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Qualitative analyses of ecological models : an automated dynamical systems approach 1995
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Title | Qualitative analyses of ecological models : an automated dynamical systems approach |
Creator |
Van Coller, Lynn |
Date Created | 2009-03-20 |
Date Issued | 2009-03-20 |
Date | 1995 |
Description | 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 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. |
Extent | 12653724 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-03-20 |
DOI | 10.14288/1.0079984 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1996-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/6299 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0079984/source |
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Q U A L I T A T I V E A N A L Y S E S O F E C O L O G I C A L M O D E L S - A N A U T O M A T E D D Y N A M I C A L S Y S T E M S A P P R O A C H B y Lynn van Coller B. Sc. University of Natal ( U N P ) , South Af r ica , 1990 B. Sc. Hons. University of Natal ( U N P ) , South Af r ica , 1991 M . Sc. University of Natal ( U N P ) , South Af r ica , 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 GRADUATE STUDIES INSTITUTE OF APPLIED MATHEMATICS DEPARTMENT OF MATHEMATICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1995 © Lynn 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 to i>ecm&e<i OE-6 (2/88) 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 re- quired 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 im- plementing 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 termi- nology 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 compli- cated 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 i i Table of Contents iv List of Tables x List of Figures x i Acknowledgements xxi 1 Introduction 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 2 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 Mode l 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 Conclusion • • • 57 4 Ratio-Dependent Mode 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 102 4.6 Conclusion 107 5 Population Genetics Mode 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 Conclusion • 136 6 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 7 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 . . 168 7.3.3 Budworm 169 7.4 Model analysis 174 vi 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 7.5 Conclusion 203 8 Conclusion 204 8.1 Main results 204 8.2 Limitations 205 8.3 Future possibilities 206 Bibliography 207 Appendices 217 A Dynamical systems theory 217 A. l Introduction 217 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 Equilibrium point 221 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 259 B.2 Theory : 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 285 C. l Modelling delays in system dynamics models 285 C. 2 Rescaling model equations 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 List 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 173 x List 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) 15 2.4 One-parameter bifurcation diagram obtained by varying e in Bazykin 's prey competit ion model with a — 0.6, b = 0.3, c = 0.4, d = 0.2 and a = 0.3. 17 2.5 Two-parameter continuation of the Hopf bifurcation shown in the previous figure 18 2.6 Two-parameter bifurcation diagram of (e, a)-parameter space for the prey competit ion model with a = 0.6, b = 0.3, c = 0.4 and d = 0.2 19 2.7 Phase portraits corresponding to the points marked with *'s in the previous figure 20 2.8 T ime plots corresponding to the phase portraits in the previous figure. . 21 2.9 (a) and (b) Two-parameter bifurcation diagrams of (a, /^-parameter space, (c) Phase portraits corresponding to regions (i), (ii), (iii) and (iv) in parts (a) and (b) 23 2.10 One-parameter bifurcation diagram obtained by varying a in the predator competit ion model with a = 0.6, b = 0.3, c = 0.4, d = 0.2 and [i — 0.06. . . 24 2.11 Diagram showing the period of the l imit cycle oscillations in the previous figure as a function of a 25 x i 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 func- tion 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) a s a function of pasture availability. 46 3.7 The pasture multiplier (PM) as a function of pasture availability (PA)- • - 48 3.8 One-parameter bifurcation diagrams obtained from varying SFCN for the new model which includes a pasture limiting multiplier 49 3.9 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. . . 53 xii 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\M2 and M2M3 planes for the original model 68 4.2 One-parameter bifurcation diagrams obtained by varying 73 in (a) the original model (a;=0, i — 1,2,3) and (b) the modified model (at=0.001, i = 1,2,3). The state variable Mi is plotted on the y-axes 72 4.3 One-parameter bifurcation diagrams obtained by varying (f>3 in the original model 74 4.4 One-parameter bifurcation diagrams obtained by varying </>3 in the modi- fied model 75 4.5 Time plots of (a) Mu (b) M 2 and (c) M3 for <f>3 = 0.16. All the other parameter values are as in the reference parameter set 76 4.6 Isoclines in the M X M 2 plane with 4>3 = 0.06 in (a) and </>3 = 0.05 in (b) and (c). 78 4.7 One-parameter bifurcation diagrams obtained by varying $73 in the original model 80 4.8 One-parameter bifurcation diagrams obtained by varying fi 3 in the modi- fied model 81 4.9 One-parameter bifurcation diagrams showing the effects of varying cf>2 for the original model 83 4.10 One-parameter bifurcation diagrams showing the effects of varying </>2 for 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 imit points and Hopf bifurcations in 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 with a t=0.002 (i = 1,2, 3) 89 4.16 T ime plots corresponding to the points that are marked with *'s in the two-parameter diagrams for the original model and the modified model wi th 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 91 4.18 One-parameter bifurcation diagrams showing the effects of varying 0 2 for (a) the original model and (b) the modified model 92 4.19 Two-parameter bifurcation diagram obtained using the modified model wi th a,-=0.001, (i = 1,2,3) and 7 l = 0.4 94 4.20 Two-parameter bifurcation diagram obtained using the modified model with a t-=0.001, (i = 1,2,3) and 71 = 0.6 95 4.21 Two-parameter bifurcation diagram obtained using the modified model wi th a t-=0.001, (i = 1,2,3) and 7 l = 1.2 96 4.22 Two-parameter bifurcation diagram obtained using the modified model wi th a,-=0.001, {i = 1,2,3) and 71 = 1.8 97 xiv 4.23 Two-parameter bifurcation diagram obtained using the modified model with a;=0.001, (i = 1,2,3) and 7 l = 2.4 98 4.24 One-parameter bifurcation diagrams obtained by varying 72 wi th 7 l = 0.4 and (a) fa = 0.07, (b) fa = 0.17 and (c) fa = 0.25 99 4.25 One-parameter bifurcation diagram obtained by varying 72 wi th 71 — 1.2 and fa = 0.2 101 4.26 Examples of isocline configurations showing different possibilities for the position of the tr itrophic equil ibr ium 104 4.27 Examples of isocline configurations at different points in (71,^2)-space. . 105 4.28 Isocline configurations with (a) a; = 0, (b) a,- = 0.001 and (c) a2- = 0.005 (i = 1,2,3) together with the reference parameter set 106 4.29 One-parameter bifurcation diagrams when a; = 0.005 (i = 1,2,3) 108 4.30 Plant 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 2 = 1.9, a 2 2 = 1-1, fen = 1.0, 612 = 0.904 and fe22 = 0.54 119 5.2 One-parameter bifurcation diagram with an = 2.1, ai2 = 1-9, a 2 2 = 1.1, 6n = 1.0 and b12 = 0.904 obtained using A U T O 121 5.3 One-parameter bifurcation diagram with a n = 2.1, a\2 = 1.9, a 2 2 = 1.1, 611 = 1.0 and b12 = 0.906. Ku = 2.1 and K12 = 2.097 123 5.4 Diagram of the (p, A r ) -plane for a n = 2.1, a12 = 1.9, a 2 2 = 1.1, bu = 1.0, b12 = 0.906 and 622 = 0.52 showing the domains of attraction for the stable phenomena 124 5.5 Two-parameter bifurcation diagram with a n = 2.1, a 1 2 = 1.9, a 2 2 = 1.1 and fen = 1.0 obtained using A U T O 125 5.6 Two-parameter bifurcation diagram including additional curves 126 xv 5.7 One-parameter bifurcation diagram with an = 2.1, a i 2 = 1-9, a22 = 1-1, 611 = 1.0 and &i2 = 0.908 obtained using A U T O 127 5.8 Dynamics in the (p, AQ-plane with an = 2.1, a12 = 1.9, a 2 2 = 1.1, &u = 1.0 and various combinations of b12 and 6 2 2 which correspond to regions A to H in the two-parameter figure 129 5.9 One-parameter bifurcation diagram with a 1 2 = 1.9, a 2 2 = 1.1, Wi = 1-0, 612 = 0.905 and fe22 = 0.525 obtained using A U T O 130 5.10 Dynamics in the (p, 7V)-plane for a 1 2 = 1.9, a 2 2 = 1.1, 6 X 1 = 1.0, &i2 = 0.905, 6 2 2 = 0.525 and (a) a n = 2.68, (b) an = 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. . . . 132 5.11 One-parameter bifurcation diagram with an = 2.1, a 2 2 = 2.1, bu = 1.0, 612 = 0.908 and &22 = 0.53 obtained using A U T O 133 5.12 Dynamics in the (p, AQ-plane for an = 2.1, a i 2 = 3.8, a22 = 2.1, bn = 1.0, &i 2 = 0.908 and 622 = 0.53 showing a stable period-4 polymorphism. . . . 133 5.13 Two-parameter continuation of the period-doubling bifurcation wi th an = 2.1, a 2 2 = 2.1, bu = 1.0 and b12 = 0.908 134 5.14 Examples of complex dynamics, (a) A n interior period-8 orbit for an = 2.3, a 1 2 = 2.9, a 2 2 = 2.5, fen = 1.0, 612 = 0.908 and 6 2 2 = 0.95. (b) A n interior chaotic attractor for an = 2.6, a i 2 = 3.1, a 2 2 = 2.5, &n = 1.0, 612 = 0.908 and fe22 = 0.95 135 6.1 t£>2 wi th r22 = 0.8 and AC22 = 0.6 142 6.2 A one-parameter bifurcation diagram obtained by varying k\\ w i th r n fixed at 0.7 ( r 2 2 = 0.8 and k22 = 0.6) 143 xvi 6.3 Diagrams of the (p, iV)-plane for a number of different values of ku wi th r n fixed at 0.7 144 6.4 Diagrams of the (rCn, rn ) -parameter space 145 6.5 Diagrams of the (p, JV)-plane corresponding to the regions A to H in the two-parameter diagram 147 6.6 Diagrams of the (p, AQ-plane corresponding to the regions N to P in the two-parameter diagram 148 6.7 Examples of the fitness functions for parameter values corresponding to (a) a stable polymorphism ( rn = 0.7,ku = 2.0, r22 = 0-8,k22 = 0.6) and (b) an unstable polymorphism ( r n = 0.4,fen = 2.0, r22 = 0.8,/c2 2 = 0.6). . 149 6.8 Curves given by w\ = w2 and w — 1 for parameter values corresponding to a stable polymorphism ( r n = 0.7, ku = 2.0, r22 = 0.8, k22 = 0.6). . . . 151 6.9 (a) Fitness functions and (b) (p, iV)-plane for r n = 1.3, ku = 0.5, r22 = 7.5 and k22 = 4.57 153 6.10 A part ial one-parameter bifurcation diagram obtained by varying r 2 2 . . . 154 6.11 Two-parameter continuations of the period-doubling bifurcation HB* in the previous figure obtained by varying (a) r n , (b) ku and (c) k22 in addit ion to r 2 2 155 6.12 (a) Fitness functions and (b) (p, AQ-plane for r n = 0.2, ku = 5.0, r 2 2 = 0.3 and £:22 = 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 in the previous figure 158 xv i i 6.15 An example of an interior chaotic attractor obtained for r n = 0.18, hi = 5.0, r 2 2 = 0.3 and k22 = 0.4 159 6.16 An example of a fitness function configuration where u>n is always superior to u>22 160 6.17 The (kn, r11)-parameter space showing the region of higher order stable polymorphic behaviour and the region corresponding to fitness function configurations of the type shown in the previous figure 160 6.18 w22 with r 2 2 = 0.25 and k22 = 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) dSL = 0.35, (c) dSL = 0.8 and (d) dSL = 0.9 178 7.4 Time plots of (a) budworm larval density, (b) foliage density and (c) branch surface area density versus time for dsL = 0.35 179 7.5 One-parameter bifurcation diagram of budworm larval density versus A t h r for dSL = 0.45 181 7.6 Two-parameter bifurcation diagram of A t h r versus dsL 182 7.7 Diagrams of budworm larval density versus foliage for the regions marked A-I in the previous figure 184 7.8 Simplified two-parameter bifurcation diagram of A t h r versus d$L 187 7.9 Two-parameter bifurcation diagram of A t h r versus dsL for the simplified model which only includes dispersal 191 7.10 Isoclines of recruitment versus budworm density 194 xvm 7.11 Two-parameter bifurcation diagram of Athr versus dsL for the predation model which includes dispersal as well as predation 196 7.12 One-parameter bifurcation diagram for pmax with d$L = 0.4 and A t h r — 5. 198 7.13 Time plots of budworm larval density for (a) pmax = 0.6 x 23 000 and (b) Pmax = 3.4 x 23 000 200 7.14 Two-parameter bifurcation diagram of pmax versus dsL 201 A. l 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 corre- sponding 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 cor- responding to p = p,\ 228 A.11 (a) Period-doubling bifurcations at Ai and A2. (b)Behaviour over time for the state variable Xi 230 A. 12 Derivation of a phase plane showing the time-dependent behaviour of two variables, Xi and x2 231 xix A. 13 (a)Bifurcation diagram of a pitchfork bifurcation and phase portraits cor- responding 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 po- sitions 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 hete- roclinic 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 bifurca- tions at /it = n*i and fx = LI*2 for a discrete system. (b)State space diagrams showing the dynamics at LI = fix, LI = \ i 2 and LI = u.3 253 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. . . 256 xxi 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, Don Ludwig , most sincerely for his invaluable suggestions and guidance. I benefited greatly from his experience in both mathematical and ecological disciplines. I would also like to thank Gene Namkoong for suggesting the population genetics models and for giving me both direction and encouragement while I studied them. I am indebted to Bard Ermentrout for writ ing 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 Col in Clark, Wayne Nagata, Gene Namkoong 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 Pe- ter Abrams, Eugene Allgower, A l a n Berryman, Eusebius Doedel, Lev Ginzburg, John Guckenheimer, Andrew Gutierrez, Alexander Kh ibn ik , L i x i n L i u , Jesse Logan, Peter Turchin, Graeme Wake and Peter Yodzis. Thank-you for your t ime and your interest. Thank-you also to my fellow graduate students in the Mathematics department for em- pathising with me through the frustrations which accompany most graduate studies. A very special thank-you to my family and non-mathematics friends for your endless encouragement and interest. Your support was a continual source of strength to me. Final ly , I would like to acknowledge the E m m a Smith , 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 General overview In this thesis I demonstrate how techniques from dynamical systems theory can be applied to ecological models in order to study their qualitative behaviour. The 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 dynamical 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 my objectives more formally. I then describe how I go about fulfilling these aims with specific references to later chapters i n the thesis. I conclude this introduction with 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. With 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 DSTOOL [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 out l ine My approach to achieve the above aims was to analyse ecological models—both con- tinuous and discrete—to demonstrate the kinds of results and information that can be obtained using dynamical systems techniques. Chapter 1. Introduction 3 I begin in chapter 2 with the analysis of three fairly simple predator-prey models which have already been studied by Bazyk in [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. Bazyk in [14] studied the models analytically. The computer packages allowed me, a novice, to reproduce and in fact improve upon his results. The chapter also illustrates how the computer packages can encourage an iterative approach to modell ing which may aid the development of more plausible models. Having demonstrated the reliabil ity of some of the computer software in 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. The model is a large one consisting of 10 ordinary differential equations and numerous parameters. Even the most knowledgeable theoretician would find an analysis of this model using pencil and paper a formidable task. The dynamical systems techniques prove to be a useful alternative to the sensitivity analyses which are tradit ional ly 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. The model is an example from a controversial area of current research known as ratio-dependent modell ing. The analysis in this chapter highlights some of the biological implausibil it ies and mathematical difficulties associated with ratio-dependent models which may be important for guiding future research. A modification to the model equations is analysed in conjunction with the original model and reveals that the latter is structurally unstable. Many 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 t ime delays in the equations [84]. As a result detailed analyses have been restricted mainly 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. Both models have fairly simple mathematical formulations involving only two alleles but they are capable of displaying complicated dynamics. I focus on the dynamics of the heterozygote. The model in chapter 5 has been part ial ly 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 with 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. The model in chapter 6 is a modif ication of that in chapter 5 but it has not been studied in detail before. This is not surprising since it is not even possible to find explicit expressions for the equi l ibr ium 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. Cr i ter ia for determining the existence and stabil ity of polymorphic equil ibria 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 budworm- forest 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 mult ip le stable states are also located. Specific conclusions relating to the particular examples are included at the end of each chapter. More general conclusions are summarised in chapter 8. Special mention should be made of two of the appendices. Appendix A contains a glossary of the basic dynamical systems concepts which are used in the main 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 l i tt le prior exposure to dynamical systems concepts. Appendix B describes how computers can be used to implement the dynamical sys- tems techniques. Descriptions of the capabilities and relative advantages and disadvan- tages of the packages that I used, as well as procedures for obtaining t ime 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 im 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. Whi le these phenomena may be of intellectual interest, they are often of l i tt le practical use. M y viewpoint is summarised by the following quote from Adler and Morris [2]: Chapter 1. Introduction 6 Only by avoiding the unthinking use of famil iar and mathematical ly con- venient models and by having the discipline to ignore interesting but dy- namical ly unimportant interactions, can we ever hope to develop predictive ecological theory. 