a few models are as follows: • transcritical and pitchfork bifurcations cannot be continued i n two parameters because the determinant of the Jacobian matrix A is zero at these bifurcation points. Other continuation and bifurcation packages w i l l have the same l i m i t a t i o n . A possible solution is to do a number of one-parameter continuations w i t h different (fixed)  Appendix  B. Numerical  details  268  values of the second parameter and then to plot the bifurcation points obtained from each continuation i n two-parameter space.  A n approximate curve can be  drawn through these points. This is done i n chapter 2. • error messages from A U T O can be misleading (as w i t h most computer packages!). In many cases the problem is solved by checking the driving program for typing errors and variable or parameter names which begin w i t h letters between h and o as such quantities are assumed to be integers by default.  B.3.2  Interactive A U T O  Capabilities In addition to the capabilities listed for A U T 0 8 6 , Interactive A U T O [117] allows the user to change the program constants interactively and to observe the development of a bifurcation diagram while a calculation is i n progress. T h e corresponding eigenvalues are shown simultaneously i n a separate graphics window. T h e advantages and disadvantages listed below relate mainly to the suitability of this package for analysing ecological models and are included for comparison w i t h the other packages. Since the packages were not set up specifically for the models that are analysed i n this thesis, these comments are not necessarily criticisms.  Advantages • T h i s package has good on-screen graphics. Different colours are used to indicate the number of unstable eigenvalues corresponding to a particular branch. T h e locations of the eigenvalues relative to the real and imaginary axes and the unit circle are also shown. Bifurcation diagrams can be viewed i n three dimensions if desired and the mouse can be used to shift diagrams and to zoom i n and out.  Appendix B. Numerical details  269  Disadvantages • B i f u r c a t i o n diagrams cannot be printed out or saved as postscript files for later printing. • E a c h time the model equations are altered the driving routine needs to be recompiled. • T h e package only runs on Personal Iris and Iris 4 D workstations (instead of on all systems supporting X-windows as for the other packages). • Interactive control of the graphics output is limited. Once a one-parameter bifurcation diagram has been generated it is not possible to view it w i t h a different variable on the y-axis. T h e scales of the axes are also not visible.  B.3.3  X P P A U T  This package (as well as the tutorial) can be obtained v i a anonymous ftp f r o m ftp.math.pitt.edu and is i n the directory pub/hardware.  Capabilities T h e A U T O interface allows most of the capabilities of A U T 0 8 6 to be enjoyed.  How-  ever, the a i m of m a k i n g A U T O easier to use has led to some restrictions which w i l l be mentioned below. T h i s package also solves systems of equations numerically (there is a choice of a l gorithms) and generates time plots and phase portraits (in two and three dimensions).  Appendix B. Numerical details  270  Hardcopies of these'diagrams can be obtained. W i t h reference to phase portraits it is possible to obtain nullclines, arrows indicating the direction of flow, as well as the positions and stabilities of equilibrium points (singular points). Other capabilities include curve-fitting of data, spreadsheet type data m a n i p u l a t i o n and generation of histograms.  Advantages • T i m e plots, phase portraits and bifurcation diagrams can all be generated using the same computer package. • T h e A U T O interface is easy to u s e — t h e number of constants that need to be altered has been reduced. • Hardcopies of bifurcation diagrams can be obtained through saving t h e m as postscript files. The data can also be saved so that it can be read into other graphics packages. • T h e package has good on-screen graphics. T h e development of a bifurcation d i agram can be seen while a calculation is in progress. T h e eigenvalues are shown simultaneously relative to the real and imaginary axes and the unit circle. It is possible to zoom in on specific regions of a diagram and the most recent oneand two-parameter bifurcation diagrams viewed are kept in memory. Once a oneparameter bifurcation diagram has been generated the variable on the y-axis can be altered. It is also possible to plot the period or the frequency of a periodic orbit as a function of the bifurcation parameter. M i n i m a as well as m a x i m a of periodic orbits can be plotted simultaneously. • A U T 0 8 6 ' s Floquet multiplier routine has been improved. • It is possible to start continuations from a numerically calculated periodic orbit.  Appendix B. Numerical details  271  • Three-dimensional phase portraits are possible. • T h e driving program is easy to write for most systems and is compiled automatically when the X P P A U T command is given. • W h e n calculating time plots and phase portraits, data information for auxiliary variables (variables which are subcomponents or composite functions of the state variables) can also be observed. • T h e package runs on any X-windows system as well as on L I N U X . • There is a comprehensive tutorial on the W o r l d W i d e Web to help the user become acquainted with the package. T h e address is: ftp://mthbard. m a t h , pitt.edu/pub/bardware/xpptut/start. h t m l .  Disadvantages • W h e n generating phase portraits only one equilibrium point is located at a time and the i n i t i a l point often has to be quite close by. • T h e A U T O interface is not set up for discrete equations and error tolerances for A U T O cannot be altered.  Also, the automatic detection of l i m i t points cannot  be turned off i n this version of A U T O . This may cause difficulties when calculating periodic orbits as at a l i m i t point of a periodic orbit two Floquet multipliers equal 1 and this affects the convergence properties of the continuation algorithm. Decreasing dsmin and dsmax may help. • There is l i m i t e d control of the appearance of printouts.  T h e stability nature of  a particular branch i n a bifurcation diagram is often obscured when consecutive continuation points are very close together. Dashed lines for unstable branches then  Appendix B. Numerical details  272  appear as solid lines and, instead of individual dots, periodic branches become thick lines. In some cases this can be overcome by increasing the m a x i m u m step size, dsmax. A l s o , the data for a diagram can be saved separately and then read into another graphics package. W h e n two-parameter bifurcation diagrams are printed out, data from the previously generated one-parameter diagram is superimposed.  B.3.4  AUT094  Capabilities This package is an updated version of and graphical interface for A U T 0 8 6 . T h e graphical interface allows program constants to be altered interactively and bifurcation diagrams can be viewed.  