1.4 Motivation Mathemat ica l models have been used to describe ecological systems for many decades. However, the interdisciplinary nature of the field has led to some conflict in opinions. Many 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 Holl ing 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 with the parameter values in an ecological model [40, 53, 115, 121], it is not sufficient to merely simulate the model equations over t ime 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, simplif ication or because information regarding the relevant dynamics is lacking. Hence, it is important to know whether altering parameter values wi l l significantly affect the predictions of the model. This is not a t r iv ia l task when large numbers of parameters are involved. As has already been mentioned, we can vary parameters across ranges of values us- ing techniques from dynamical systems theory. In this way we can obtain information Chapter 1. Introduction 7 regarding the presence and nature of attractors 1 in the system. Whereas transient dy- namics vary with the in i t ia l values of the state variables and the t ime period over which solutions are calculated, the techniques in this thesis are concerned with the behaviour of the system once the in i t ia 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. The application of the dynamical systems techniques to a system of nonlinear equa- tions can be a formidable task for a mathematician, let alone a non-mathematician. It is also t ime-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 stabil ity analysis". Although analytical methods can provide remarkable results, they have two strong l imitations [111]. F i rst , 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 smal l ' distances where 'sufficiently smal l ' 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 tradit ional t ime plots and phase portraits are used to display results as well as bifurcation diagrams. The latter diagrams provide a concise way of summarising the effects of different parameter values on the behaviour of the system. 1 See section A . 2 . 5 for a definit ion of this te rm. Chapter 1. Introduction 8 1.5 The bigger picture Qualitat ive analyses are not new and can be traced back at least to Poincare 2 . In eco- logical circles names such as Lotka and Volterra [73, 120], Rosenzweig and M a c A r t h u r [76, 103], Hol l ing [61], May [79, 80, 82, 83] and G i lp in [44], and many others, are well - known 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, al l these models are fairly simple theoretical models be- cause of the mathematical difficulties encountered with more complicated models. The introduction of various computer packages since the mid-1980's has allowed dy- namical systems techniques to be applied with 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. The papers by Collings [23], Collings and Wol lk ind [24], Collings et al. [25] and Wol lk ind et al. [127] study a fairly practical biological control model of mite interactions. However the mathematics is st i l l complicated and difficult for the reader wi th l i t t le 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 il lustrate how conclusions regarding model sensitivity and resilience can be drawn from bifurcation diagrams. They also derive meaningful ecological implications from their results. A l l four 2 A H o p f bi furcat ion (see section A .2 .10 ) , which is an impor tan t concept i n d y n a m i c a l systems theory, is also k n o w n as a Po inca re -Andronov-Hopf bifurcat ion and the Po inca re -Bend ixon theorem is funda- menta l to qual i ta t ive analyses. Fur ther details can be found i n A r n o l d [7] and W i g g i n s [124]. Chapter 1. Introduction 9 papers note that many of their conclusions would not have been obtained without the use of AUT086 [28]. In particular, Collings and Wol lk ind [24] obtained three previously undiscovered possibilities for qualitative behaviour using a predator-prey model of the type studied by Bazyk in [14]. In this thesis I hope to take the road less travelled by showing how dynamical 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 in the next chapter with 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 This 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 rel iabil ity 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]. Dynamical systems techniques are applied to three fairly simple predator- prey 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 in different regions of these diagrams is explained using phase por- traits. The 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 with which qualitative analyses may be done using the abovementioned computer packages allows a fairly quick determination of the effects of model alterations. This can facil itate the formulation of more plausible models. These predator-prey models have already been studied analytically by Bazyk in [14]. I have included his results for comparison with those obtained by the computer software. The latter are in 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. The next two sections discuss two modifications to the basic model. Many of the phenomena that occur in 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 al l the basic terminology as well as the conventions I use in 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 Lotka and Volterra in the 1920's (see [34] for a de- scription of the model and its analysis). The model equations are where x represents prey density and y predator density. A l l the parameters are real and positive. The term ax describes the exponential growth of the prey population in the absence of predators and — cy describes the exponential decline in the predator population in the absence of prey. The terms — bxy and dxy describe the interaction between predator and prey. From 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 arbitrari ly small perturbation to the model can change its qualitative dynamics. For example, replacing the exponential growth of the prey with logistic growth changes the dynamics from (neutral) cycles to a stable equi l ibr ium (see A number of modifications to this model have been studied since the 1920's. In par- t icular, Bazykin 's [14] work is well-known among ecologists because of his comprehensive x = ax — bxy V = ~cy + dxy (2-1) [34]). Chapter 2. Preliminary Example 12 qualitative analyses of the models and his accompanying diagrams, which summarise the different possible behavioural regimes. One of Bazykin's modifications to the Lotka- Volterra equations (2.1) is given by the system y = + & (2-2) He justified using Michael is -Menten interaction terms by analogy wi th the mechanism of enzyme reactions. The denominators of these terms prevent unl imited predation of prey and unl imited growth of the predator population with the growth of prey density, respectively. The Hol l ing 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-tr iv ial (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 equil ib- r ium 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 in figure 2.1. As can be seen from this figure, there are no bifurca- tions (see section A.2.2) and the equil ibr ium point remains unstable as a is varied. But the figure does show how the equil ibr ium value of x changes with 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 equil ibr ium point with 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 Bazyk in [14] Chapter 2. Preliminary Example 13 12 9 x 6 3 0 .0 0.1 0.2 0.3 0.4 0.5 a Figure 2.1: One-parameter bifurcat ion d iagram obtained by vary ing a i n system (2.2) w i t h a = 0.6, b = 0.3, c = 0.4 and d = 0.2. T h e state variable x is p lot ted on the y-axis . (see figure 2.2) and verify that figure 2.1 summarises Bazykin's [14] results. The phase portraits in figure 2.2 show that x increases indefinitely for al l values of a. This is an obvious shortcoming of the model and led Bazyk in to introduce further modifications. 2.3 A d d i n g intraspecific competition among prey To improve the model Bazyk in added a term to the prey equation to take into account intraspecific competit ion among prey. Here competit ion refers to a decrease in reproduc- tion or an increase in death rate with an increase in prey density. The assumption that competit ion is l inearly dependent on prey density results in the system of equations • . bxy 2 X — dX ' -. , £ X 1+ax V = -cy + T^-x- (2-3) We can create one- and two-parameter bifurcation diagrams (see section A .2 .1 for a de- 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. Chapter 2. Preliminary Example 14 x x Figure 2.2: Phase por t ra i ts obtained by B a z y k i n for model (2.2). Qua l i t a t i ve ly s imi la r d iagrams can be obtained by t ak ing a = 0.6, 6 = 0.3, c = 0.4, d = 0.2 and (a) a = 0.1, (b) a = 0.2, (c) a = 0.3 and (d) a = 0.55. Bazykin's [14] results for this system are shown in figure 2.3. 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)1 we expect a bifurcation to occur as the line with 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 J T h e or ig in is also an equ i l ib r ium point in both cases but only non t r iv ia l equ i l i b r i um points hav ing x > 0 and y > 0, (x, y) ^ (0, 0) are considered in detai l . In region ( i i i ) y < 0 at the e q u i l i b r i u m point A and hence this poin t is not of biological interest. Chapter 2. Preliminary Example 15 Figure 2.3: T h e fol lowing diagrams are adapted from B a z y k i n [14]. (a) Two-parameter bifurcat ion d iagram of (e, o)-parameter space, (b) Phase portrai ts corresponding to regions (i) , (ii) and ( i i i ) in 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 equi l ibr ium point. B y choosing values of 0.3 for a and 0.1 for e, we can either determine such a point analytically (as Bazyk in 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 in t ime unt i l we are near the equil ibr ium point. Using X P P A U T this is done by choosing the menu option I N T E G R A T E followed by G O . Choosing the SINGular P O I N T option then finds the equil ibr ium point and indicates whether it is stable or unstable. A separate window appears with this information. The state variable values at the equi l ibr ium point are then entered as the in i t ia l point in the in i t ia l point window. Since Bazyk in 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 main 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 DS in 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 command, 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 ( l imit 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 imi t cycles surrounding an unstable equil ibr ium point. Since a is fixed in figure 2.4, this one-parameter diagram describes the dynamics along a horizontal line at, say, a = cx\ in figure 2.3(a). In figure 2.4 I have labelled the continuation branches A and B to indicate which equi l ibr ium 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 with e but the Chapter 2. Preliminary Example 17 x 0.09 0.15 Figure 2.4: One-parameter bifurcat ion d iag ram obtained by vary ing e i n system (2.3) w i t h a = 0.6,6 = 0.3, c = 0.4, d = 0.2 and a = 0.3. T h e labels A and B m a r k the cont inua t ion branches corresponding to the equ i l i b r i um points given i n equations 2.4, H B stands for H o p f b i furcat ion and B P for b i furcat ion point ( t ranscri t ical i n this case). Exp lana t ions of the various l ine types can be found i n section A . 2 . 1 . In par t icular , the curves of sol id circles mark the m a x i m a and m i n i m a of 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 , c • da(d — ac) — ec . A x = i — > y = Tu—V~ d — etc o [d — acy B ( x = ±y = 0). (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 I 0.1 0.2 0.3 e Figure 2.5: Two-parameter cont inuat ion of the H o p f bifurcat ion shown i n figure 2.4. point and B is again a saddle point. This configuration corresponds to region (i) in figure 2.3. For e > 0.12 B is now stable and A is a saddle point, but the numerical output f rom 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. We would also like to reproduce Bazykin's two-parameter bifurcation diagram shown in 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). The result is figure 2.5. The first observation we can make from this diagram is that the curve of Hopf bifur- cations is very different from Bazykin's straight line in 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 imit cycle behaviour) is only possible if e < 0.0515 and a < 0 .5 2 , 2 A U T O slows down considerably as a increases toward 0.5 and e tends to 0 and never ac tua l ly reaches this point , a l though the curve does get very close i f A U T O is left to run for a sufficiently long t ime per iod . It can be verified ana ly t i ca l ly that the curve does pass through (0,0.5). However, complex behavioura l changes occur at this point wh ich is why A U T O has computa t iona l difficulties. 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 Figure 2.6: Two-parameter bifurcat ion d iagram of (e, a)-parameter space for mode l (2.3) w i t h a = 0.6, 6 = 0 . 3 , c = 0 . 4 and d — 0.2. T h e regions (i) , (ii) and ( i i i ) 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 in 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 in 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 with the Hopf bifurcation continuation. Mode l (2.3) is simple enough for the curves in figure 2.6 to be determined analytically although the algebra is rather messy. It can be shown that transcrit ical bifurcations occur along the straight line e d a = + - a c and Hopf bifurcations occur along the curve (—ac2)a2 + (acd — ec2)a — edc = 0. (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 in figure 2.6 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 MAPLE 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 in figure 2.3(b). 0 6 10 15 20 0 5 10 15 20 x x Figure 2.7: Phase portrai ts corresponding to the points marked w i t h *'s in figure 2.6. It is also informative to view the temporal dynamics of a model. Time plots corre- sponding to figure 2.7 for initial values x = 10 and y = 3 are shown in figure 2.8. 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) 10 x 5 (c) 10 x 5 100 T i m e (b) 10 x 5 (d) 10 x 5 - 100 T i m e 100 T i m e 200 Figure 2.8: T i m e plots corresponding to the phase portrai ts in figure 2.7. T h e i n i t i a l poin t x — 10, y = 3 was used in 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 Example 22 2.4 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 — ^ y V = -cy + T^-x-W2. (2.5) 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 DSTOOL 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 AUTO'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 23 Figure 2.9: These diagrams are adapted from [14]. (a) and (b) Two-parameter bifurcat ion diagrams of (a , /z)-parameter space, (c) Phase portrai ts corresponding to regions (i) , ( i i ) , ( i i i ) and ( iv) in parts (a) and (b). Chapter 2. Preliminary Example 24 60 x 0 0.04 0.08 0.12 0.16 0.2 a Figure 2.10: One-parameter bifurcat ion d iag ram obtained by vary ing a i n system (2.5) w i t h a = 0.6, 6 = 0.3, c = 0.4, d — 0.2 and fi = 0.06. T h e labels A and C mark the cont inuat ion branches for the two non t r i v i a l e q u i l i b r i u m points . H B marks the H o p f bifurcat ion and L P the l i m i t po in t . Such a steep increase in 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 in A U T O to locate an orbit of high period. The required approximation is obtained by continuing this orbit of fixed period in /J, as well as a. The resulting two-parameter diagram in (a, ^)-space is shown in figure 2.12. (I would have l iked 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 with 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 with figure 2.9(a). Phase portraits and t ime plots corresponding to the points marked with *'s in figure 2.12 are shown in 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 imit Chapter 2. Preliminary Example 25 Figure 2 . 