Advantages • There is a help menu which describes the functions of the program constants. • Demonstration examples show how the various capabilities of the package can be used. • M o d e l equations can be changed interactively. • T h e package runs on any X-windows system. • It is possible to start a continuation from a numerically calculated periodic orbit. • T h e routine for calculating Floquet multipliers has been improved.  Disadvantages • Printouts of bifurcation diagrams cannot be obtained.  Appendix B. Numerical details  273  • T h e on-screen graphics are more cumbersome to use than those of Interactive A U T O and X P P A U T . Changes to a bifurcation diagram can only be viewed at the end of a calculation using a separate command instead of while the program is running. T h e way i n which bifurcation points are labelled also makes the diagrams difficult to read. T h e disadvantages were the m a i n reason why I d i d not make use of A U T 0 9 4 .  B.3.5  DSTOOL  This package can be obtained v i a anonymous ftp from macomb.tn.cornell.edu  and is i n the pub/dstool directory. Capabilities This package generates time plots and two-dimensional phase portraits for both discrete and continuous systems of equations. It calculates equilibrium points and their stabilities as well as stable and unstable manifolds for saddle points. There is provision for extensions to the package—three-dimensional graphics as well as continuation and bifurcation routines m a y be incorporated i n the near future.  Advantages • A mouse can be used for specifying starting points for time plots and phase portraits.  This is very convenient and speeds up the generation of these diagrams  considerably. • T h e package is good at locating periodic points for discrete models.  Appendix B. Numerical details  274  • Printouts of diagrams can be obtained. • For ordinary differential equation models trajectories can be calculated forwards or backwards. • T h e package runs on any X-windows system as well as on L I N U X .  Disadvantages • Three subroutines need to be modified each time a new model is entered. • T h e package sometimes crashes when calculating equilibrium points for difficult parameter values.  B.3.6  Other packages  Some other packages are available for analysing dynamical systems. Part of the following list can be found i n [4]. 1. A L C O N [27]. This is a continuation method for algebraic equations f ( x , A) =  0.  L i m i t points and simple bifurcations can be computed on demand. 2. B I F P A C K [112]. This is an interactive program for continuation of large systems of nonlinear equations. Bifurcation points are also detected. 3. D Y N A M I C S  [96]. This package iterates maps, solves differential equations, and  plots trajectories. It runs on both I B M P C ' s and U N I X workstations which support X-windows. 4. L O C B I F [68]. This is an interactive program designed for multiparameter bifurcation analysis of equilibrium points, l i m i t cycles and fixed points of maps. A t  Appendix B. Numerical details  275  present it is set up for I B M P C ' s but a U N I X version is in process and is being incorporated into D S T O O L . 5. P A T H [66]. This software package for dynamical systems can apparently handle m u c h larger systems of ordinary differential equations than A U T O . 6. P H A S E R [54]. This package generates phase portraits for continuous and discrete d y n a m i c a l systems. It runs on I B M P C ' s . 7. P I T C O N [100, 101]. This is a Fortran subprogram for continuation of e q u i l i b r i u m points and for detecting l i m i t points. A s has already been mentioned, no comprehensive comparison of different techniques has been done because of the enormity of the task. None of the abovementioned packages analyses a model at the touch of a button and complementary analytical techniques, as well as other numerical techniques, are still needed i n order to obtain a complete understanding of any model. Parameter studies are as much an art as a science [111]. In the next section I explain how to generate time plots, phase portraits and bifurcation diagrams using some of the packages I have mentioned.  B.4  U s i n g the packages  A l t h o u g h this section gives some guidelines for using the various packages, it is important that anyone wishing to use these packages reads the relevant user manuals to find out the exact commands for performing various tasks. M a n y of the manuals have introductory examples for the reader to work through and these are invaluable for getting acquainted w i t h the capabilities of the package. There is a comprehensive tutorial for using X P P A U T on the W o r l d W i d e Web. The address is: ftp://mthbard. m a t h , pitt.edu/pub/bardware/xpptut/start. h t m l .  Appendix B. Numerical details  276  Time plots These can be obtained using either D S T O O L or X P P A U T . T h e user simply chooses time as the variable to be plotted on the x-axis and one of the state variables for the y-axis. After entering i n i t i a l values for the state variables and the time period over which to simulate the model, the 'run' or 'go' command can be chosen. The process may be repeated after changing one or more i n i t i a l values or parameter values. These steps can all be done interactively. B o t h packages have a number of choices for the numerical algorithm that is used to calculate solutions. M i n i m u m and m a x i m u m or fixed stepsizes can be chosen as well as error tolerances. In most cases the default choices are quite adequate. If the mouse is used to choose an i n i t i a l point, then only the i n i t i a l time and the value of the state variable on the y-axis w i l l be altered automatically. Values for the other state variables w i l l remain unchanged unless new values are typed i n . W h e n analysing a discrete system using D S T O O L , changing the time increment to 1 i n the 'orbit' window and using the 'continue' icon allows the user to determine the period of a cycle. The amplitude of the cycle can be deduced by looking at the state variable values i n the 'settings' window that correspond to the m a x i m u m and m i n i m u m of the cycle.  Phase portraits These can again be generated using either D S T O O L or X P P A U T . A l t h o u g h threedimensional portraits are possible i n X P P A U T , I w i l l only discuss two-dimensional portraits as I find them much easier to interpret and, hence, find them more useful in most situations.- (Note that the dimension of the system can be greater than two but the results are projected into a two-dimensional plane.)  Appendix  B. Numerical  details  277  In order to generate a phase portrait one of the state variables needs to be chosen for plotting on the x-axis and another for the y-axis. For systems of dimension greater than two the solution trajectories w i l l be projected into this plane. Initial values can be typed i n manually or the mouse may be used. O n l y the values of the state variables shown on the axes w i l l be altered when using the mouse. T h e user can again choose between the various numerical methods for calculating solution trajectories. In addition to solution trajectories the positions of equilibrium points can be shown in phase diagrams. W h e n asked to find equilibrium points (also called fixed points or singular points) both D S T O O L and X P P A U T automatically calculate the eigenvalues and hence the stability of these points. X P P A U T tends to locate one singular point at a t i m e and often the i n i t i a l point has to be fairly close to the fixed point if the search is to be successful. D S T O O L uses a Monte Carlo technique to generate a specified number (the default is 10) starting points and then searches for fixed points beginning at these starting values. This method is fairly efficient and choosing the 'find' option i n the 'fixed point' window a few times generally locates all the relevant points. If desired, the user can choose the i n i t i a l point from which to begin a calculation instead of using the M o n t e Carlo technique. For discrete models D S T O O L also finds periodic points. For continuous systems both packages also calculate one-dimensional stable and u n stable manifolds associated with saddle points i n the plane. These help delimit domains of attraction. X P P A U T can calculate nullclines (curves showing where each differential equation is equal to zero) and display these i n the phase portrait. E q u i l i b r i u m points are located at intersections of nullclines corresponding to different state variables.  One-parameter bifurcation diagrams For systems of ordinary differential equations one-parameter bifurcation diagrams can be generated using Interactive A U T O , A U T 0 9 4 and X P P A U T . For discrete systems either  Appendix  B. Numerical  details  278  of the former two packages can be used.  The first step is to choose the bifurcation  parameter to be plotted on the x-axis. One of the state variables can be chosen for the y-axis. T h i s choice must be specified in the driving program for Interactive A U T O but can be done interactively i n the other two packages. In addition to choosing the scales for the axes there are a number of other program constants which need to be set.  These govern, for example: the length and type of  continuation; the output to the screen; the detection of l i m i t points; error tolerances, steplengths and mesh intervals for the various numerical routines; and a number of other aspects of the computation. The various possibilities are listed i n the A U T 0 8 6 manual [30] as well as i n the H E L P menu in A U T 0 9 4 . The example or demonstration programs also give an idea of appropriate values. In X P P A U T some of these constants have been preset to simplify the use of A U T O . This is very convenient in most situations but can be restrictive i n others. In order to begin generating a bifurcation diagram a fixed point (equilibrium point), corresponding to a particular parameter set, is required. Such a point can be determined analytically (where possible) or numerically using X P P A U T or D S T O O L . A continuation can then be started from this point.  A t transcritical, pitchfork and period-doubling  bifurcations A U T O automatically calculates the various intersecting branches. A t Hopf bifurcation points the user must specifically initiate the calculation of periodic orbits by choosing the relevant restart label. In Interactive A U T O and A U T 0 9 4 one of the program constants also needs to be altered. In order to extend any continuation branch across a wider range of parameter values, the endpoint of the branch can be chosen as the restart value. For discrete systems period-doubling bifurcations are labelled as Hopf bifurcations by A U T O but the period-2 branches emanating from this bifurcation cannot be calculated directly. T h e second iterate of the model must be used for this purpose. Since a (period-1)  Appendix  B. Numerical  details  279  e q u i l i b r i u m point of the original model is also an equilibrium point of the second iterate of the model, both the period-1 and the period-2 equilibrium point branches w i l l be traced out using this latter model. However, for higher order period-doublings higher order iterates of the model are required and the process is more tedious. In such cases as well as for complicated discrete models whose second iterate is difficult to calculate, D S T O O L can be used to generate an approximate bifurcation diagram. E q u i l i b r i u m points can be calculated at regular intervals across a range of parameter values and their coordinates recorded.  These points can then be plotted using some  other graphics package, such as G N U P L O T [125], to obtain an approximate bifurcation diagram. T h i s approach is used i n chapter 7.  Two-parameter bifurcation diagrams Once a l i m i t point or Hopf bifurcation point has been detected in a one-parameter continuation, it can be continued i n a second parameter.  This is done by choosing the  appropriate restart value corresponding to the bifurcation point, a second parameter for the y-axis, and either altering the relevant program constant i n Interactive A U T O and A U T 0 9 4 or choosing the two-parameter option i n X P P A U T . L i m i t cycles of fixed period can also be continued in two parameters. T h i s is useful for approximating homoclinic or heteroclinic bifurcation curves since homoclinic and heteroclinic orbits have infinite period. Designating a U S Z R function to locate an orbit of high period, say period=100 or 1000, when constructing a one-parameter bifurcation diagram allows a two-parameter continuation of this orbit to be done . T h e resulting 2  two-parameter curve gives the required approximation to the curve of homoclinic or heteroclinic bifurcation points. This technique is used i n section 2.4. W h e n continuing 2  A p o i n t satisfying the U S Z R f u n c t i o n w i l l be given a l a b e l w h i c h can be used as a restart value for  the t w o - p a r a m e t e r c o n t i n u a t i o n .  Appendix  B. Numerical  details  280  these curves i n two parameters, detection of l i m i t points should be turned off as it may lead to spurious bifurcation points. This is done automatically i n X P P A U T . A n y bifurcation points should also be checked against the numerical output to see whether an eigenvalue or Floquet multiplier has, i n fact, changed sign.  B.5  Pointers and warnings  As with a l l computer packages, the ones that I have discussed do not work exactly as one might wish i n a given situation. E a c h model is unique and requires a slightly different approach whereas a computer package is built for more general use. In this section I list some problems and suggestions arising from m y experiences w i t h the packages. M a n y of these w i l l only make sense once the examples i n the m a i n chapters of the thesis have been studied, but I have included them here so that most of the computer technicalities are confined to one place for ease of reference.  • W h e n calculating fixed points using D S T O O L , the package sometimes hangs and w i l l not respond to input.  