1 2 : Two-parameter bifurcat ion d iagram of (a , p)-space for a = 0.6, 6 = 0.3, c = 0.4 and d = 0.2. H B marks the H o p f bifurcat ion cont inuat ion, L P marks the l i m i t poin t cont inuat ion and per iod=30 marks the cont inuat ion of the orbit of fixed per iod . Chapter 2. Preliminary Example 26 (a) (b) (c) (d) y 5 - Figure 2.13: Phase portrai ts corresponding to the points marked w i t h *'s in figure 2.12. cycle behaviour is very small for these parameter values. As 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 imit cycle behaviour is negligible. We can investigate figure 2.12 further by generating one-parameter bifurcation dia- grams 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. As 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. As 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 Example 27 (a) 10 x 5 (b) 10 (c) 10 x 5 100 Time 200 * 5 www\AAAA/W 100 Time (d) 10 x 5 Figure 2.14: T i m e plots corresponding to the points marked w i t h *'s in figure 2.12. T h e i n i t i a l point 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 C o n c l u s i o n This chapter describes how qualitative analyses of three predator-prey models may be done using various computer software. The numerical results are compared with analyt i - cal results obtained by Bazykin [14]. For model (2.3) which has intraspecific competit ion among prey, the numerical results are more accurate than Bazykin's approximate ana- lyt ic results. This conclusion is possible as I obtained an exact analytical expression for Chapter 2. Preliminary Example 28 (a) 60 x (b) 60 (c) F i g u r e 2 . 1 5 : One-parameter bifurcat ion diagrams for (a) fi = 0.1, (b) fx = 0.074 and (c) / i = 0.02. These correspond to the hor izonta l dot ted lines i n figure 2.12. Chapter 2. Preliminary Example 29 the two-parameter Hopf bifurcation curve for which the discrepancy in results arises. The analysis of model (2.5) results in a numerical approximation to a curve which Bazyk in did not describe analytically. The position of this curve results in a very small two-parameter region corresponding to l imit cycle behaviour. A knowledge of the relevant mathematical techniques, such as centre manifold the- ory 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. This is espe- cially 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 faci l i tating an iterative approach to modell ing 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 dy- namical systems techniques. In this case the analysis suggests an improvement in the formulation of the model equations. Chapter 3 S h e e p - H y r a x - L y n x M o d e l 3.1 Int roduct ion This chapter investigates a more complicated model having 10 state variables and a large number of parameters. Analyt ica l work done by hand and isocline analyses are of l i t t le use in 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 in the model and lead to an improvement in the formulation of the equations. The model I have chosen is an example of a system dynamics model and has four main components—sheep, hyrax, lynx and pasture. I begin in section 3.2 with some background to the systems modell ing approach for those who may be unfamil iar with it . I also discuss the tradit ional 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 in 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 l imitations of a tradit ional sensitivity analysis and highlights dynamics which are 1 See section 3.2 for a descript ion of sensi t ivi ty analyses. 30 Chapter 3. Sheep-Hyrax-Lynx Model 31 biologically implausible, namely that pasture growth is unl imited when sheep densities are low. A modification to the pasture growth term is discussed in section 3.5.3. The analysis is completed by a two-parameter study of the effects of cull ing rates on farmers' revenue. This is followed by section 3.5.5 which interprets the main results of the analysis from a biological viewpoint. 3.2 Dynamic models and systems analysis—some background The systems approach to modell ing was made popular by Forrester [39] in the early 1960's. This 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. The methodology was originally applied to industr ial , urban, and world population systems but its ut i l i ty 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 in the ecological context: The formulation of the models allows for considerable freedom from con- straints and assumptions, and allows for the introduction of the non-l inearity and feed-back which are apparently characteristic of ecological systems. This ease of formulation and flexibility are important for modell ing 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 . Kowal [69] notes that an analysis of dynamic models can provide approximations to ecosystem dynamics long before tradit ional 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 The problem with complex models comes at the t ime of analysis. Dynamic models usually involve a large number of equations (generally ordinary differential equations) and parameters which makes their behaviour difficult to predict (Patten [99]). We need to find suitable ways of analysing the dynamics of these models. The tradit ional approach has been to use numerical routines to obtain solutions over t ime for a given set of parameter values. Optimisation routines are also often employed (Maynard Smith [85]) to determine the 'best' possible strategy with respect to a cost or revenue function. These routines are implemented using computers. A computer's speed of computation and abil ity 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]). Whi le 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 arbitrari ly small change in the solution given by the model if we are to have any faith in the predictions of the model (Hadamard [53]). This corresponds to Hadamard's concept of a well-posed problem with respect to part ial 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 in Bry l insky [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 imited in that only a single, small perturbation of each parameter is considered. This 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 modell ing which I wi l l use to il lustrate 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 Af r ica . Two main 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 in the region. The second problem is the predation on sheep by lynx. The principal food for lynx is hyrax, but from t ime to t ime lynx prey on sheep. It is the latter problem that is of direct concern to farmers—they tend to be more tolerant of the competit ion with hyrax. The model in [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 cull ing strategies for hyrax and lynx. There are 10 state variables in the model. The sheep, hyrax and lynx populations are each divided into three classes—juveniles, female adults and male adu l t s—and there is one variable representing pasture. The quantity of most interest to farmers is revenue. This auxil iary variable is a function of the state variables and is made up of wool sales, mutton sales, the value of sheep stock, and the cost of cull ing hyrax and lynx. The differential equation for each state variable is formulated by adding and sub- tracting quantities representing the processes affecting that variable. For example, the equation for hyrax juveniles is as follows: Chapter 3. Sheep-Hyrax-Lynx Model 34 Rate of change of ^ hyrax juveniles j births - maturation - deaths - predation - cull ing where • b i r t h s depend on the number of hyrax female adults and decrease with increasing hyrax density, • m a t u r a t i o n represents the number of juveniles that mature to become adults in 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 wi th in - creasing hyrax density, • p r e d a t i o n (by lynx) varies with the relative number of hyrax and lynx (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 mathematical terms the above equation becomes dHj JJ TT u II u —- — tijB — tijM — njr) — Hjp — tljc dt where Hj is the number of hyrax juveniles, HJB is the number of hyrax births in 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 ki l led by lynx during the year, and HJC is the number of hyrax juveniles that are culled. A ful l description of the mathematical formulation of these terms can be found in [116]. B y way of example, Chapter 3. Sheep-Hyrax-Lynx Model 35 HJM — Hj X HJMN where HJMN (hyrax juvenile maturation normal) is the fraction of juveniles that become adults each year and • 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 ki l led per lynx per year, • LPM is the lynx predation multipl ier (lynx functional response) which is an increas- ing 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 'normal ' ratio, representing the usual level of abundance under typical environmental conditions. The last term in equation (3.1) adjusts the total amount of predation so that only the number of juveniles ki l led is taken into account in this equation. A possible choice for the lynx predation multipl ier is shown in figure 3.1. A n S-shaped functional response is used as the lynx can only eat a l imited amount even when prey (3.1) where Chapter 3. Sheep-Hyrax-Lynx Model 36 2 1.5 LpM 1 0.5 0 0 1 2 3 AP 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. The other equations in the model are formulated in a similar manner to the above example for hyrax juveniles. Table 3.1 shows the processes affecting each animal 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 in the model are averaged using first order delays. A description of how this is included in the model is given in appendix C. Swart and Hearne [116] used tradit ional methods to study this model. They used optimisation routines to find the hyrax and lynx cull ing rates which gave m a x i m u m profitabil ity in terms of revenue. They also investigated the sensitivity of the system to parameter perturbations. They found that lynx culling is essential and that substantial increases in both sheep numbers and revenue are possible by simultaneously cull ing hyrax and lynx. Opt imal cull ing rates in terms of revenue are around 30 percent per annum Chapter 3. Sheep-Hyrax-Lynx Model 37 State variable G r o w t h processes D e a t h processes Hyrax juveniles (HJ) Hyrax female adults (Hp) Hyrax male adults (HM) Lynx juveniles (Lj) Lynx female adults (Lp) Lynx male adults (LM) Sheep juveniles (SJ) Sheep female adults (SF) Sheep male adults (SM) Pasture (P) births (HJB) juvenile maturation (HJM) juvenile maturation (HJM) births ( L j B ) juvenile maturation (LJM) juvenile maturation (LJM) births (SJB) juvenile maturation (SJM) juvenile maturation (SJM) production (Pp) deaths (Hjp>),maturation (HJM), predation (ifjp),culling (HJC) deaths (HFD),predation (Hpp), culling (HFC) deaths (ifM.o),predation (HMP), culling (HMc) deaths (LJD),maturation (LJM), culling (LJC) deaths (LFD),culling (Lpc) deaths (ZMB ) ,cul l ing (LMC) deaths (SJD),maturation (SJM), predation (Sjp),culling (SJC) deaths (S^£>),culling (SFC) deaths (5M£»),culling (SMC) grazing (Pq) Table 3.1: Tab le showing the processes affecting each state variable and some of the abbreviat ions used i n the mode l equations. for both hyrax and lynx. From a policy point of view the model is robust with 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 i l lustrate the usefulness of the dynamical systems software in this setting. Chapter 3. Sheep-Hyrax-Lynx Model 38 3.4 Technical details A few minor modifications to the model are required to facilitate the use of this software. The first involves scaling the state variables so that they al l have the same order of magnitude. Combining quantities of very different magnitude may lead to computer round-off errors [43]. I chose values close to the in i t ia l values in [116] as scaling constants. In other words, I replaced each state variable V{ by the quantity S j U j 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 ith population we can mult iply this new Vi by s; . To prevent the scaling from altering the dynamics of the model, the differential equation for ut- is divided through by S{. The above manipulations are made clearer in appendix C. Secondly, in order for the computer packages to generate continuous bifurcation dia- grams, al l functions in the model need to be continuous. The original model represents farmers' sheep cull ing strategies using two step functions. I replaced these with continu- ous 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 equi l ibr ium behaviour than with day to day variations, I did not include the seasonality functions. Solving the system of equations numerically over t ime is st i l l the best way to study seasonal variation in most cases. The various versions of A U T O only allow a state variable or the _L 2 -norm of the state variables to be displayed on the y-axis of the bifurcation diagrams they generate. The Z-2-norm of a vector v = ( u l 5 . . . , vm) is given by 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 Bard Ermentrout makes this possible. Let v be the vector of existing state variables and let h(v) be the revenue function. We can add the equation dR _ -R+h(v) ~dt ~ 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 wi l l not affect system equil ibria since at these points R = h(v) and hence ^ = 0 (as required for an equi l ibr ium value). The existence and stabil ity of phenomena such as periodic orbits ( l imit cycles) are also not affected. Since r acts as a delay t ime, we would like it to be small so that R is a close approximation to revenue. However, care must be taken in 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 M o d e l a n a l y s i s 3 . 5 . 1 R e f e r e n c e p a r a m e t e r va lues We need to choose an ini t ia l set of parameter values before we can determine the effects of varying parameters across ranges of values. Most of the values are given in [116]. Only values for the hyrax and lynx cull ing 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 equi l ibr ium. In Chapter 3. Sheep-Hyrax-Lynx Model 40 5 4 ^ £ C A T = 0 - 1 5 , 0 . 2 , . . . 3 Revenue 2 1 - 0 0 0.35 0.7 1.05 1.4 HCN Figure 3.2: Three one-parameter bifurcat ion diagrams w i t h revenue plot ted as a funct ion of HCN- E a c h curve corresponds to a different (fixed) value of 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 in X P P A U T to integrate the system of equations unt i l an equil ibr ium point was reached. Using this equi l ibr ium point as the in i t ia l point, I then used the A U T O interface to vary HCN- This 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. This diagram shows how the equil ibr ium revenue changes as HCN varies (see figure 3.2). The 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 opt imal (in terms of revenue) for al l values of LCN- I chose this value for HCN and then used X P P A U T to vary LCN in order to find the corresponding opt imal value for LCN- This gave figure 2 A complete descript ion of the available commands can be found i n the X P P A U T documenta t ion as wel l as i n the interact ive tu to r ia l that is available—see appendix 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 in revenue curves ly ing below that in figure 3.3 and higher values resulted in 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 opt imal and have the added advantage that small perturbations wi l l not have much effect on revenue, since the revenue surface appears to be fairly flat in a region surrounding these values. A t these reference values equi l ibr ium 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 mi l l ion Rand so the above values need to be mult ip l ied 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 in this model which could be used to il lustrate the dynam- ical systems techniques. Those affecting population growth rates are l ikely to have the Chapter 3. Sheep-Hyrax-Lynx Model 42 Revenue 0.2 0.4 SFCN (c) 10 P 5 - 0.6 (b) 0.8 LF 0.4 0.2 0.4 SFCN 0.2 0.4 SFCN 0.6 0.6 Figure 3.4: One-parameter bifurcat ion diagrams obtained from vary ing SFCN ( nomina l value = 0 .28/yr) w i t h L f j v = 0 . 7 / y r . In (a) revenue is plot ted on the y-axis , 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. greatest influence on the dynamics. I have chosen to study two such parameters in this chapter—the sheep female cull ing 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. As before I used X P P A U T to integrate the system numerically, using the reference parameter values, unt i l an equil ib- r ium was reached. Using these equil ibr ium 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. The results are shown in 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. The effects of varying a parameter with 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 puudj) 1 2 LFN Figure 3.5: One-parameter bifurcat ion diagrams obtained f rom vary ing LFN ( nomina l value = 0 .7 /yr) w i t h S ' jrc'jv=0.28/yr. In (a) revenue is p lot ted on the y-axis , 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. shows the effect of the parameter on equil ibr ium revenue, the second the effect on the equi l ibr ium number of lynx females, and the third the effect on the equi l ibr ium amount of pasture. A number of observations can be made from figures 3.4 and 3.5. The 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 imi t ing points, the causes of these difficulties can be determined. Output 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 lynx 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 imi t ing values there is no equil ibr ium at which al l three populations are present and, hence, A U T O cannot continue the equil ibr ium branch any further. A second observation can be made by looking at the bifurcation diagram for pasture in figure 3.4(c). As the equi l ibr ium number of sheep declines (as SFCN increases and approaches the value corresponding to sheep extinction), the equi l ibr ium value for pasture increases unchecked. Clearly this is unrealistic and suggests that some modification should be made to the model equations to l imit 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. Aga in 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 unt i l an equil ibr ium was reached. V iewing the numerical output in the data window lent some insight into the behaviour of the model . I found that as LFN increases through 0.76, prey abundance 3 at equi l ibr ium passes through a threshold value above which lynx begin to supplement their diet with lambs. This loss of lambs explains the decrease in revenue. A n important point to note is that if a tradit ional sensitivity analysis had been done using opt imal equi l ibr ium values for the state variables and a nominal value of L F i V = 0 . 