I have not been able to locate the cause of this but  quitting D S T O O L and restarting the package, although frustrating, does solve the problem. • W h e n starting a fixed point continuation i n A U T O , the error tolerances cannot be set lower than the accuracy of the state variables that have been given as the starting point. If the starting points are only known to low accuracy, then a short continuation can be done on low accuracy (high error tolerance). T h e continuation can then be restarted at higher accuracy using the values calculated by A U T O . (Accuracy is increased by decreasing dsmax, the m a x i m u m step length, decreasing the error tolerances, or increasing ntst, the number of discretisation points.)  Appendix  B. Numerical  details  281  • A U T O w i l l generate bifurcation diagrams faster if the accuracy is lower. In many cases the results are still sufficiently accurate, however some bifurcation points may need to be checked w i t h greater accuracy calculations. It is a good idea to check the eigenvalues and Floquet multipliers in the numerical output files created by A U T O to see whether a bifurcation point has been correctly labelled.  A l s o , if  there is a sudden j u m p in a continuation, or a curve becomes very jagged, then the continuation should be repeated using greater accuracy. • T h e choice of step size depends on the extent of the parameter range over which changes i n behaviour occur. In most cases ds = 0.02 and dsmax = 0.05 are good choices.  However, I used ds = 0.0001 and dsmax  between 0.001 and 0.01 for  the population genetics models as the bifurcations take place w i t h i n fairly small parameter ranges. Scaling the equations and parameters (as is done i n chapters 3 and 4) circumvents this problem. • Bifurcation diagrams can become very complicated even when studying simple models. However, not all continuation branches are always of interest. For example, some may correspond to negative or zero values of the state variables. A l s o , when studying a particular aspect of a model, some branches may be superfluous. It is important to look at the numerical output which is printed to the screen during a continuation so that one can keep track of which branches are relevant and which are not. • It is a good idea to generate a number of starting points at a variety of parameter values to ensure that a complete bifurcation diagram is obtained and that isolated branches have not been overlooked. D S T O O L is convenient for this purpose.  Appendix  B. Numerical  details  282  • Results can often be checked using more than one package. For example, a b i furcation diagram generated by A U T O can be checked by choosing a number of parameter combinations corresponding to different qualitative regions and generating phase portraits for these combinations (using X P P A U T or D S T O O L ) to check the dynamics.  This is most convenient i n X P P A U T where the bifurcation dia-  grams and phase portraits can be generated within the same package. T i m e plots and phase portraits can be checked by choosing different numerical methods to calculate solutions. B o t h D S T O O L and X P P A U T offer a variety of methods.  By  looking at the eigenvalues corresponding to an equilibrium point one can check that the stability of the point has been correctly labelled. • It is easy to get side-tracked into studying the bifurcation structure of a model and into studying complicated behavioural changes instead of concentrating on phenomena which are of biological interest. In general, sharp boundaries and the exact values at which bifurcations occur are not important as biological parameters are hardly ever known w i t h certainty.  T h e general behaviour and the types of  changes that can occur are of greater practical interest. For example, l i m i t cycles of small amplitude w i l l be indistinguishable from equilibrium points due to the natural variation of field data.  Also, a system w i l l easily be perturbed from a  stable node w i t h a small domain of attraction and, hence, such a phenomenon may be of minor interest. It is also important to look at time plots corresponding to fixed parameter combinations to see how quickly a system approaches the l i m i t i n g behaviour indicated i n a bifurcation diagram. If a system takes a long t i m e to attain its equilibrium configuration then the transient dynamics may be more important than the final equilibrium values.  Appendix  B. Numerical  details  283  • It is sometimes enlightening to generate bifurcation diagrams for a larger range of parameter values than is of direct interest as there may be hysteresis phenomena or bifurcation points outside the range of interest which affect the behaviour inside this range. Other suggestions which are best described with reference to a particular model are included i n the relevant chapters.  Clearly, the more one utilises the techniques and  packages that have been introduced, the more familiar one becomes w i t h t h e m , and the more creative one can be in their use. The above comments w i l l be most useful to those researchers actively involved in analysing their own models.  Appendix C  Mathematical details for the sheep-hyrax-lynx model  C.l  Modelling delays in system dynamics models  A system does not always respond immediately to a change i n one of its components. Often there is a time lag between the i n i t i a l change and the response of the system. For example, a change i n hyrax population density w i l l only affect the size of the hyrax population the following season. It takes time for an increase i n population density to affect the reproductive success of the hyrax and the survival of their offspring. In order to represent this i n the model, averaged or smoothed versions of certain quantities are used i n calculating growth rates. Three quantities are averaged i n this m o d e l — h y r a x density (Hp), prey abundance ( A p ) , and the grazing multiplier (GM)- For example, when sheep have been grazing more than their usual amount their condition is expected to improve resulting i n a decline i n juvenile mortality and a rise i n fecundity (Swart and Hearne [116]). However, this will only occur after a prolonged increase i n the average amount of pasture consumed. Hence, sheep fecundity and juvenile mortality are functions of the grazing multiplier average, GM, instead of GM- A first order distributed delay (see M a c D o n a l d [77] or M a y [81]) is used to calculate GM- T h a t is, <1GM _ GM — GM  dt  tdei  where t^ei is the average delay time. Examples of this type of delay equation can be found i n Forrester [39] and a detailed explanation of the mathematics underlying the above differential equation can be found i n M a c D o n a l d [77]. T h e above ordinary differential 284  Appendix  C. Mathematical  details for the sheep-hyrax-lynx  model  285  equation is added to the original system of model equations thus increasing the dimension of the system to be solved. Fortunately this increase i n dimension does not pose a problem when solving the system numerically. Further discussions on delays i n biological systems and their effects on system behaviour can be found i n [78, 81]. For the sheep-hyrax-lynx model the delays were not found to affect stability even for large average delay times.  C.2  Rescaling model equations  Suppose the differential equation for the state variable u,- is given by dvi  Let  Vi = S j U j ,  then Vi = — Si  and 1  dvi dt  dvi  Si dt =  fi(siVi,  • • • , SiVi, . . . ,  S V ). m  m  Si  Dropping the bars for convenience we get  ^1 - Lf.( . dt  —  Ji\S^V\,  \ . . . , SiVi, . . . , S V m  m  I  Si  as required. T h e state variables have been replaced by s,-u,- and the differential equation for Vi is divided through by s,- as stated i n section 3.4.  Appendix D  Mathematical details for the budworm-forest model  D.l  Derivation of new foliage equation in spruce budworm model  We expect the amount of new foliage consumed by an individual larva to depend on the availability of new foliage per larva, that is, to depend on ^ Fb  represents the  (Fi/Kp  F  L  amount of foliage that is new foliage). T h e larvae prefer new foliage to old and thus, i n most circumstances, w i l l eat all the new foliage before moving on to old foliage. A s s u m i n g that larval densities do not get so low that there is an overabundance of new foliage, we have new foliage consumption/larva = — ^ — — .  Lb However, when larval densities are high, competition among budworm for new foliage becomes significant and larvae eat old foliage more readily than before. W e can model this competition (see Starfield and Bleloch [113]) by including the factor  L V  dL \ Fb/K ) 0  b  F  where do is the m a x i m u m foliage consumption rate per larva. T h i s factor is close to 1 when there is abundant new foliage per larva (that is, when  F  b  ^  F  is large) and close to  0 when there is an overabundance of larvae resulting i n intraspecific competition. T h e new equation is n  .  ..  F  n  b  new foliage consumption/larva = — — —  (  1  KpLb V 286  doKpL \ —  Fb  h  .  Appendix  D. Mathematical  details for the budworm-forest  287  model  T h i s factor is rather severe and could lead to negative values for very large larval densities.  Instead we can use a negative exponential function [113] which approaches  zero when there is very little new foliage available per larva. T h i s gives Ff,  (  d  new foliage consumption/larva = — — — I 1 — e KFLI,  O  K  F  >>  L  h  F  V  T h e total amount of new foliage consumed is then /„  Fb  new foliage consumption  KL F  d  1 - e  O F b b K  L  F  h  and this gives  remaining new foliage  =  —r  ;T~~(1 ~  KF  KF  e  A  )  -A b F  where  _  D.2  d^KpLb  Summary of model equations  T h e subscript b denotes the i n i t i a l or base value for the state variable and subscript e denotes the new value after one year.  Foliage (F)  F,  ^  where  Fi  =  -^[e-  A  KF  +  (K -l)e- }F , B  F  b  Appendix  D. Mathematical  details for the budworm-forest  A  —  model  K  F  doL  t  (A-l  B  + e- ), A  Cl  cj = 0.357.  B r a n c h surface area (S)  rsSi  S  e  where Si = [1 - d (l s  -  ^) }S . 2  b  •Tb  B u d w o r m (L)  L  — dsh-rrG Ks  2  e  L  &  where  G  =  H  =  Le  =  H(2-H), Fi  -n. A  I  1  -  1 1  , Pm. + E< A L\ r  ) S> L  sr  E =  u =  Pm  (EiW  Athr 1/s  -  E )A L , 2  sr  4  Ei = 165.64 and E = 328.52, 2  W AFI  — Api (1 = 34.1, A  + A F2  F 2  ( / ^ - 1) (1 F  = 24.9 and B  F  = -3.4,  Appendix  D.  Mathematical  details for the budworm-forest mo>  L  4  =  (A + p  3  =  p  3  = 0.473 and B  A L  B -^-)L ,  p  p  e~ L , D  2  S (PsatF b  k  L  Li  =  = 0.828,  (1 -  + LIY  2 b  = 0.425, qmaxe~ )L , C  b  C = 0.003L . b  Reference values for the parameters are given in table 7.1.  Listings Examples of the calling programs for the various models are listed here. T h e entire calling program does not need to be rewritten each time. Once one file has been created, only those lines which define the model equations and set the parameter values and program constants need to be altered.  Bazykin model with prey competition—XPPAUT # M o d e l equations. dx/dt=a*x-b*x*y/(l+alp*x)-eps*x*x dy/dt=-c*y+d*x*y/(l+alp*x) # Initial values. x(0)=0.2 y(0)=0.2 # Parameters and nominal values. param a=0.6,b=0.3,c=0.4,d=0.2,alp=0.3,eps=0.001 done  290  System dynamics m o d e l — X P P A U T # Sheep, hyrax, l y n x and pasture model w i t h pasture l i m i t i n g function.  # # Differential equations for the state variables. dpas/dt=(temp*ppn*area*plm(pai)-tssu*gn*gm(pai))/pmax dhj/dt=(hfmax*hf*hfn*hdfm(hda)-hjmax*hj*hjdn*hjdm(hda)-ahjdm(pai) -hjmax*hj*hjmn-hp*hjmax*hj/(hjmax*hj-r-hfmax*hf-|-hmmax*hm))/hjmax dhf/dt=(0.5*hjmax* hj*hjmn-hfmax*hf*hfdn-hfdm(pai)-hfmax*hf*hcn -hp*hfmax*hf/(hjmax*hj-(-hfmax*hf-|-hmmax*hm))/hfmax dhm/dt=(0.5*hjmax*hj*hjmn-hmmax*hm*hmdn-hmdm(pai)-hmmax*hm*hcn -hp*hmmax*hm/(hjmax*hj+hfmax*hf-(-hmmax*hm))/hmmax dlj/dt=(lfmax*lf*lfn*lfm(paa)-ljmax*lj*ljdn*ljdm(paa)-ljmax*lj*ljmn)/ljmax dlf/dt=(0.5*lj*ljmax*ljmn-lfmax*lf*lfdn-lfmax*lf*lcn)/lfmax dlm/dt=(0.5*ljmax*lj*ljmn-lmrnax*lm*lmdn-lmmax*lm*lcn)/lmmax dsj/dt=^(sfmax*sf*sfn*sdfm(gma)-sjmax*sj*sjdn*sjdm(gma)-sjmax*sj*sjmnmax(0,l-lpm(pa))*lut*sjpn-sjmax*sj*sjcn*sjcm)/sjmax dsf/dt=(0.5*sjmax*sj*sjmn-sfmax*sf*sfdn-sfmax*sf*sfcn)/sfmax dsm/dt=(0.5*sjmax*sj*sjmn-smmax*sm*smdn-smmax*sm*smcn*smcm)/smmax dhda/dt=(hd-hda)/dell dpaa/dt=(pa-paa) / del2 dgma/dt=(gm(pai)-gma)/del3 d t r / d t = ( - t r - f (mutwool-culling-cssu*ssu+ssu*ssuval)/trmax) / tau  # # State variables and i n i t i a l values. pas(0)=0.66,hj(0)=0.7,hf(0)=0.525,hm(0)=0.525 lj(0)=0.4,lf(0)=0.6,lm(0)=0.6,sj(0)=0.80670,sf(0)=0.75567,sm(0)=0.50379 hda(0)=1.0,paa(0)=1.0,gma(0)=1.0,tr(0)=3.9069  # # F i x e d variables—these are quantities which are used repeatedly # i n the model equations. M a k i n g them fixed variables simplifies # the appearance of the calculations considerably. hut=hjmax*hjr*hj+hfmax*hf+hmmax*hm lut=ljr*ljmax*lj-|-lmmax*lm+lfmax*lf ssu=sjr*sjmax*sj+sfmax*sf+smmax*sm hd=hut/hun pa=(hut/lut)/(hun/lun) 291  pai=pmax*pas/pav tssu=ssu+max(0,(hut-hun)/hs) hp=lut*lpn*lpm(pa) sjcm=(2.01-argf(1.01,0.01,1.2,gma))/2 smcm=2.01-argf(1.01,0.01,1.2,gma)  # # F i x e d variables needed to define revenue (mutton and wool sales and cost # of culling). mutwool=sjmv*sjmax*sj*sjcn*sjcm+sfmv*sfmax*sf*sfcn-|-smmv*smmax *sm*smcn*smcm-|-sjwv*sjmax*sj+sfwv*sfmax*sf-f-smwv*smmax*sm culling=ccl*(lfmax*lf*lcn+lmmax*lm*lcn)+cch*(hfmax*hf*hcn+hmmax*hm*  # # A u x i l i a r y variables—these are quantities other than the state # variables whose values we would like to appear in X P P A U T ' s # data window. Here we have revenue and total revenue. # Total revenue is the quantity referred to in the analysis as revenue, aux rev=mutwool-culling-cssu*ssu aux totrev=mutwool-culling-cssu*ssu+ssu*ssuval  #  .  .  .  # In order to view the values of the fixed variables i n the data window # we need to have the following statements: fhut=hut flut=lut fssu=ssu fhd=hd fpa=pa fpai=pai ftssu=tssu fhp=hp fsjcm=sjcm fsmcm=smcm  # # Parameters and nominal values. param hjr=0.5,ljr=0.5,sjr=0.67,hun=700000,lun=:700,pav=6.6e7 param hs=18,ppn=332,area=200000,gn=365,sub=0.000005,temp=1.0 param hfn=1.5,hjdn=0.5,hfdn=0.1,hmdn=0.1,hjmn=1.0,lpn=84.11,hcn=0.0 param lfn=0.7,ljdn=0.5,lfdn=0.13,lmdn=0.13,ljmn=1.0,lcn=0.0 292  param param param param param param param  #  sfn=0 .75,sjdn=0.1,sfdn=0.02,smdn=0.02,sjmn=0.5 sjpn=90.0,sjcn=0.09,sfcn=0.28,smcn=0.28 sjmv=55.0,sfmv=79.0,smmv=75.0,sjwv=5.0,sfwv=7.0,smwv=9.0 ccl=30.0,cch=1.0,cssu=2.0,ssuval=200.0,pmax=1.0e8 hjmax=500000,hfmax=500000,hmmax=500000,ljmax=500,lfmax=500 lmmax=500,sjmax=100000,sfmax=100000,smmax=100000 dell=0.9,del2=0.6,del3=0.9,tau=0.05,trmax=1.0e7  # User functions—these are the multiplier functions. argE(A,B)=(A/B)-l argC(A,B)=ln(argE(A,B)/(A-l)) argr(A,B,slope)=slope*A/((A-l)*argC(A,B)) argf(A,B,slope,x)=A/(l+argE(A,B)*exp(-argC(A,B) *exp(argr(A,B,slope)*ln(max(x,sub))))) plm(x)=3.25*(x+0.001)*exp(-1.2*(x+0.001)) gm(x)=argf(1.5,0.1,0.8,x) hdfm(x)=2.5-argf(2.4,0.1,1.3,x+0.4) hjdm(x)=argf(1.8,0.1,1.0,x) lpm(x)=argf(1.6,0.8,0.5,x) lfm(x)=argf(1.5,0.01,0.7,x) ljdm(x)=2.0-argf(1.9,0.05,l.l,x+0.2) sdfm(x)=argf(2.8,0.4,1.3,x) sjdm(x)=12.0-argf(11.5,0.1,2.7,x+1.6) ahjdm(x)=max(0,exp(-3*x)*(hut-hun)*hj*hjmax/(hjmax*hj-|-hfmax*hf-|-hmmax*hm)) hfdm(x)=max(0,exp(-3*x)*(hut-hun)*hfmax*hf/(hjmax*hj+hfmax*hf-t-hmmax*hm)) hmdm(x)=max(0,exp(-3*x)*(hut-hun)*hmmax*hm/(hjmax*hj-|-hfmax*hf-|-hmmax*hm))  # done  293  Ratio-dependent m o d e l — X P P A U T # M o d e l equations. dMl/dt=gaml*(l-exp(-omegal/Ml)-MlAbl)*Ml(l-exp(-omega2*Ml/M2))*phi2*M2/omega2 dM2/dt=gam2*(l-exp(-omega2*Ml/M2)-M2Ab2)*M2(l-exp(-omega3*M2 / M3)) *phi3*M3/omega3 dM3/dt=gam3*(l-exp(-omega3*M2/M3)-M3Ab3)*M3 # Initial values. Ml(0)=0.8 M2(0)=0.2 M3(0)=0.2 # Nondimensionalised parameters and values, param omegal=14.0,omega2=20.0,omega3=16.67 param gaml=0.65,gam2=0.4,gam3=0.4 param phi2=0.07,phi3=0.06 param bl=0.02,b2=0.02,b3=0.0 done  294  Population genetics model I—Interactive A U T O . c Note: lines beginning with c are comments, c c Population genetics model with exponential fitness functions, c PROGRAM AUTO c IMPLICIT D O U B L E PRECISION (A-H,0-Z) c c N O T E : parameters l i w and lw are V E R Y I M P O R T A N T . T h e y set aside c space for A U T O i n the work arrays I W and W . In Interactive A U T O c they must be "hardwired" into the code. If you begin to have problems c w i t h large continuations (e.g. periodic solutions using a big N T S T ) , t r y c setting l i w and lw larger and recompiling the executable, c P A R A M E T E R ( l i w = 10000) P A R A M E T E R ( l w = 250000) dimension I W ( l i w ) , W ( l w ) . dimension ipar(50),rpar(50),icp(20) character*10 params(20) character*50 name c call dfinit c c N D I M (number of state variables) ipar( 1)=2 c I P S (+1 for ode's, - 1 for maps) ipar( 2 ) = - l c IRS ipar( 3)=0 c ILP ipar( 4)=0 c NTST ipar( 5)=15 c NCOL ipar( 6)=4 c IAD 295  ipar( 7 =3 ISP ipar( 8^ =1 ISW ipar( 9^ =1 IPLT ipar(10 )=0 NBC i p a r ( l T =0 NINT ipar(12} =0 IADS ipar(13 =1 NMX ipar(14^ =100 NUZR ipar(15^ =0 NPR ipar(16} =50 MXBF ipar(17) =5 IID ipar(18) =2 ITMX ipar(19) =8 ITNW ipar(20) =5 NWTN ipar(21) =3 JAC ipar(22) =0 ICP(i) icp(l) = 1 icp(2) = 2 D O 1 1==1,2 ipar(30+i)=icp(  1 CONTINUE DS r p a r ( l ) = :0.0001d0 DSMIN rpar(2)=:0.000002d0 DSMAX rpar(3)=:0.01d0 RLO rpar(4)==0.0 RL1 rpar(5) =4 . 0 AO rpar(6) = -10.0 Al rpar(7) = 250.0 EPSS rpar(8) == l . d - 6 E P S L ( i ) , i =1,20 rpar(9) == l . d - 6 EPSU rpar(10) = l . d - 6 nparams=number of parameters that you want to vary nparams = 6 declaration of parameter names params(l) = ' a l l ' params(2) = ' a l 2 ' params(3) = 'a22' params(4) = ' b l l ' params(5) = ' b l 2 ' params(6) = 'b22' name = ' Genetics model ' call autool(ipar,rpar,iw,liw,w,lw,params,nparams,name) stop end 297  c SUBROUTINE  VECFLD(ndim,u,icp,par,ijac,f,t)  c c T h i s subroutine evaluates the right hand side of the first order system c c input parameters : c n d i m - dimension of u and f. c u - vector containing u. c par - array of parameters i n the differential equations. c icp - par(icp(l)) is the i n i t i a l 'free' parameter. c par(icp(2)) is a secondary 'free' parameter, c for subsequent 2-parameter continuations. c ijac - =1 if the jacobians dfdu and dfdp are to be returned, c =0 if only f(u,par) is to be returned in this call. c t - current time. c c value to be returned : c f - f(u,par) the right hand side of the ode. c i m p l i c i t double precision (a-h,o-z) c dimension u(ndim),par(30) dimension f(ndim) c Parameters a l l = par(l) a l 2 = par(2) a22 = par(3) b l l = par(4) b l 2 = par(5) b22 = par(6) c State variables P = u(l) e N = u(2) c Fitness functions w l l = exp(all-bll*eN) w l 2 = exp(al2-bl2*eN) w22 = exp(a22-b22*eN) 298  c M e a n fitnesses wlmarg = P*wll+(1-P)*wl2 w2marg = P*wl2+(1-P)*w22 fitmean = P * w l m a r g - f ( l - P ) * w 2 m a r g c c DIFFERENCE EQUATIONS c f ( l ) = P*wlmarg/fitmean f(2)= fitmean*eN c return end c S U B R O U T I N E PARDER(ndim,u,icp,par,ijac,dfdu,t) c c this subroutine evaluates the derivatives c of the first order system and with respect to (u(l),u(2)). c Not included for this model hence ijac=0 in the first subroutine. c return end c S U B R O U T I N E DFDPAR(ndim,u,icp,par,ijac,dfdp,t) c c this subroutine evaluates the derivatives c of the first order system and with respect to free parameters. c Not included for this model. c return end c S U B R O U T I N E STPNT(ndim,u,par) c c i n this subroutine the steady state starting point must be defined, c (used when not restarting from a previously computed solution), c the problem parameters (par) may be initialized here or else in init. c 299  c c c c c c  n d i m - dimension of the system of equations, u - vector of dimension n d i m . upon return u should contain a steady state solution corresponding to the values assigned to par. par - array of parameters i n the differential equations. i m p l i c i t double precision (a-h,o-z)  c dimension u(ndim),par(30) c c initialize the problem parameters. par(l)=2.1d0 par(2)=1.9d0 par(3)=l.ld0 par(4)=1.0d0 par(5)=0.904d0 par(6)=0.524d0 par(14) = D B L E ( 1 ) c initialize the steady state. u(l)=0.5d0 u(2)=2.1008d0 c return end c S U B R O U T I N E S P R O J (ndim, u, isw, icp, par, vaxis, pt) c c this subroutine can be used to define a special projection c i n the bifurcation window. This subroutine is called c when the ' S P ' is toggled O N (issue the command successively c to t u r n the toggle from O N to O F F , and vice versa). c c input values: c n d i m - dimension of u. c u - vector containing coordinates of current solution. c isw - the number of parameters being used i n the current c continuation. 