6 / y r , then no change in equi l ibr ium revenue would have been seen for a 10 percent increase in LFN- However, using a nominal value of 0.7/yr, a 10 percent increase places LFN at 0.77/yr. This is above the threshold point and thus a decrease in equi l ibr ium revenue occurs. The only stabil ity 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 equi l ibr ium point becomes unstable and a periodic orbit is init iated. However, oscillations are only 3 See the explana t ion under equation (3.1). Chapter 3. Sheep-Hyrax-Lynx Model 45 associated with the sheep-pasture subsystem. The lynx and hyrax populations do not cycle. In this case the l imit 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 in this range of parameter values while the hyrax and lynx 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 imit pasture growth as suggested earlier, it would be informative to study the roles of the existing density-dependent functions. This can be viewed as another form of sensitivity analysis in 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 in 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 multipl ier which is a function of P^ , the pasture availabil ity index. PA is given by P/P where P is an average pasture density. The grazing mult ipl ier has the form shown in figure 3.6. Chapter 3. Sheep-Hyrax-Lynx Model 46 1.5 0 0 1 2 3 PA Figure 3.6: T h e grazing mul t ip l i e r function (GM) as a function of pasture avai labi l i ty . In order to remove a density-dependent function from the model , the simplest ap- proach is to replace it by a constant function. For example, we can set GM = 1- This was done for each multipl ier function in turn and bifurcation diagrams obtained from the altered model were compared with those from the original model in each case. Removing the grazing mult ipl ier , GM, had the greatest effect on model dynamics. Even after altering a number of parameter values the system did not reach equi l ibr ium. The sheep population either increased indefinitely or decreased to extinction for each parameter set that was tried. This is not surprising since the grazing mult ipl ier affects pasture grazing and sheep fecundity as well as sheep juvenile deaths. Without GM the density-dependence of pasture grazing and sheep dynamics on pasture availabil ity (GM is a function of PA) is removed. The results show that this density-dependence is cr i t ical for regulating the sheep-pasture subsystem. The effects of the other functions in the model were quantitative rather than qualita- tive. That 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 equi l ibr ium, but did not alter the qualitative dynamics. In relative terms however, the effects of the fecundity mult ipl ier functions were more noticeable than those of the death and predation multipl iers. The above comments do not imply that the latter multipl ier functions play an in - significant role in the model. They 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 cr it ical in determining model behaviour, they are important for making the model more realistic. It was observed earlier that the model lacks a feedback relationship that would l imit pasture growth when sheep densities are very low. We also know that pasture availabil ity 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 . This modif ication is discussed in the next section. 3.5.3 A d d i n g density-dependence to pasture growth The reader wi l l probably have noticed that equi l ibr ium pasture values only become un- 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 smal l range of values. A lso , improving model realism in extreme regions may affect the dynamics corresponding to more normal values and so play an important part in understanding system behaviour. The formulation of a model also affects statistical parameter f itt ing routines. If important relationships are left out then these routines may give misleading results or fai l 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 imi t pasture growth, I included a pasture multipl ier (PM) in equation (3.2) as follows: 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 in 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 in figure 3.7. (3.3) 1 0 0 2 4 PA Figure 3.7: T h e pasture mul t ip l i e r (PM) as a function of pasture ava i lab i l i ty (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 Model 49 SFCN Figure 3.8: One-parameter bifurcat ion diagrams obtained f rom vary ing SFCN ( n o m i n a l value = 0 .28/yr) for the new mode l which includes a pasture l i m i t i n g mul t ip l i e r . In (a) revenue is p lot ted on the y-axis , i n (b) the state variable for l y n x females is used, and i n (c) the state var iable for pasture is used. In figure 3.8(a) total revenue declines to zero as SFCN increases. This is more math - 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 pos- sible. Figure 3.8(c) shows the max imum pasture density which occurs when a positive sheep equi l ibr ium (and hence a positive value for revenue) is impossible. This density is lower than in figure 3.4(c) but depends on the exact nature of the pasture mult ipl ier . Another observation from figure 3.8 is that instead of A U T O being unable to converge at very low SFCN (sheep female cull ing normal) values, a Hopf bifurcation occurs and the bifurcation diagrams show that no stable equi l ibr ium at which al l three populations Chapter 3. Sheep-Hyrax-Lynx Model 50 (a) Revenue (c) 0.5 1 1.5 LFN Figure 3.9: One-parameter bifurcation diagrams obtained f rom vary ing LFN (nomina l value = 0 .7 /y r ) for the new mode l which includes a pasture l i m i t i n g multiplier. In (a) revenue is p lo t ted on the y-axis , 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 sup- port the high sheep population. Thus, introducing the pasture multipl ier has improved the dynamics at low values of SFCN as well as high values and has solved the problem of revenue increasing rapidly as in figure 3.4(a). In figure 3.9 a l imit point (see section A.2.13) has replaced the Hopf bifurcation of figure 3.5. The l imi t point bifurcation clearly shows that for LFN > 1-491 no equi l ibr ium at which al l three populations coexist is possible. Again , using X P P A U T to integrate the system numerically gives insight into the dynamics corresponding to the different regions in the bifurcation diagrams. In particular, the temporal dynamics show that the pasture mult ipl ier slows and l imits 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 imi t ing pasture growth has a stabilising influence (see figure 3.10). The oscillatory approach to equi l ibr ium by the original model (figure 3.10(a)) is replaced by a smooth approach in figure 3.10(b). 40 60 Time 40 60 Time Figure 3.10: T i m e plots obtained using (a) the or ig ina l mode l and (b) the modif ied m o d e l w i t h SFCN = 0.5. The l imi t ing value for pasture in figure 3.8(c) is st i l l rather high but, since I had no experimental data on which to base the form of the pasture mult ipl ier , I did not think it worthwhile to fiddle with the function to obtain a more plausible value. The effects of introducing the multipl ier have already been adequately demonstrated. A closer look at the behaviour exhibited by the model may suggest further modif ica- tions to the equations. The above is just one example of how bifurcation diagrams can help in the process of model building. Another example can be found in chapter 4. The 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 cul l ing both hyrax and l ynx The model in this chapter was originally developed to study the effects of cull ing hyrax and lynx. As was done in [116] and earlier in this chapter, opt imal values (with respect to Chapter 3. Sheep-Hyrax-Lynx Model 52 5 4 3 Revenue 2 1 0 0 0.1 0.2 0.3 0.4 LCN Figure 3.11: 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 l imit point. revenue) were found for the hyrax and lynx culling normals. However, only one parameter was varied at a t ime. 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 cull ing normal LCN have the form shown in figure 3.11. For LCN below the l imit point the sheep population dies out as there is too much predation by lynx. Using A U T O the l imi t point can be continued in HCN, the hyrax culling normal, as well as LCN- That is, we can see how the position of this l imit point varies as a function of both HCN and LCN • This gives the two- parameter bifurcation diagram in figure 3.12. From this figure it can be seen that sheep become extinct as a result of the combined effect of lynx predation and competit ion with 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 imi t point in the corresponding LCN bifurcation diagrams (see figure 3 . 3 ) — A U T O Chapter 3. Sheep-Hyrax-Lynx Model 53 0.4 0.3 HCN 0.2 0.1 0 0 0.1 0.2 0.3 0.4 LCN Figure 3.12: Two-parameter cont inuat ion of the l i m i t point i n figure 3.11. (To determine the behaviour corresponding to a par t icular region i n this two-parameter d iagram, choose values for HCN and LCN i n this region and then use X P P A U T to integrate the system numerical ly .) signalled non-convergence and stopped calculating. Combining figure 3.12 with the observation that the lynx population dies out for LCN > 0.37 for al l 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 equi l ibr ium for a large set of cull ing rates. The diagram would be of even greater use if we knew the revenue value corresponding to each point in this two parameter space. This can be done by recording information given by X P P A U T and using some other graphics package to plot a three- dimensional 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 equi l ibr ium point numerically. I Sheep die out Chapter 3. Sheep-Hyrax-Lynx Model 54 0.4 HCN 0.2 |- Sheep die out A l l three populat ions coexist at equ i l i b r i um L y n x die " out 0.1 0.2 LCN 0.3 0.4 Figure 3.13: Two-parameter bifurcat ion d iagram of the HCN and LCN parameter space for the modif ied m o d e l . then used the A U T O interface to vary LCN in both directions. Using the G R A B feature of X P P A U T to move along the branch of equi l ibr ium 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 in 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 in the cull ing rates in this region. This is a desirable property when it comes to developing management strategies. 3.5.5 B io log ica l interpretat ion of results The analysis of the previous sections has led to a number of insights into the sheep- pasture, hyrax- lynx 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. Cul l ing too many ewes wi l l obviously cause sheep numbers to decline. More important is the effect on revenue. Significant increases in 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 cull ing fewer ewes results in a lower cash flow. Another trade-off results from the decrease in pasture availability which accompanies a larger sheep stock. This is already reflected in 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 wi l l affect the returns from wool and mutton 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 lynx fecundity is very high (LFN > 1-5) then the sheep population wi l l not be able to survive. However, if lynx 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. Further decreases in lynx fecundity at these values have no effect on revenue. For intermediate values (0.78 < LFN < 1.5) .considerable increases in revenue are possible if lynx fecundity is decreased. This favours the cull ing of lynx females in particular. In section 3.5.2 we found that density-dependence of sheep and pasture dynamics on pasture availabil ity is crit ical for regulating the sheep-pasture subsystem. In fact this encouraged the modification of pasture growth to include density-dependence. This mod- 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 inal ly , the effects of cull ing both hyrax and lynx were summarised using two- parameter diagrams. In particular, figure 3.14 shows that the model is robust to changes Chapter 3. Sheep-Hyrax-Lynx Model 57 in cull ing rates provided these rates are sufficiently high. 3 .6 C o n c l u s i o n This chapter has i l lustrated a number of potential uses of bifurcation analyses using packages such as X P P A U T . F i rst , 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 tradit ional 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. As a result these diagrams can indicate where model relationships are incomplete and can thus aid in model formulation. Another example of this can be found in chapter 4. The analysis also showed that the model is quite robust in a qualitative sense—the stabil ity 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. The 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 in this chapter can be useful for uncovering regions of more complex behaviour in such cases. Chapter 4 Ratio-Dependent Mode l 4.1 Introduction Despite having a large number of state variables and parameters, the system dynamics model in 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 may be approached using dynamical systems techniques, a dual a im of the chapter is to highlight some of the difficulties associated with ratio-dependent models. These models are currently a topic of considerable controversy in ecological circles. The 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 laim makes estimation of parameter values from experimental data fairly straightforward. However, there are a number of correction factors in 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 may 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 in 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 with ratio-dependent models. After choosing a set of parameter values which give rise to a stable tr itrophic equil ib- r i um (that is, a stable equi l ibr ium at which the plant, herbivore and predator populations are al l nonzero) I begin the analysis by varying each parameter value in 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 prel iminary 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 il lustrate 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 mult ip le stable states. In some cases a stable tritrophic equi l ibr ium coexists with a stable l imi t cycle suggesting the possibility of an abrupt change in the behaviour of the system if it is sufficiently perturbed (see section A.2.7) . To complete the study the l imits of isocline analysis 1 in a three-dimensional setting are demonstrated. Gutierrez et al. [52] used this technique in their analysis of the model . A l though isocline analyses have been employed in many settings and with considerable success [36, 38, 44, 56, 74, 92, 103], in more complicated higher dimensional models for which the categorisation of variables as slow versus fast2 is not possible, their application is l imited. A n isocline analysis allows at most two variables to vary simultaneously. This means that for the current model one variable is held fixed which results in a part ly static * A descript ion of this technique together w i t h examples can be found i n [34]. 2 I f the state variables i n a mode l vary on different t ime scales i t is often possible to approximate the system by a two-dimensional mode l representing either the slow or the fast dynamics . A n isocline analysis can then be done using the reduced system. Chapter 4. Ratio-Dependent Model 60 representation of the dynamics. The dynamical systems techniques allow al l three state variables to vary simultaneously thus permitt ing 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 with fresh interest in ecological theory in recent years (Berryman [17]). A m o n g the advantages of these types of models are that they prevent the paradoxes of enrichment 3 and biological control 4 predicted by classical models [17]. Exper imental 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 Ginzburg and Akcakaya [45] and McCar thy et al. [88] conclude from 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, in the top consumer is sufficient to preclude the paradox of enrichment from classical models. From his work on whether patterns among trophic levels are a reliable way of distinguishing between prey- and ratio- dependence, Sarnelle [107] concludes that the ratio-dependent approach should only be applied when the predator and prey are the top two trophic levels in an ecosystem. Abrams [1] argues that patterns and experimental results that have been used in support 3 C lass ica l models predict that enriching a system w i l l cause an increase i n the e q u i l i b r i u m density of the predator but not the prey and w i l l destabilise the communi ty equ i l i b r i um (see B e r r y m a n [17]). 4 C l a s s i c a l models predict that i t is not possible to have bo th a very low and 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]). 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 Fryxel l [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 crit icisms. The 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 in these regions. As a result even when prey (re- source) 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 Hanski [55] this is against intuit ion 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 in a mathematical sense and that in biological terms the result would be that both species increase init ial ly and then predators consume all the prey and both species become extinct. Since prey-dependent models cannot predict this outcome they are pathological in 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 in the next section predicts oscillations of large amplitude in these 'pathological' regions (see figure 4.3). Whi le 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 ratio- dependent model . Akcakaya 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 equil ibria, l imit 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 in 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 in this debate, I introduce a small mod- ification to the ratio-dependent model of Gutierrez et al. [52] and study this modified model in conjunction with 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 The model equations are functionally homogeneous (that is, al l 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 al l organisms as they seek to satisfy their metabolic requirements. Details of the formulation of the model can be found in Gutierrez et al. [52]. The final equations are: Chapter 4. Ratio-Dependent Model 63 dMx ' <xxM0" rDxMx. ) Dx - rxMx h Mx dt D2M2 dM2 D2 - r2M2b* M2 dt D3M3 dM3 D3 - r3M3b3 M3 (4.1) dt where Mx is the biomass of plants, M2 is the biomass of herbivores, and M3 is the biomass of predators. The 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 - l imitation. MQ represents a biomass equivalent of the light energy incident in the growing space of the plants, and (f>Vtx and d)u>x (which both lie between 0 and 1) scale the potential photosynthesis rate to the realised rate. The respiration term, r 4 M , 1 + f > ' , requires further explanation. Respiration usually in - creases wi th 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 smal l , the authors chose the simpler formulation r , - M , 1 + 6 ' where bi has a value between 0.02 and 0.05. The disadvantage of this choice is that r; must have rather unusual units 5 which depend on bi so that the term r ; M / + b ' has the same units as (namely, g.day-1 where g are the units of Mi). This dependence of the units on b{ is not satisfactory from a mathematical viewpoint but since bi is small I chose to ignore this init ial ly. In the next section I discuss a small alteration to the model 5 T h e units of ri are g~b'day~1 where g are the units of M ; . Chapter 4. Ratio-Dependent Model 64 which takes care of the difficulty. Gutierrez et al. [52] use parameter values corresponding to a cassava 6 -mealybug 7 - parasitoid 8 system in Afr ica and claim that their analysis demonstrates that the para- sitoid Epidinocarsis lopezi (De Santis) can control the mealybug (except on poor soils) whereas Epidinocarsis diversicornis (Howard) and native natural enemies cannot. M y first a im 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. Only after changing some of the parameter values by several orders of magnitude did I succeed in producing qualitatively similar diagrams. This haphazard approach of fiddling with parameter values is not satisfactory. Scaling the equations would give a better idea of the relative magnitudes of the parameters. The procedure involved is discussed in the next section. 4.4 N o n d i m e n s i o n a l i s a t i o n The technique of nondimensionalising or scaling is commonly used to simplify a system of equations as it has a number of other advantages. It i l luminates which parameters are most important in 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 al l have the same order of magnitude, say between 0 and 1. This is important when solving the equations numerically as very different magnitudes can lead to computer round- off errors (Gerald and Wheatley [43]). In the cassava-mealybug-parasitoid system the 6Manihot escuhnta C ran tz 7Phenacoccus manihoti Ma t . -Fe r r . 8 T h e larvae of a paras i to id feed on l i v i n g host tissue such that the host is not k i l l e d u n t i l l a rva l development is finished. 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 ta l . Natura l scalings 9 for the state variables are given by their carrying capacities when resources are abundant. Gutierrez et al. [52] calculate these to be 'BiDi Ki = i = 1 ,2 ,3 . Replacing Mi by KiMi (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) 1 0 . dMi dMx dMx dt dt* dMx dt dt* = T (OxDx (l - exp ax M0 DxKxMx r^ExMiP Mi dM2 ~d~F dM3 dt* —T ^1 — exp T (e2D2 (I - — r 1 — exp cx2KxMx D2K2M2 exp a2K1M1 D2K2M2 a3K2M2 D3K3M3 D3K3 K2 r2{K2M2)b2 M2 M3 r 63D3 1 - exp a3K2M2 D3K3M3 - r3(K3M3)b* M3. (4.2) Note that riK.\* = 6iDi. Choosing the dimensionless combinations of parameters ji = rOiDi i — 1 ,2 ,3 <f>i = TOLi 2 = 2,3 n etiKi-x a - = t t 1 ! = 1 ' 2 ' 3 9 For an in t roduc t ion to nondimensional is ing systems of ordinary differential equations see [34]. 1 0 I replaced the product #i</>r),i^aj,i by 0 \ since a l l three parameters have the same effect on the dynamics and, thus, do not need to be considered separately. Chapter 4. Ratio-Dependent Model 66 where KQ = M0, gives the nondimensionalised equations (4.3) where I have replaced Mi by Mi and t* by t for convenience. dMi dt dM2 dt dM3 dt 7i ^ 1 - e x p = 7 2 ^ 1 - e x p = 7s ( ( 1 - exp "~M\ n2Mx M2 £l3M2 M10i)M1-^-[l-exp - M2b* )M2-^-(l- exp 0 2 M X n3 M2 n3M2 Mo M3 M3 M3 - M3h 1 M3 (4.3) The 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 in converting , the resource into biomass. fa can be thought of as the availabil ity of the resource to the consumer or perhaps the nutrit ional value of the resource. 0 ; is made up of a ratio of quantities. The 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. More simply, Cli gives a measure of the ratio of supply to demand. The results in [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 with those in [52]. I. mentioned earlier that the parameters r,- in the original model have units which depend on bi. This can be prevented by replacing the terms r i M i 1 + b i wi th terms of the form r i M ^ Y 1 ) 6 ' where Ti has the same units as Mi. T can be thought of as a threshold value above which self - l imitation becomes noticeable. W i t h this modification the units of r,- are day-1 and rt- can be interpreted as a respiration rate as was originally intended. The 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 in system (4.3) once again. Thus the problems with the respiration term can be ignored in the rest of the analysis. Having scaled the equations we need to choose values for the parameters. A n ad- vantage of scaling is that there are now 11 parameters instead of the original 18. For comparison with the results of Gutierrez et al. [52] I wanted to find values which resulted in isocline configurations similar to those in their paper. X P P A U T calculates and dis- plays isoclines in 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 competit ion 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 2 - p e a k . In fact, using the techniques in [52] it would not have been possible to conclude that the tr itrophic equi l ibr ium is stable for the isocline configuration shown in 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 imit cycle. The in i t ia l values of Mi, M2 and M 3 determine whether the system approaches the stable equi l ibr ium or the l imi 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 equi l ibr ium point, it is a convenient starting point. Before beginning an analysis of the model I would like to introduce a small mod- ification to the equations. Since the problems associated with ratio-dependent models Chapter 4. Ratio-Dependent Mode] 68 Figure 4 . 1 : Isoclines in the (a) MiM2 and (b) M2M3 planes for 7! = 2.0, j2 = 0.4, 73 = 0.1, £2i = 9.0, fi2 = 8.0, Q3 = 10.0, fa = 0.4, <j>3 = 0.05, and b{ - 0.02 ( i= l , 2 ,3 ) . B o t h the stable equ i l i b r i um point and the l i m i t cycle are shown. M3 is fixed at 0.550 in (a) and M\ is fixed at 0.602 in (b). These values correspond to the equ i l ib r ium point . 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. Abrams [l] states that modifications to ratio-dependent models cannot be made biologically realistic be- cause 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. The ratios in model (4.3) have the form Mi ' The difficulties are experienced when Mi approaches zero. Add ing a constant in the denominator, that is replacing the ratio by *T*< <4'4> a,- + Mi would alleviate the problem. Although this addition may appear difficult to justify Chapter 4. Ratio-Dependent Model 69 biologically, Gutierrez [51] used an exponent of this form in his functional response term. That 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. When the exponent is small we have k (1 — exp Mi Mt « k(i-(i-9M^))Mi = kiliMi-x. In order to preserve this property when using the modified term (4.4), I replaced (4.5) by k (1 — exp ftM-i (a; + Mi) cii + Mi. where a,- is a small constant, say 0.001. The resulting equations are: dMx dt dM2 ~dT dM3 ~dT 71 ^1 - exp - j £ l l - e X P ax + M: ft2Mx 72 [1 - exp fa f, - r U 1 _ e x p 73 (1 - exp a2 + M2 n2Mx I a2 + M2 VL3M2 a3 + M3 n3M2 1 r - ] ) ( a 1 + M 1 ) - 7 l M 1 1 + 6 1 (a2 + M2) a2 + M2) - >y2M21+b* (a3 + M3) a3 + M3)-73M31+b\ (4.6) a3 + M3\ If a t =0 (i = 1,2,3) then the above model is equivalent to system (4.3). We are now in a position to begin the analysis. Chapter 4. Ratio-Dependent Model 70 4.5 M o d e l analysis 4.5.1 One-parameter studies Since a part ial qualitative analysis of the original model was done by Gutierrez et al. [52] but using different techniques (namely, isocline analysis), it wi l l be informative to compare some of the results. The choice of dimensionless parameters was done with this in mind . 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 wi l l be done for all the parameters except the 6;'s due to the observation that intraspecific competit ion at the ith trophic level now increases as bi decreases since Mj now lies between 0 and 1 as a result of the scaling. This was overlooked in 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 st i l l increases with biomass as required.) Since one of the main conclusions in 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 fi3. I then discuss the remaining parameters. Both the original model corresponding to a t =0 (i = 1,2,3) in 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 in 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 in table 4.1. The results, together with possible biological interpretations, are described in the next section. For generality I wi 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 71 P a r a m e t e r Descr ip t ion V a lue 7 i potential growth rate per unit plant 2.0 72 potential growth rate per unit herbivore 0.4 73 potential growth rate per unit predator 0.1 <j>2 availability of plant to herbivore 0.4 4>3 availability of herbivore to predator 0.05 fix supply of resources/demand by plants 9.0 supply of plants/demand by herbivores 8.0 supply of herbivores/demand by predators 10.0 6i degree of self-limitation for plants 0.02 b2 degree of self-limitation for herbivores 0.02 b3 degree of self-limitation for predators 0.02 Table 4.1: Reference parameter set. In the subsequent figures i n this chapter, on ly those parameters which are exp l i c i t l y ment ioned have been altered. The values for a l l the other parameters correspond to the ones i n this table. Analys is of the predator parameters The parameter 73 can be thought of as the potential predator biomass growth rate when prey are abundant, or as the predator's conversion efficiency in the presence of abundant prey. A n important observation is that the value of 73 does not affect the isoclines. The M3 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 in [52] would not give any insight into how this parameter influences the behaviour of the system. Bifurcat ion 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 unt i l it was close to an equil ibr ium and then X P P A U T was used to find the exact location of the equil ibr ium 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 equi l ibr ium point does not vary Chapter 4. Ratio-Dependent Model 72 LIS3,oooYoo*LP H B P D 8 _ LPs i < « « »0«i'i*LP i (b) M i (a) 1 0.8 0.6 M i 0.4 0.2 0 0 0.05 0.1 0.15 0.2 73 F i gure 4.2: One-parameter bifurcat ion diagrams obtained by vary ing 73 i n (a) the o r ig ina l mode l (a;=0, i = 1,2,3) and (b) the modif ied mode l (a;=0.001, i — 1,2,3) . T h e state variable M\ is plot ted on the y-axes. H B denotes a H o p f bifurcat ion, LP a l i m i t point and PD a per iod-doubl ing bifurcat ion. with 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 equi l ibr ium point is unstable for very low values of 73 (low predator growth rate) and the stable attractor is a l imit 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 at t ractors—a sink and a stable l imit cycle (such as in figure 4.1). The in i t ia l values of the state variables determine which final state is reached. Also, perturbations to the system may cause a jump 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 imi t cycles in figure 4.2(a). The range of values is so smal l , however, that it is not of much 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 t ime taken to reach equi l ibr ium. 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 equi l ibr ium biomasses. The 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). Thus, 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 availabil ity (or nu- tr i t ional value) of the herbivore to the predator. Bifurcation diagrams for the original and modified models respectively and for al l three state variables are shown in figures 4.3 and 4.4. F rom these figures we can see that as fa increases there is a general increase in the Mi equi l ibr ium value or l imit cycle max imum. The larger fa the greater the availabil ity 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 equi l ibr ium. As the Mi equi l ibr ium value approaches the Mi carrying capacity in the original model , a Hopf bifurcation (see section A.2.10) occurs at fa = 0.09 (see figure 4.3). The periodic orbit associated wi th 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 in 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). As fa increases above the upper Hopf bifurcation the m i n i m a of the M2 and M% cycles in particular get very small (of the order of 1 0 - 1 5 and lower according to X P P A U T ' s data window). From a practical viewpoint these populations would be considered extinct due to statistical variation in which case the plant population would increase to its carrying Chapter 4. Ratio-Dependent Model 74 0 0.05 0.1 0.15 <f>3 (b) 0.6 | 1 1 1 (c) 0.15 0.15 fa F i g u r e 4.3: One-parameter bifurcat ion diagrams obtained by vary ing <f>3 i n the o r ig ina l mode l . T h e state variables M\, M2 and M3 are plot ted on the y-axes i n (a), (b) and (c) respectively. O n l y the posi t ions of the H o p f bifurcations have been indicated. T h e changes i n s tab i l i ty of the l i m i t cycles occur at l i m i t points or per iod-doubl ing bifurcations but these have not been marked i n the d iagrams. Chapter 4. Ratio-Dependent Model 75 Figure 4.4: One-parameter bifurcation diagrams obtained by varying <f>3 in the modified model. T h e state variables Mi, M2 and M 3 are plotted on the y-axes in (a), (b) and (c) respectively. Chapter 4. Ratio-Dependent Model 76 (a) 1.2 (b) Time (c) 1- 0.8 0.6 M3 0.4 Time Figure 4.5: Time plots of (a) M i , (b) M 2 and (c) M 3 for <p3 = 0.16. A l l the other parameter values are as in the reference parameter set. Chapter 4. Ratio-Dependent Model 77 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. This wi l l be discussed in more detail later. Another problem with these low m i n i m a is that they lead to numerical difficulties. This occurs as a result of the way in which the model is formulated—the dependence of many of the terms on the ratio in 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. Manual ly 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 imit 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>3 has a significant effect on the equi l ibr ium values of all three state variables. This is in agreement with 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 - isoci ine. It is true that if fa is decreased the M2 - isocline widens (see figures 4.6(a) and (b)). But this is provided that M3 is constant. If the system is integrated and M3 is allowed to vary unt i l a new equi l ibr ium is reached and the isoclines are plotted with this new equi l ibr ium M3 value, then the final M2 - isocline may in fact have a narrower C-shape than before (see figure 4.6(c)). The thi rd parameter which directly affects the predator is Q 3 . It is more difficult to 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) M 2 1 0.8 0.6 0.4 0.2 0, / - - - 0 M l (b) M 2 1 0.8 0.6 0.4 0.2 0. - "A 0 03. OA OS 01 M l (c) 0.2 0.4 0.6 0.1 M l Figure 4.6: Isoclines i n the M i M 2 plane. In (a) <f>3 = 0.06 and i n (b) and (c) <f>3 = 0.05. A l l the other parameter values are fixed. In (a) and (b) M3 has the same value—the value corresponding to the stable t r i t roph ic e q u i l i b r i u m point i n (a) when <j>3 = 0.06. In (c) M3 has the value corresponding to the e q u i l i b r i u m point when <f>3 — 0.05. Chapter 4. Ratio-Dependent Model 79 capacities. Hence it reflects how l imit ing 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 fi3 there is no change in the qualitative behaviour which in this case is given by stable l imit cycles. Even the amplitudes of these cycles do not alter much although those for