300  icp - p a r ( i c p ( l ) ) is the i n i t i a l 'free' parameter. par(icp(2)) is a secondary 'free' parameter, for subsequent 2-parameter continuations. par - array of parameters in the differential equations. vaxis - controlled by program constant I P L T (see A U T 0 8 6 User Manual), this is the second number per line written i n unit 7 (file fort.7). return values: pt - array whose 1st, 2nd and 3rd elements are plotted the x, y and z axes, respectively. i m p l i c i t double precision (a-h,o-z) dimension icp(20) dimension u(ndim), par(30), pt(3) if (isw.eq.l) then p t ( l ) = par( 1) pt(2) = u(2) pt(3) = vaxis else if (isw.eq.2) then p t ( l ) = par( 1 ) pt(2) = par( 2 ) pt(3) = vaxis endif return end S U B R O U T I N E S P J A X S (ndim, isw, icp, axes ) this subroutine defines the names of the axes used i n the projection defined in the subroutine sproj. input value: n d i m - dimension of u. 301  c I A B S ( i s w ) - the number of parameters being used in the current c continuation. c icp - p a r ( i c p ( l ) ) is the i n i t i a l 'free' parameter, c par(icp(2)) is a secondary 'free' parameter, c for subsequent 2-parameter continuations, c return value: c axes - character string array w i t h the x, y, and c z axes names, respectively. c integer*4 n d i m , isw, icp(20) character* 10 axes(3) c if (isw.eq.l) then axes(l) = ' i c p ( l ) ' axes(2) = ' N ' axes(3) = ' ' else if (isw.eq.2) then axes(l) = ' i c p ( l ) ' axes(2) = 'icp(2)' axes(3) = ' ' endif c return end c c*** graphics initializations for interactive A U T O c S U B R O U T I N E G P H D F T ( ldebug, lintog, labtog, ldsplt, @ leigen, lfltog, lsavpt, @ lgraph, lvideo, lsproj, nproj, @ n d m p l t , delay, sclbif, scldis, @ sclev, filext ) c logical ldebug, lintog, labtog, ldsplt, leigen, lfltog logical lsavpt, lgraph, lvideo, lsproj integer*4 nproj, ndmplt real*8 delay 302  real*8 sclbif(6), scldis(6), sclev(6) character*10 filext(2) c c*** toggles c ldebug: T = > debugging output c lintog: T = > plotting of lines between points c labtog: T = > plots two-character identifier at bifurcation points c ldsplt: T = > open optional U N I X G r a p h window c leigen: T = > open eigenvalue plotting window c lfltog: T = > illuminates points temporarily as they are plotted c lgraph: T = > use graphics (F = > can then run jobs i n the background) c lvideo: T = > reposition windows in botton 1/4 of screen for videotaping c lsproj: T = > plot special projection denned i n subroutine sproj c i n the bifurcation window c lsavpt: T = > eigenvalues saved i n fort. 11 ('svaut *' moves this to m.*) c D O N O T A L T E R the following lines. ldebug = .false. lintog = .false. labtog = .true. ldsplt = .false. leigen = .true. lfltog = .false. , lcomfl = .false. loutfl = .false. lgraph = .true. lvideo = .false. lsproj = .false. lsavpt = .false. c c*** default window scales c ... Bifurcation window scales c eg. when isw = 1 = > default plot axes x,y,z = p a r ( i c p ( l ) ) , u ( l ) , u(2); c when isw = 2 = > default plot axes x,y,z = p a r ( i c p ( l ) ) , par(icp(2)), u ( l ) c T h e following lines can also be changed interactively. sclbif(l) = 0.5d0 sclbif(2) = 0.56d0 sclbif(3) = O.OdO 303  sclbif(4) = l.OdO sclbif(5) = O.OdO sclbif(6) = lO.OdO c ... Eigenvalue window scales sclev(l) = -2.0d0 sclev(2) = 2.0d0 sclev(3) = -2.0d0 sclev(4) = 2.0d0 c c*** set files names c comfil = 'input ' outfil = 'output ' c c*** file strings for saving, deleting, etc. c fTlext(l) = 'gen ' filext(2) = 'gen2 ' c c*** other stuff c nproj = number of projections to be plotted (up to nine) c n d m p l t = dimension of the bifurcation window plot c delay = factor for duration of flash display c nproj = 2 ndmplt = 3 delay = O.OdOO c return end  304  Population genetics model I — D S T O O L . # include <model_headers.h> /* Note: T h e symbols /* and */ denote the beginning and end of comments respectively. */  / * -  ;  Required function used to define the vector field or map. T h e values of the vector field mapping at point x with parameter values p are returned i n the pre-allocated array f. For vector fields, the last components of both f and x are time components. A l l arrays are indexed starting from 0.  :  :  */  int genetics_def(f,x,p) double *f,*x,*p;  { double a l l , a l 2 , a 2 2 , b l l , b l 2 , b 2 2 , P , N ; double w l l , w l 2 , w 2 2 , m a r g w l , m a r g w 2 , m e a n f i t ; /* Parameters whose values can be changed interactively. */ a l l = p[0]; al2 = p[lj; a22 = p[2]; b l l = p[3]; b l 2 = p[4]; b22 = p[5];  /* STATE VARIABLES */ P = [ 0 ] ; /* Frequency of allele A l */ N = x[l]; /* Population density */ x  /* FITNESS FUNCTIONS */ w l l = exp(all-bll*N); w l 2 = exp(al2-bl2*N); w22 = exp(a22-b22*N);  /*  305  M A R G I N A L FITNESS FUNCTIONS */ margwl = P*wll+(1-P)*wl2; margw2 = P*wl2+(1-P)*w22; meanfit = P*margwl-|-(1-P)*margw2;  /* DIFFERENCE EQUATIONS */ f[0] = P*margwl/meanfit; f[l] = meanfit*N; } /* E n d of model equations. */ /*; ; — O p t i o n a l function used to define the Jacobian m at point x w i t h parameters p. The m a t r i x m is pre-allocated (by the routine d m a t r i x ) ; A t exit, m[i][j] is to be the partial derivative of the i'th component of f w i t h respect to the j ' t h component of x.  =  */  /* int user_jac(m,x,p) double * * m , *x, *p;  { }  */  /*  ;  ;  O p t i o n a l function used to define the inverse or approximate inverse y at the point x w i t h parameter values p. The array y is pre-allocated.  r  :  */  int user_inv(y,x,p) double *y,*x,*p; { }  */ /*  O p t i o n a l function used to define aux functions f of the variables x and parameters p. The array f is pre-allocated. T i m e is available as the 306  last component of x.  /* int user_aux_func(f,x,p) double *f,*x,*p;  { } */ /* Required procedure to define default data for the dynamical system. N O T E : Y o u may change the entries for each variable but P L E A S E D O N O T change the list of items. If a variable is unused, N U L L or zero the entry, as appropriate. : — :  */  int genetics_init() { /* define the dynamical system i n this segment */ int n_varb=2; /* d i m of phase space */ static char *variable_names[]={"P","N".}; /* list of phase varb names */ static double variables[]={0.,0.}; /* default varb i n i t i a l values */ static double variable_min[]=={0.,0.}; /* default varb m i n for display */ static double variable_max[]={l.,5.}; /* default varb max for display */ static char *indep_varb_name="time"; /* name of indep variable */ double indep_varb_min=0.; /* default indep varb m i n for display */ double indep_varb_max= 100.; /* default indep varb m a x for display */ int n_param=6; /* d i m of parameter space */ static char * p a r a m e t e r _ n a m e s [ ] = { " a l l " , " a l 2 " , " a 2 2 " , " b l l " , " b l 2 " , " b 2 2 " } ; /* list of p a r a m names */ static double parameters[] = {2.1,1.9,1.1,1.0,0.904,0.56}; /* i n i t i a l parameter values */ static double parameter_min[]={0.,0.,0.,0.,0.,0.}; /* default p a r a m m i n for display */ static double parameter_max[]={2.,2.,2.,3.,3.,3.}; /* default param m a x for display */  307  int n_funct=0; /* number of user-defined functions */ static char *funct.names[] = {""}; /* list of funct names; {""} if none */ static double funct_min[] = {0.}; /* default funct m i n for display */ static double funct_max[] = {0.}; /* default funct max for display */ int m a n i f o l d _ t y p e = E U C L I D E A N ; /* P E R I O D I C (a periodic varb) or E U C L I D E A N */ static int periodic_varb[] = { F A L S E , F A L S E } ; /* if P E R I O D I C , which varbs are periodic? */ static double period_start[] = {0.,0.}; /* if P E R I O D I C , begin fundamental dom a i n */ static double period_end[]={l.,l.