M 3 decline slowly as f i 3 is increased. It is only at low values of 0 3 that a change in dynamics occurs. It is also at these low values that D 3 has the greatest effect on equi l ibr ium magnitudes. Low values of tts suggest a restricted supply in relation to demand and are reflected in both low M2 and low M 3 equi l ibr ium values. Since there are relatively fewer herbivores available (in relation to predators) the predators are able to control them better but the lower herbivore equil ibr ium also restricts the number of predators that can survive. As expected, lower equil ibr ium values for M2 correspond to higher equi l ibr ium values for M i . The corresponding bifurcation diagrams for the modified model are shown in 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 in the amplitudes of the cycles followed by a stable tr itrophic equi l ibr ium as ££3 is increased. Again the introduction of the a,-'s has had a stabilising influence on the dynamics. From figure 4.8 we can clearly see that high values of fi3 are, however, undesirable as the M x equi l ibr ium value is low while that for M2 is relatively high. The above analysis has shown that the properties of the predator affect the stabil ity of the system as well as the equil ibr ium magnitudes of the herbivore (directly) and the plant (indirectly). The extent of these effects depends on the properties of both 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 Model 80 Figure 4.7: One-parameter bifurcat ion diagrams obtained by vary ing Q3 i n the o r ig ina l mode l . T h e state variables M\, M2 and M 3 are plot ted on the y-axes i n (a), (b) and (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 , M2 and M 3 are plotted on the y-axes in (a), (b) and (c) respectively. Chapter 4. Ratio-Dependent Model 82 Analysis of the plant and herbivore parameters The parameter (f>2 can be thought of as the availability or the nutrit ional value of the plant to the herbivore. The bifurcation diagrams for the original and modified models respectively and for Mi, M2 and M 3 are shown in figures 4.9 and 4.10. For both models the values of M2 and M 3 at equi l ibr ium remain constant for the most part as (f)2 is varied (up to a l imit ing point) while the Mi equi l ibr ium value declines. A 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>2 value. This enables the herbivore to have an even greater impact on the plant. If (f>2 is sufficiently high then the herbivore can send the plant population to extinction. This is suggested by the very low cycle min ima for the original model and is even clearer for the modified model where the Mi equi l ibr ium value is very low for (j>2 > 1.061 (the location of the Hopf bifurcation). Of 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)2 to the right of the upper Hopf bifurcation (4>2 > 1.061) 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. The modified model does not have this problem. Suppose we vary 71, the assimilation or conversion efficiency of the plant when re- sources are abundant. The bifurcation diagrams are shown in figure 4.11. Comparing these diagrams with figures 4.9 and 4.10 we see that they are almost mirror images. That is, decreasing 71 has a very similar effect to increasing <f>2. Both parameters can be 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. As a result the Chapter 4. Ratio-Dependent Model 83 F i g u r e 4.9: One-parameter bifurcat ion diagrams showing the effects of va ry ing <f>2 for the o r ig ina l mode l . T h e state variables M i , M 2 and M 3 are plot ted on the y-axes i n (a), (b) and (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, M2 and M 3 are plotted on the y-axes in (a), (b) and (c) respectively. Chapter 4. Ratio-Dependent Model 85 (a) M i (b) M i Figure 4.11: One-parameter bifurcat ion diagrams showing the effects of va ry ing 71 for (a) the o r ig ina l mode l and (b) the modif ied mode l . T h e state variable M i is p lo t ted on the y-axes. T h e l i m i t points ( L P ) m a r k the endpoints of the region of hysteresis (see section A.2.11). detrimental effect of the herbivore on the plant is greater. Increasing fa, the availabil - ity (nutrit ional value) of the plant to the herbivore, achieves the same result but more directly. We can generate two-parameter diagrams in ( 7 i , 0 2 ) - s p a c e by continuing the l imit points and the Hopf bifurcations in two parameters (see sections A .2 .1 and B.4). The results are shown in figure 4.12—sol id lines indicate Hopf bifurcation continuations and dotted lines indicate l imit point continuations. These diagrams show clearly that decreas- ing 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 imit 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 imit cycles and/or mult iple stable states are possible. Both these phenomena are important biologically. We can make a number of observations by comparing figures 4.12(a) and (b). F i rst , Chapter 4. Ratio-Dependent Model 86 (a) 0.75 fa 0.5 0.25 1 E q u i l i b r i u m values 1 I for a l l three _ popula t ions tend •"' — to zero _. • V ^ ^ \ T w o stable . -'li* at tractors M X Stable j . t r i t rophic J Stable .jy^ Stable t r i t rophic equ i l i b r i um / c y c l e s ^ ^ e q u i l i b r i u m - ( l o w plant / (high plant value) value) i i i (b) 1 0.75 E q u i l i b r i u m values for a l l three _ popula t ions tend to zero fa 0.5 - 0.25 Stable t r i t rophic equ i l i b r i um -( low plant value) _. 2 7i Stable t r i t rophic equ i l i b r i um (high plant value) 7i F i g u r e 4.12: Two-parameter bifurcat ion diagrams showing the effects of va ry ing bo th 71 and fa on the posi t ions of the l i m i t points and H o p f bifurcations i n figure 4.11. T h e d iag ram i n (a) corresponds to the or ig ina l m o d e l and that i n (b) to the modif ied mode l . So l id lines indicate H o p f b i furcat ion continuat ions and dotted lines indicate l i m i t point continuations. 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 equi l ibr ium values for the state variables are close to zero in the upper left triangle of the two-parameter space and this results in numerical problems when using the original model. X P P A U T also has difficulty calculating zero isoclines in 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 tr itrophic equil ibria while stable cycles occur in the other. One of the regions of stable equil ibria has high equi l ibr ium values of M i but the other has low equil ibr ium values—an important distinction ecologically. The 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 min ima when 72 and <j)3 are varied independently (compare figures 4.3, 4.4 and 4.13). We can investigate this relationship in more detail by continuing the Hopf bifurcations in figure 4.13 in fa as well as 72. The results are shown in figure 4.14. 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 72 72 Figure 4.13: One-parameter bifurcation diagrams showing the effects of varying 72 for (a) the original model and (b) the modified model. Again , in general decreasing fa has a similar effect to increasing 72. However there are a few more Hopf bifurcations associated with these parameters. A U T O stops at the Chapter 4. Ratio-Dependent Model 88 Figure 4.14: Two-parameter bifurcation diagrams showing the effects of vary ing bo th and <p3 on the posi t ions of the H o p f bifurcations in figure 4.13. Chapter 4. Ratio-Dependent Model 89 point marked M X when using the original model since the equi l ibr ium values for M 2 and M 3 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 model , are more stable in general. The regions corresponding to tr itrophic equil ibria are larger and the cycles in the upper half of the two-parameter space are less complex. These claims are made clearer in 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 with a t =0.002 [i = 1,2,3) . The regions of stable equil ibria are even larger than in figure 4.14(b) resulting in 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 2=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 .15 and (ii) 72=0.7, </>3=0.3. (c) These plots were obtained using the modified model with a«=0.002 (i = 1,2,3) and (i) 7 2=0.8, <p3=0.2 and (ii) 72=0.8, <^3=0.3. Chapter 4. Ratio-Dependent Model 91 Figure 4.16 shows the t ime plots corresponding to points marked with *'s in the left- hand section of the upper region of cycles in figures 4.14(a),(b) and 4.15. The left -hand portion of this region is where the values of M2 and M 3 are low and, hence, where the nonzero a 8 's have most effect. Clearly nonzero a,-'s reduce the complexity of the cycles (even for very small values) and increasing their value also reduces the cycle amplitude. A n addit ional 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 rom an ecological viewpoint. The only parameters that have not been discussed so far are f i i and fl2. B i furca- t ion diagrams corresponding to O i are shown in figure 4.17. Only the diagrams for Mi ( a ) Mi (b) Mi 10 20 30 40 50 10 20 30 40 50 fli Figure 4.17: One-parameter bifurcat ion diagrams showing the effects of va ry ing fli for (a) the o r ig ina l mode l and (b) the modif ied mode l . T h e state variable Mi is p lot ted on the y-axes. have been shown but those for M2 and M3 have the same shape—the carrying capacity equil ibria are however different. As can be seen from figure 4.17, in both cases fli does not affect the stabil ity of the equi l ibr ium point and only affects equi l ibr ium magnitudes when it drops to low values. Similar conclusions can be made regarding ft2 (see figure 4.18) except that at low Q2 values there is a change in dynamics to stable l imit cycles for both models although the Chapter 4. Ratio-Dependent Model 92 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 p l a n t a n d h e r b i v o r e d y n a m i c s 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. The two-parameter diagrams in 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 wi th 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, wi 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 imi 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 in conjunction with these diagrams. As 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 imit point continuations (dotted lines). Let us consider the l imit point continuations first. Figures 4.24(a), (b) and (c) show one-parameter bifurcation diagrams obtained by varying 72 wi th 71 = 0.4 and <f>3 = 0.07, 0.17 and 0.25 respectively. These diagrams correspond to the horizontal dotted lines in figure 4.19. The l imit points in these diagrams demarcate the region of hysteresis, that is, the range of parameter values giving rise to mult iple equil ibria. We can see that this range of values increases as <f>3 increases. The existence of a hysteresis phenomenon does not imply the existence of two stable equil ibria. This depends on other factors such as the occurrence of Hopf bifurcations. In figure 4.24(a) there are two stable equil ibria for 72 between the two l imi t points (0.18 < 72 < 0.30). One of these stable equil ibria corresponds to a low equi l ibr ium Mi value and the other to a high equi l ibr ium Mi value. The unstable equi l ibr ium intermediate to these stable equil ibr ia 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 equil ibria occurs between the lower l imit point and the Hopf bifurcation on the upper branch of equil ibr ia (0.41 < 72 < 0.57 in (b) and 0.60 < 72 < 0.65 in (c)). Col lat ing the information obtained from figure 4.24 allows us to classify the different regions in figure 4.19 according to the qualitative behaviour that is found there. That is, the region enclosed by the l imit point continuations (region of hysteresis) can be divided into a region of two stable equil ibria and a region of only one stable equi l ibr ium having a Chapter 4. Ratio-Dependent Model 94 1 / 1 /Hys teres is / / (one stable-" i i Stable equi l . / (high plant .• value) / N equil . )- ' Stable t r i t rophic equ i l i b r i um (low plant value) - - - - ^ j T w o stable equ i l ib r ia - i i i i 0 0.5 1 1.5 2 2.5 Figure 4.19: Two-parameter bifurcat ion d iagram obtained using the modif ied mode l w i t h a ,=0.001, (i = 1, 2, 3) and 71 = 0.4. T h e H o p f bifurcat ion continuations are indica ted by sol id lines and the l i m i t poin t cont inuat ions by dotted lines. Chapter 4. Ratio-Dependent Model 95 Figure 4.20: Two-parameter bifurcat ion d iag ram obtained using the modif ied mode l w i t h 0^=0.001, (i = 1, 2, 3) and 71 = 0.6. T h e H o p f bifurcat ion continuations are indicated by so l id lines and the l i m i t point cont inuat ions by dotted lines. Chapter 4. Ratio-Dependent Model 96 Figure 4 .21 : Two-parameter bifurcat ion d iagram obtained using the modif ied mode l w i t h a t =0.001 , (i = 1, 2, 3) and 71 = 1.2. T h e H o p f bifurcat ion continuations are indica ted by sol id lines and the l i m i t point cont inuat ions by dotted lines. Chapter 4. Ratio-Dependent Model 97 Figure 4 . 2 2 : Two-parameter bifurcation d iagram obtained using the modified model wi th a ,=0.001, (i = 1,2,3) and ji = 1.8. T h e H o p f bifurcation continuations are indicated by solid lines and the l i m i t point cont inuat ions by dotted lines. Chapter 4. Ratio-Dependent Model 98 Figure 4.23: Two-parameter bifurcation d iagram obtained using the modified mode l w i t h a;=0.001, (i = 1,2,3) and 71 = 2.4. T h e Hopf bifurcation continuations are indicated by sol id lines and the l i m i t point cont inuat ions by dotted lines. Chapter 4. Ratio-Dependent Model 99 F i g u r e 4 . 2 4 : One-parameter bifurcat ion diagrams obtained by vary ing 72 w i t h 71 = 0.4 and (a) <f>3 = 0.07, (b) c/>3 = 0.17 and (c) ^3 = 0.25. (These correspond to the hor izonta l dot ted lines i n figure 4.19.) Chapter 4. Ratio-Dependent Model 100 low value for Mi. The regions outside the l imit point continuations have one equi l ibr ium point and it is stable. Those to the left of the hysteresis region have high equi l ibr ium Mi values and those to the right have low equil ibr ium Mi values. The 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 imit point continuations. The one-parameter bifurcation diagram in figure 4.25 corresponds to the horizontal dotted line in 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 imit cycles. Starting at 72 = 1.5, as 72 is decreased we have a single stable equi l ibr ium wi th a low Mi value. A t the (lower) Hopf bifurcation this stable equi l ibr ium is replaced by stable l imit cycles which increase in amplitude as 72 is decreased. As the upper Hopf bifurcation is approached the cycles undergo some period-doubling bifurcations and then rapidly decrease in amplitude. Due to a hard loss of stabil ity associated with 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 equi l ibr ium 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 imi t cycles when ^3 = 0.2. Using this information together with results obtained from one-parameter diagrams at different fixed values of fa allows us to determine the regions in figure 4.21 which give rise to stable cycles. Such regions also occur in figures 4.20, 4.22 and 4.23. Al though figure 4.19 contains Hopf bifurcations, the cycles associated with them are unstable (see, for example, figure 4.24) and occur over such a small parameter range that they are not of much biological interest. Having classified the qualitative dynamics in the various regions of the two-parameter Chapter 4. Ratio-Dependent Model 101 • -LP - • • _ L P - i i l B . » »i » 0 0.5 1 1.5 72 Figure 4.25: One-parameter bifurcat ion d iagram obtained by vary ing 72 w i t h 71 = 1.2 and <f>3 = 0.2. (Th i s corresponds to the hor izonta l dot ted l ine i n figure 4.21.) diagrams in figures 4.19-4.23 we can make a number of general observations. F i rst , as 71 increases the l imi t point continuation curves move closer together unt i l in figure 4.23 there are no l imi t points at al l . Hence, increasing 71 reduces the amount of hysteresis and the possibility of mult ip le stable states. Also, the positions of the l imi t point continuations in two-parameter space change as 71 is increased resulting in relatively smaller regions of low M\ equi l ibr ium 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). Comparing 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 unt 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 in figures 4.21-4.23. The size of this region first increases and then decreases unt i l , for sufficiently high 71 values, it ceases to exist. Mi 1 0.8 0.6 0.4 0.2 n Chapter 4. Ratio-Dependent Model 102 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 availabil ity of plants for survival. From the above observations we can deduce that when plant resources are l imited (at low values of 71) the potential for having two stable equi l ibr ia is greater and the region corresponding to low M i equi l ibr ium values is larger. When 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 (72,</>3)-space are very similar for all values of 71 (only the lower region of cycles decreases in area) suggesting that the system is no longer l imited 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 be- haviour or a stable equi l ibr ium occurs as well as the magnitude of the plant biomass at the equi l ibr ium. Low values of fa (the availability or nutrit ional 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 maintain fairly high biomasses. A t intermediate values of these parameters population"cycles may occur provided the value of fa is sufficiently high. The 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 From their isocline analysis of model (4.1) Gutierrez et al. [52] conclude that the par- asitoid E. lopezi could control the cassava mealybug while E. diversicornis could not. Chapter 4. Ratio-Dependent Model 103 However, wi th three state variables all having similar t ime scales, these deductions are not as straightforward as they may seem. F i rst , the equi l ibr ium isocline configuration in the MiM2 (M2M3) phase plane depends on the value of M 3 ( M i ) as well as on the parameter values. This was shown in figure 4.6. Thus, 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\M2 plane corresponds to a tr itrophic equi l ibr ium. Figure 4.26 shows three possibilities. Two of these (namely, (b) and (c)) appear in [52] but it was assumed that the equil ibr ium point in (b) was unstable. Even if the exact position of the equil ibr ium point is known, it is not possible to tel l from the qualitative structure of the isoclines whether this point is stable or unstable and whether or not l imi 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 imit cycles and high values to a stable equil ibr ium (see figure 4.2). Thus, numerical computation is needed to determine the exact location as well as the local stabil i ty of an equi l ibr ium point for the current model. The 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 imit point continuations) are crossed. In general, it is the proximity of the tr itrophic equil ibr ium point to the peaks of the M\ and M2 isoclines in the M\M2 and M2M3 planes respectively that is important for determining the robustness of model behaviour with respect to parameter perturbations. If the equi l ibr ium 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 equi l ibr ium isocline Chapter 4. Ratio-Dependent Model 104 (a) M2 0 02 04 075 OS 1 Ml 0 02 04 06 08 1 M2 (b) 0 02 04 06 08 1 Ml (c) M2 0 02 OA 06 08 1 M2 Figure 4.26: Examples of isocline configurations showing different possibi l i t ies for the pos i t ion of the t r i t roph ic e q u i l i b r i u m . 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: Examples of isocline configurations at different points in (71 ,<?!>2)-space. B y inference the above criticisms have all noted that if the exact positions of the isoclines and the tritrophic equil ibrium were known in 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) . Comparing figures 4.28(a) and (b) we can see that introducing the a;'s prevents the M\ and M2 isoclines from passing through the origin. Hence the equi l ibr ium 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 in 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 Model 106 (a) M2 M3 (b) M2 1 0.8 0.6 0.4 0.2 0, 02 OA 05 08" Ml 0 02 04 06 08 M2 (c) M2 M3 1.4 1.2 1 0.8 0.6 0.4 0.2 0, ,4 / / / / / " / / . / / / 0 02 OA M 0T8 1 M2 Figure 4.28: Isocline configurations for mode l (4.6) w i t h (a) aj = 0, (b) a< = 0.001 and (c) az- = 0.005 (z = 1,2,3) together w i t h the reference parameter set. Chapter 4. Ratio-Dependent Model 107 The result is an even more robust model. The stabilising influence of increasing the values of the aj's is i l lustrated by the bifurcation diagrams in 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 st i l l smal l , this suggests that the model is structurally unstable and hence predictions from ratio-dependent models should be treated with caution. Another consequence of introducing nonzero aj's is that we no longer get abrupt changes in the qualitative shapes of the isoclines. Consider the plant isocline. For 7 i < 4>2 the plant isocline has the hump shape shown in figure 4.30(a)(i). 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 in figure 4.30(b) (i) to the asymptotic curve in figure 4.30(b)(ii). The same applies to the herbivore isocline in the M2M3 plane. 4.6 C o n c l u s i o n In this chapter a part ial analysis of a tritrophic ratio-dependent model has been done. Large differences in 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. The bifurcation analysis revealed a number of cases of mult iple stable states. Many of these phenomena occur over very small parameter ranges and are therefore not of much 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. Many of the instances of mult iple stable states arise from a hard loss of stabil ity associated with a Hopf bifurcation. This means that a stable equi l ibr ium and Chapter 4. Ratio-Dependent Model 108 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 <f>2 ^3 Figure 4 . 2 9 : One-parameter bifurcat ion diagrams when a; = 0.005 (i = 1 ,2 ,3) . Chapter 4. Ratio-Dependent Model 109 (i) M2 0 2 04 06 08 M l (a) (ii) M2 2 1.6 1.2 0.8 0.4 0, 0 02 0 4 0 6 0 8 M l 1 1.2 M2 (b) (ii) M2 2 1.6 1.2 0.8 0.4 0. 0 02 OA 0 6 0 8 M l 1 1.2 Figure 4.30: P l a n t isoclines for (a) the o r ig ina l and (b) the modif ied mode l . T h e M\M2 plane is shown i n (a)(i) and (b)(i) and the M2M3 plane is shown i n (a)(ii) and (b)( i i ) . Chapter 4. Ratio-Dependent Model 110 a stable l imit 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 imi t cycles of large amplitude and a stable equi l ibr ium. A point worth noting is that if the Hopf bifurcation associated with the l imit cycles had been studied analytically, the algebra required to identify the hard loss of stability would have been very t ime-consuming and possibly too difficult to do by hand. Only the in i t ia 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 ^ n a c [ t n e m o s t significant effect. This is in general agreement with Gutierrez et al. [52] (in terms of their model the ratios and ^j^ 2 - were found to be important) although they did not explicit ly describe these effects as was done in 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. Both trophic levels affect the magnitude of M i at equi l ibr ium, 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 equi l ibr ium M i values and larger regions of mult iple 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. The parameters affecting predator dynamics also affect the equi l ibr ium biomasses. In the course of the analysis some previously noted criticisms of ratio-dependent mod- els were highlighted as they caused numerical difficulties and biological implausibil it ies. In particular, such models are not valid when a state variable which occurs in the de- nominator 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. Whi le this modif ication had very l i t t le effect on the dynamics for parameter values corresponding to reasonable equi- l ib r ium 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 in regions where the state variables are small . Even the addition of very small terms (a,- = 0.001) reduced the complexity of the cycles. The 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 im is to suggest what k ind of predator can keep herbivore numbers low. F inal ly , the zero isocline configurations in the M\M2 and M2M3 planes were inves- tigated and some l imitations of the three-dimensional setting were discussed. In order to obtain useful information from the isoclines their exact equi l ibr ium positions need to be determined so that the position and nature of the tr itrophic equi l ibr ium 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 in a very general sense. However, it is informative to see the kind 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 wi l l look at discrete-time models. The model in this chapter is a population genetics one in which both population size and gene frequency are state variables. It is a single locus, two allele model with density-dependent fitness functions and it has already been studied in Asmussen [8] and Namkoong et al. [93]. Although the model equations are fairly simple, interesting dynamics arise as a result of the discreteness. No new theoretical results are obtained in this study but the dynamical systems techniques prove useful in a number of ways: first, the theoretical results are demonstrated fairly easily and without having to struggle with the mathematical details. The 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 tr ial and error to try and locate the desired dynamics. In addit ion, while the theoretical results note the existence of various types of qualitative behaviour, they do not give information regarding the extents of the regions in 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. The latter information is important from an ecological perspective as it influences the amount of attention that is given to various possibilities. It wi 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 in 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 intuit ively or analytical ly as the bifurcation point at which they are init iated does not coincide with changes in the relative carrying capacities of the genotypes. Also, the period-2 cycle is ini t ial ly unstable and a further bifurcation is required before it becomes stable. Numerical techniques are indispensable in such situations. I begin in the next section with a list of new terminology that is used in this chapter as well as in chapter 6. I then summarise some background information on population genetics models and the main theoretical results relating to the particular model that is studied in the rest of the chapter. Section 5.4 describes the model equations and is followed by the model analysis. The focus in the latter section is on cycling or periodic behaviour as this behaviour is the most difficult to study by hand. The analysis begins with one-parameter studies which investigate the effects of altering relative carrying capacities. A two-parameter bifurcation diagram is then obtained. This diagram divides the two-parameter space into regions corresponding to different qualitative behaviour. The region of stable period-2 polymorphisms is larger than that for stable polymorphic equil ibr ia for the particular case studied and intersects regions of heterozygote inferiority, superiority as well as regions of partial dominance. Final ly , section 5.5.4 studies higher period cycles. Polymorphic 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 (equil ibrium or periodic cycle). Questions for further study are included in the conclusion. Chapter 5. Population Genetics Model I 114 5.2 N e w terminology Some new terminology is required in this chapter as well as in chapter 6 to explain both the discrete dynamics and the biological significance of the results. • pe r iod -k cycle or orbit : For discrete models we do not get l imi 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 equi l ibr ium point (see section A.2.6) . For further details on discrete models refer to section A .3 .5 . • per iod -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 A2. Then there are three possible genotypes: A\A\, A\A2 and A2A2. • fitness: The 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 repro- duction of that genotype [110]. • homozygote: The genotypes A-^Ai and A2A2 are homozygotes. • heterozygote: The genotype A\A2 is a heterozygote. • fixed or homomorphic equi l ib r ium: This is an equi l ibr ium point at which only one allele is present, that is, at which only a homozygote is present. Chapter 5. Population Genetics Model I 115 • polymorphic equilibrium: This is an equil ibr ium point where more than one allele is present, that is, where a heterozygote is present. • carrying capacity: This refers to the equi l ibr ium population density correspond- ing to a particular genotype when only that genotype is present. It is denoted by K{j for the genotype AiAj in this thesis. • heterozygote superiority: Heterozygote superiority or overdominance occurs when the heterozygote's carrying capacity is greater than those for the homozy- gotes, that is K\2 > K\\,K22 for the case of two alleles. Heterozygote inferiority 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, Kn < Ki2 < K22 or K22 < 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 t ime scale than changes in population size (Roughgarden [104]). How- ever, once it was realised that gene substitution could occur in the same length of t ime as that needed by a population to reach an equi l ibr ium, the dangers of this separation were acknowledged. Since the late 1960's a number of models in which both population size and gene frequency are variables have been studied. The classical one locus, two allele selection models in this category used constant viabilities for the genes and pre- dicted monotonic population convergence to a unique stable equi l ibr ium (Asmussen [8]). However, incorporating density-dependent selection can have a dramatic effect on the Chapter 5. Population Genetics Model I 116 dynamics—both regular and chaotic cycles can arise. Studies of the effects of density-dependence for discrete generation organisms can be found in [8, 9, 93, 104]. In particular, Asmussen and Feldman [9] and Asmussen [8] show that in such situations local stabil ity analyses may be inadequate to explain the global behaviour related to changes in gene frequency and population size. In addit ion to fixed and polymorphic stable equil ibria, Asmussen [8] found regular and chaotic cycles when using monotone decreasing density-dependent fitness functions. In certain situations equil ibr ia and cycles exist simultaneously. Asmussen [8] also shows that stable periodic polymorphisms may occur in the absence of heterozygote superior i ty—the latter condi- t ion being necessary for polymorphic equil ibria 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. Thus, overdominance in 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 equi- l ibr ia and stable l imi t cycles to persist for higher growth rates than would be possible with the model's purely ecological counterpart. The above conclusions were arrived at analytically for linear (logistic) monotone de- creasing fitness functions. However, when the density-dependence is modelled using ex- ponential fitness functions, Asmussen [8] comments that the mathematics becomes very difficult. Numerical solutions then become necessary. In [8, 93] it is also noted that when more complex, higher order behaviour (such as a cycle) is present, intuit ion and local stabil ity analyses break down. Again numerical techniques are required. Chapter 5. Population Genetics Model I 117 5.4 Model equations Suppose we have a single population and two alleles, A± and A?. A t t ime t the population size is denoted by Nt and the frequency of allele A\ is denoted by pt. The fitness of genotype A{Aj = 1,2) at t ime t is denoted by w\-. The marginal fitness of Ai is thus ™\ = PMI + ( i - Pt)«4 (*' = 1>2) and the mean population fitness is wt = ptw\ + (1 - p t)w Differences in fitness among genotypes may be interpreted as the result of different re- sponses to ecological pressures. Following Asmussen [8] and Namkoong et al. [93] expo- nential density-dependent fitness functions of the form w ,•• = exp(a,j - bijNt) a;j,6,j > 0, i,j = 1,2 (5.1) are used in 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 equa- tions for p and N are: w\ Pt+i = Pt — wl Nt+1 = w*Nt (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 wi l l be looking for attracting boundaries at p = 0 or p = 1, interior (polymorphic) equil ibria having 0 < p < 1, period-2 and also higher period sta- ble orbits. Att ract ing boundaries correspond to situations where one allele survives at the expense of the other, while at interior attractors both alleles persist thus maintain - 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 associ- ated relative carrying capacities of the homozygotes and heterozygote. The methods of analysis and the results are discussed in the next section. 5 . 5 Model analysis 5 . 5 . 