}; /* if P E R I O D I C , end of fundamental dom a i n */ int m a p p i n g _ t o g g l e = T R U E ; /* this is a map? T R U E or F A L S E */ int inverse_toggle=FALSE; /* if so, is inverse F A L S E , A P P R O X J N V , */ /* or E X P L I C I T J N V ? F A L S E for vec field */ /* In this section, input N U L L or the name of the function which contains.. */ int (*def_name)()=genetics_def; /* the eqns of motion */ int (*jac_name)()=NULL; /* the jacobian (deriv w.r.t. space) */ int (*aux_funcjtiame)()=NULL; /* the auxiliary functions */ int (*inv_name)()=NULL; /* the inverse or approx inverse */ int (*dfdt_name)()=NULL; /* the deriv w.r.t time */ int (*dfdparam_name)()=NULL; /* the derivs w.r.t. parameters */ /* end of dynamical system definition */ # include <ds_define.c>  }  308  Spruce budworm model—DSTOOL. # include <model_headers.h> /* Note: T h e symbols /* and */ denote the beginning and end of comments respectively. */  /*-  ;  Required function used to define the vector field or map. T h e values of the vector field mapping at point x w i t h parameter values p are returned i n the pre-allocated array f. For vector fields, the last components of both f and x are time components. A l l arrays are indexed starting from 0.  _  */  int budworm_def(f,x,p) double *f,*x,*p;  { int npara,nfood,npred,nsurv,nhist,mdisp,ndisp; double slsurv,sdie,predmax,defsat,dsearch; double fgrow,fmax,smax,sgrow,predsat,sharp; double e x f r a c , e x t h r , A , B , f o l n e w , f o l o l d , f o l t o t , C , S l , B L l ; double B L 2 , D , B L 3 , B L 4 , B L 5 , B L 6 , W , E l , H , G , B L 7 , B L 8 , F O , S , B L , t e m p l ; /* Parameters whose values can be changed interactively. */ slsurv = p[0]; sdie = p[l]; predmax = p[2]; defsat = p[3]; exfrac = p[4]; exthr = p[5]; predsat = p[6];  /* STATE VARIABLES */ F O = x[0]; /* Foliage */ S = x[l]; /* B r a n c h surface area */ B L = x[2]; /* B u d w o r m density */  /*  FLAGS 309  */ npara = 1; nfood = 1; npred = 1 ; nsurv = 1 ; nhist = 1; mdisp = 1; ndisp = 1;  /* CONSTANTS */ fgrow = 1.5; fmax = 3.8; smax = 2.4E4; sgrow = 1.15; sharp = 4; dsearch = 1.0; /* PRELIMINARY  EQUATIONS  Foliage dynamics */ A = defsat*BL*fmax/FO; folnew = exp(-A)*FO/fmax; B = 0.357*(A-l+exp(-A)); folold = exp(-B)*FO*(fmax-l)/fmax; foltot = folnew+folold; /* Surface area dynamics * / SI = (l-sdie*(l-foltot/FO)*(l-foltot/FO))*S; /* B u d w o r m dynamics */ /* T h e if-then statements i n this section are for the switches. */ C = 0.003*BL; if (npara = = 1) 310  B L 1 = (l-0.4*exp(-C))*BL; else BL1 = BL; if (nfood = = 1) B L 2 = (0.425*(FO-foltot)/(defsat*BL))*BLl; else B L 2 = 0.425*BL1; if (npred = = 1 ) {tempi = predsat*FO*FO+BL2*BL2; D = (predmax*2.3E4)*BL2/(S*templ);  } else D = 0; B L 3 = exp(-D)*BL2; if (nsurv = = 1 ) B L 4 = (0.473+0.826*BL3/BL)*BL3; else B L 4 = 0.825*BL3; if (nhist = = 1) { t e m p i = 24.9*(fmax-l)*(l-exp(-B))/A; W = 34.1*(l-exp(-A))/A+templ-3.4; if ( W > 0) { B L 5 = (166.0*exp((log(W))/3)-329.0)*0.46*BL4; } else B L 5 = 96.0*BL4;  } else B L 5 = 96*BL4; if ( B L 5 < = 20.0*BL4) B L 5 = 20.0*BL4; if (mdisp = = 1) { E l = 0.46*BL4/exthr; if ( E l > 0) { t e m p i = exp(sharp*log(El)); BL6 = (l-(exfrac*templ)/(l+templ))*BL5;  } 311  else BL6 = BL5;  } else BL6 = BL5; if (ndisp = = 1) { H = (foltot/fmax)*dsearch; G = H*(2-H); B L 7 = slsurv*S*G*G*BL6/smax; } else B L 7 = slsurv*BL6; B L 8 = (S1/S)*BL7;  /* DIFFERENCE  EQUATIONS  */ f [0] = fgrow*foltot / (1+(fgrow-1) *foltot/fmax); f[l]= sgrow*Sl/(l+(sgrow-l)*Sl/smax);. f[2]= B L 7 ; } /* E n d of model equations. */ /*_  .  .  _  Optional function used to define the Jacobian m at point x w i t h parameters p. The m a t r i x m is pre-allocated (by the routine d m a t r i x ) ; A t exit, m[i][j] is to be the partial derivative of the i'th component of f w i t h respect to the j ' t h component of x.  */  /* int user_jac(m,x,p) double * * m , *x, *p;  { } */ /*; ; ; — O p t i o n a l function used to define the inverse or approximate inverse y at the point x with parameter values p. The array y is pre-allocated. */ 312  /* int user_inv(y,x,p) double *y,*x,*p;  { }  */ /*-  ;  O p t i o n a l function used to define aux functions f of the variables x and parameters p. The array f is pre-allocated. T i m e is available as the last component of x. "  int  */'  '  budworm_aux(f,x,p)  double *f,*x,*p; { if (x[2] > 0) f[0] = log(x[2]); else f[0] = 0; }  ;  / * -  Required procedure to define default data for the dynamical system. N O T E : Y o u may change the entries for each variable but P L E A S E D O N O T change the list of items. If a variable is unused, N U L L or zero the entry, as appropriate.  :  ~  7  int b u d w o r m J n i t Q { /* define the dynamical system i n this segment */ int n_varb=3; /* d i m of phase space */ static char *variable_names[]={"F","S","B"}; /* list of phase varb names */ static double variables[]={0.,1.68E4,0.}; /* default varb i n i t i a l values */ static double variable_min[]={0.,0.,0.}; /* default varb m i n for display */ static double variable_max[] = {5.,40000.,350.}; /* default varb m a x for display */ static char *indep_varb_name=="time"; /* name of indep variable */ double indep_varb_min=0.; /* default indep varb m i n for display */ 313  double indep_varb_max=1000.; /* default indep varb m a x for display */ int n_param=7; /* d i m of parameter space */ static char *parameter_names[]={"slsurv","sdie","predmax","defsat", "exfrac","exthr","predsat"}; /* list of param names */ static double parameters[]={0.28,0.75,1.0,0.0074,0.5,5.0,0.085}; /* i n i t i a l parameter values */ static double parameter jnin[]=:{0.,0.,0.,0.,0.,0.,0.}; /* default param m i n for display */ static double parameter_max[]={l.,l.,3.,l.,l.,20.,l.}; /* default param m a x for display */ int n_funct=l; /* number of user-defined functions */ static char *funct_names[]={"lnB"}; /* list of funct names; {""} if none */ static double funct_min[]=={-2.0}; /* default funct m i n for display */ static double funct_max[]={3.0}; /* default funct m a x for display */ int m a n i f o l d _ t y p e = E U C L I D E A N ; /* P E R I O D I C (a periodic varb) or E U C L I D E A N */ static int periodic_varb[] = { F A L S E , F A L S E , F A L S E } ; /* if P E R I O D I C , which varbs are periodic? */ static double period_start[]={0.,0.,0.}; /* if P E R I O D I C , begin fundamental domain */ static double period_end[]={l.,l.,l.}; /* if P E R I O D I C , end of fundamental domain */ int mapping_toggle=TRUE; /* this is a map? T R U E or F A L S E */ int inverse_toggle=FALSE; /* if so, is inverse F A L S E , A P P R O X J N V , */ /* or E X P L I C I T J N V ? F A L S E for vec field */ /* In this section, input N U L L or the name of the function which contains...  */ int int int int int  (*def_name)()=budworm_def; /* the eqns of motion */ (*jac_name)()=NULL; /* the jacobian (deriv w.r.t. space) */ (*aux_func_name)()=budworm^aux; /* the auxiliary functions */ (*inv_name)()=NULL; /* the inverse or approx inverse */ (*dfdt_name)()=NULL; /* the deriv w.r.t time */ 314  int (*dfdparam_name)()=NULL; /* the derivs w.r.t. parameters */ /* end of dynamical system definition */ $ include <ds_define.c>  }  315  

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