1 Approach Equi l ib r ia and their associated stability properties have been studied analytical ly for models such as the one described above. These theoretical results help predict the condi- tions under which stable equil ibria 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 ut i l i ty of the available software. I used D S T O O L to solve the system over t ime and to gener- ate starting points for A U T O . The bifurcation diagrams were obtained using Interactive A U T O . The A U T O interface in X P P A U T is not yet set up to deal with 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 in behaviour occur over small parameter ranges. Namkoong et al. [93] state that alleles that affect seedling survival can increase car- rying capacity and simultaneously destabilise population growth dynamics. This 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, fei2 = 0.904 and varying fe22. The results agree with those in [93]. I then go on to determine the effects of simultaneously varying the heterozygote parameter, fei2, by generating a two-parameter bifurcation diagram in (fe22, &i 2 ) -parameter space. I conclude 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 &2 2 > 0.526. A n example is shown in figure 5.1 for the value fe22 = 0.54. In this figure there is a 5 . , . N 0 I , , , I 0 p 1 Figure 5.1: D y n a m i c s i n the (p, 7V)-plane for mode l (5.2) w i t h a n = 2.1, a i 2 = 1.9, a 2 2 = 1.1, & n = 1.0, &12 = 0.904 and 6 2 2 = 0.54. T h i s d iagram was obtained 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). The 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, in this case both boundaries are repelling and there is an interior attracting period-2 orbit as expected. Chapter 5. Population Genetics Model I 120 Comparing the carrying capacities K~u = 2.100, K\2 = 2.102 and K22 = 2.037 we see that we are just within the region of heterozygote superiority. We can now use A U T O to vary b22. It is noted in appendix B that for discrete systems A U T O can detect period-doubling bifurcations but cannot continue the resulting period- 2 orbits, and hence cannot detect higher period orbits. This is clearly a disadvantage in the present situation where we are specifically interested in the period-2 orbits. A way of overcoming this problem is to study the second iterate of the model since period-2 orbits wi l l become equil ibria in this new model (see section A.3.5) . In terms of the original model A U T O wi l l then be able to detect period-1 equil ibria (since these are also period-2 equil ibria) , 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. From equations (5.2) we obtain Pt+2 = Pt+i—TT - Pt — ^nr Wx+L w w1+l Nt+2 = wt+1Nt+1 = wt+1w*Nt (5.3) where u>i+1 = Pt+Mi1 + {1 ~ Pt+i)w$i - p t ^ e x p ( a i l - bi^Nt) + (1 - p t - ~ ) exp(a i 2 - bi2wtNt) and wt+1 = pt+1wl+l + (l-pt+1)w2+l Figure 5.2 shows the results obtained from varying 622 using the second iterate of the Chapter 5. Population Genetics Model I 121 Figure 5.2: One-parameter bifurcat ion d iagram of mode l (5.3) w i t h a n = 2.1, ai2 = 1.9, a 22 = 1 1 , & i i = 1.0 and &12 = 0.904 obtained using A U T O . (Behaviour for smaller and larger values of 622 than indica ted i n the figure can be found by ext rapola t ing the curves and lines at the boundaries of the figure. T h e period-2 orbi ts are indicated i n the figure. T h e phenomena corresponding to boundary values of p, namely p — 0 or p = 1, are label led. Branches marked w i t h a * correspond to inter ior values of p, namely 0 < p < 1. H B stands for H o p f bifurcat ion. T h i s is really a per iod-doubl ing b i furca t ion but A U T O marks i t w i t h the H B symbol . ) model 1 . I plotted N versus 622 instead of p versus 622 as the period-2 orbits can be seen with 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. Also, a large number of bifurcations occur at p = 1. Plott ing p versus 62 would result in many branches ly ing on top of one another. For 622 < 0.523 we can see that the boundary at p = 0 is attracting wi th a single 1 S o m e very compl ica ted bifurcat ion diagrams can be generated when s tudy ing this m o d e l , however not a l l cont inua t ion branches are of interest. Those branches which have p < 0 , p > l o r i V < 0 are not p rac t ica l ly significant but i t is good practice to continue such branches w i t h i n the parameter range under s tudy i n case a bifurcat ion occurs and they re-enter the ranges of interest. Chapter 5. Population Genetics Model I 122 equi l ibr ium point. A t b22 = 0.523 the genotype A2A2 loses superiority in carrying ca- pacity and we move into the region (b22 > 0.523) of heterozygote superiority. Whi le the difference in carrying capacities is not too large we st i l l have a unique equi l ibr ium (an interior equi l ibr ium this time) but as b22 increases, causing the carrying capacity of A2A2 to decline and the instabil ity of A i A i to have a greater influence on the dynamics, this equi l ibr ium bifurcates to become an attracting interior period-2 orbit. As 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 in [93] and are in agreement with previous findings that heterozygote supe- riority in equi l ibr ium carrying capacity is a necessary condition for a stable polymorphic equi l ibr ium when fitness is a decreasing function of population size. In addit ion, figure 5.2 shows clearly that heterozygote superiority is not a sufficient condition for a stable polymorphic equi l ibr ium 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 equi- l ibr ia , can exist in the absence of overdominance. Suppose we set b\2 = 0.906, then K\2 = 2.097 which means that we no longer have any regions of heterozygote superior- ity. The 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 b22 once again. In this case the dynamics are a l i tt le more complicated. We sti l l have an equi l ibr ium 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 al l values of b22. For 0.520 < b22 < 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 with 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 N 2.9 2.5 - 2.1 - 1.7 1.3 P=l P = 0 p=l P = 0 p=l 1. l i m i t point 2. per iod-doubl ing -[ 3. per iod-doubl ing 4. bifurcat ion point 0.5 0.51 0.52 0.53 622 0.54 0.55 0.56 Figure 5.3: One-parameter bifurcat ion d iagram of model (5.3) w i t h a n = 2.1, a i 2 = 1.9, 022 = 1.1, 611 = 1.0 and 612 = 0.906 obtained using A U T O . Ku = 2.1 and K12 = 2.097. Branches marked w i t h a * correspond to interior values of p, namely 0 < p < 1. the period-doubling bifurcation which initiates the interior period-2 orbit occurs when 622 — 0.519 but the orbit is init ial ly 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 polymorphism 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 equil ibr ia since these phenomena separate the domains of attraction of the stable phenomena. Figure 5.4 shows the (p, AQ-plane corresponding to 62 = 0.521 and indicates the domains of attraction for this particular value of 62 - The region denoted by a is the domain of attraction for the sink at p = 0, b is the domain of attraction for the period-2 polymorphism, 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 Chapter 5. Population Genetics Model I 124 5 N 0 Figure 5 .4: T h e (p, 7V)-plane using mode l (5.3) for an = 2.1, ai2 = 1.9, a22 — 1-1, 6 n = 1.0, &12 = 0.906 and 622 = 0.52 showing the domains of a t t rac t ion for the stable phenomena. equil ibria 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 with 612 as well as 62 - We can find out more about this dependence on 6 1 2 by generating a two-parameter bifurcation diagram. 5 . 5 . 3 Two-parameter bifurcation diagram Using A U T O we can trace the paths of period-doubling bifurcations 2 and l imit points in two-parameter space. Points 2 in figure 5.3 are period-doubling bifurcations and points 1 are l imi 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 2 A U T O labels per iod-doubl ing bifurcations as H o p f bifurcations for discrete models (see section A . 3 . 5 ) . T h u s points marked H B in the figures are really per iod-doubl ing bifurcations. 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 Figure 5.5: Two-parameter bifurcat ion d iag ram of mode l (5.3) w i t h a u = 2.1, a i 2 — 1.9, (Z22 = 1 . 1 and 611 = 1.0 obta ined using A U T O . Curve 1 is the l i m i t point cont inuat ion and curve 2 the per iod-doubl ing bi furcat ion cont inuat ion . on the one obtained for the second iterate of the model by using the R E A D P command in Interactive A U T O . We could choose any one of the parameters a i i , a i2 , a 2 2 , 611 or 612 as the second parameter to vary. Since the parameter 6 2 2 corresponds to one of the homozygotes, it may be interesting to vary one of the heterozygote parameters. The parameter 612 was chosen for this i l lustration to compliment the results in the previous section. Figure 5.5 shows the results obtained from continuing points 1 and 2 in figure 5.3 in two-parameter space. The two-parameter continuation of point 3 in figure 5.3 can be ob- tained from the original model as mentioned earlier. A U T O cannot continue transcrit ical bifurcation points such as point 4 in figure 5.3. However, we can obtain a good approx- imat ion to the relevant line or curve by doing a number of one-parameter continuations as in section 5.5.2 and noting the coordinates at which the bifurcation at p = 0 occurs. We 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 with Chapter 5. Population Genetics Model I 126 0.909 0.908 0.907 - A C 0.906 0.905 0.904 - H F 0.5 0.51 0.52 0.53 0.54 0.55 '22 Figure 5.6: Two-parameter bifurcat ion d iagram of mode l (5.3) w i t h a n = 2.1, a\2 = 1.9, 022 = 1.1 and 611 = 1.0 obta ined using A U T O . Curve 1 is the l i m i t point cont inuat ion and curve 2 the per iod-doubl ing bi furcat ion cont inuat ion . Curve 3 is the per iod-doubl ing cont inuat ion obta ined using the o r ig ina l mode l (5.2) and curve 4 is the curve where the bifurcat ion 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 in figure 5.6 but only those of interest for the present discussion have been drawn. The dotted line has been included to demarcate, together with line 4, the regions of heterozygote superiority and inferiority 3 . In regions F and G there is heterozygote superiority and in regions A , B and D there is heterozygote inferiority. C, E and H are regions of part ial dominance. These distinctions in relative carrying capacities wi 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 3 T h e s e lines can be obtained ana ly t ica l ly using the definit ion of heterozygote superiori ty. T h e dotted line could also have been obtained from one-parameter bifurcat ion diagrams where 612 is varied instead of 622 as it is along this l ine that the period-2 orbi t at p = 1 changes its s tab i l i ty properties. Chapter 5. Population Genetics Model I 127 2.9 1 1 i p=i i ——J^?i p=i - *,p=i NT : H B .•* p=i - i i i i i 0.5 0.51 0.52 0.53 0.54 0.55 0.56 622 Figure 5.7: One-parameter bifurcat ion d iag ram of model (5.3) w i t h a n = 2.1, 012 = 1.9, a 2 2 = 1.1, 6 n = 1.0 and 612 = 0.908 obtained using A U T O . Branches marked w i t h a * correspond to inter ior values of p , namely 0 < p < 1. H B marks a per iod-doubl ing bifurcat ion. ./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 1 2 at 0.908. From 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 imi t point of such a phenomenon, occurs. Also from figure 5.6, we expect the equi l ibr ium at p = 0 to change stabil ity as we pass through line 4. Using A U T O to generate a one-parameter diagram with 6 1 2 fixed at 0.908 gives figure 5.7 which is just as we predicted. The above exercise can be repeated for other values of 6 1 2 as well. A complementary way of obtaining insight into figure 5.6 is to choose points in regions A to H and to display the dynamics in 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 in figure 5.8. Str ict ly 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. Whi le Asmussen [8] and Namkoong et al. [93] documented the occurrence of stable period-2 polymorphisms in such regions, they did not investigate the extent of the regions corresponding to such phenomena. From figure 5.6 we can see that regions D and E occupy a fairly small region in the (622, 612) parameter space and thus may have l imited ecological significance. Another observation is that the region of stable polymorphic equil ibr ia (region G) is very smal l . Thus, in the region of heterozygote superiority, stable period-2 polymor- phisms are much more likely than stable polymorphic equil ibria. It appears that the instabil i ty 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 period four (and higher) Consider first orbits of period 4. Since I did not find such orbits by varying 6 1 2 and 6 2 2 , I chose b\2 and 622 to lie in region E, a region where more complex behaviour already Chapter 5. Population Genetics Model I 129 ( A ) ( C ) (E) (G) ( B ) (D) (F) ( H ) N 2.11* 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, a22 = 1 1 , &u = 1 0 and various combina t ions of 612 and 622 which correspond to regions A to H in figure 5.6. Chapter 5. Population Genetics Model I 130 4 - N 1. l i m i t poin t 2. per iod-doubl ing 3. per iod-doubl ing 4. b i furcat ion point 5. per iod-doubl ing 2.7 Figure 5.9: One-parameter bifurcat ion d iag ram of mode l (5.3) w i t h ai2 = 1.9, a22 = 1.1, bu = 1.0, &12 = 0.905 and 622 = 0.525 obtained using A U T O . Branches marked w i t h a * correspond to inter ior values of p, namely 0 < p < 1. occurs in the form of period-2 cycles, and then varied the other parameter values in turn. Consider &i 2 = 0.905 and 6 2 2 = 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 map, 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 rom studies of other discrete systems (for example, [71]) it is l ikely that there wi 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 in the (p, AQ-plane for fixed values of a n . Some examples are given in Chapter 5. Population Genetics Model I 131 figure 5.10. Two-parameter continuations of the period-doubling bifurcation in 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 instabil ity of its dynamics. This supports the comment in [93] that alleles that affect seedling survival can increase carrying capacity and simultaneously destabilise population growth dynamics. Interior period -4 orbits But what about the dynamics of the heterozygote? It seems likely that complex polymor- phic behaviour would exist in regions where both homozygotes exhibit unstable dynamics. In order to investigate this question I chose both an and a22 to be greater than 2.0. Using the parameter values a n = 2.1, a12 = 1.9, a22 = 2.1, bu = 1-0, 612 = 0.908 and b22 = 0.53, I found a set of starting points using D S T O O L and then varied each parameter in turn using A U T O . The only interior period-doubling bifurcation was found by increasing a i 2 . Figure 5.11 shows the results. There are period-2 orbits at both p = 0 and p = 1 for al l values of a i 2 . This is expected since both a n and 022 are greater than 2.0. The period-2 orbit at p = 1 is attracting for a i 2 < 1.907 (the region of heterozygote inferiority) and the orbit at p = 0 is attracting for a i 2 < 3.598. A t this latter point the heterozygote becomes dominant, a bifurcation occurs and a stable interior period-2 orbit is init iated. As ai2 is increased further, a period-doubling bifurcation occurs at ai2 = 3.750. Al though 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 in the (p, iV)-plane for 012 = 3.8. F rom 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 and (a) a n = 2.68, (b) a n = 2.69 and (c) a n = 2.75. In (a) there is a period-8 a t t ractor at p = 1. In (b) this changes to a period-16 attractor and i n (c) we have what appears to be a chaotic at t ractor . Chapter 5. Population Genetics Model I 133 1 1 *. •" ' - H B •• ' / - - * - v - H B - . - 1 1.5 2.5 3.5 4.5 622 Figure 5.11: One-parameter bifurcat ion d iag ram of mode l (5.3) w i t h a n = 2.1, 022 = 2.1, bu = 1.0, 6 1 2 = 0.908 and 622 = 0.53 obtained using A U T O . Branches marked w i t h a * correspond to inter ior values of p, namely 0 < p < 1. H B marks a per iod-doubl ing bifurcat ion f rom period-2 to period-4 orbits . Figure 5.12: D y n a m i c s i n the (p, A^)-plane for a n = 2.1, a i 2 = 3.8, a 2 2 = 2.1, & n = 1.0, 6 i 2 = 0.908 and 6 2 2 = 0.53 showing a stable period-4 p o l y m o r p h i s m . Chapter 5. Population Genetics Model I 134 Figure 5.13: Two-parameter cont inuat ion of the per iod-doubl ing bifurcat ion ( H B ) i n figure 5.11 w i t h a n = 2.1, a 2 2 = 2.1, bn = 1.0 and 6 i 2 = 0.908. rates, a; j , determine the type of attractor, that is, whether the attractors are equil ibr ia 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 though we have located a stable polymorphic period-4 orbit, the values of a12 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 in the region of heterozygote superiority that polymorphic attractors exist in figure 5.11 and since K22 = 3.962 is large, this region is only entered when a12 is large. Hence, by decreasing A^2 2 we may be able to find stable interior period-4 orbits for lower values of a 1 2 . Figure 5.13 shows a two-parameter continuation of the period-doubling bifurcation labelled 5 in figure 5.11. The parameter b22 is varied in addition to a\2- F rom figure 5.13 it can be seen that the period-doubling bifurcation point is reduced to a\2 = 2.831 if b22 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 a4j values and observe the results for particular parameter combinations. Some examples of more complex dynamics are shown in figure 5.14. (a) (b) Figure 5.14: Examples of complex dynamics , (a) A n interior period-8 orbi t for a n = 2.3, a i 2 = 2.9, « 2 2 = 2.5, fen = 1.0, fci2 = 0.908 and fc22 = 0.95. (b) A n interior chaotic a t t ractor for a n = 2.6, a i 2 = 3.1, a 2 2 = 2.5, &n = 1.0, & i 2 = 0.908 and 6 2 2 = 0.95. Chapter 5. Population Genetics Model I 136 5.6 Conclusion In this chapter a part ial analysis of a population genetics model with monotone density- dependent fitness functions has been done. Al though most of the theoretical results had been obtained by Asmussen [8] and Namkoong et al. [93], this study gave rise to a two- parameter bifurcation diagram which indicates the relative frequency of occurrence of the various types of dynamical behaviour in addition to proving their existence. Using 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 equil ibria and higher period orbits I concluded that the parameters 6,j (i,j = l,2) have the greatest influence on the stabil ity 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 rst , could the numerical results be used as a basis for arriving at analyt ical 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 ful l 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 equil ibr ia 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 ful l 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 ime non-monotone fitness functions are considered. The model has not been studied in 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 monotoni- cally decreasing fitness functions because of the detrimental effects of population density on growth [110]. However, this assumption is not biologically realistic for al l population densities [110]. A t low densities, increases in density may benefit both reproduction and survival. In this chapter I consider the same basic model as in chapter 5 but with non- monotone density-dependent fitness functions which have a single hump. This model is more difficult to study than that in the previous chapter because of the increased com- plexity in 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. The terminology described in section 5.2 is used again in this chapter. As in chapter 5 the study focusses mainly on stable polymorphic behaviour—both equil ibr ia and higher period orbits. Such phenomena correspond to the maintenance of genetic diversity in a population. From an ecological perspective we would l ike to know how common 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 equil ibria but that stable periodic polymor- phisms are not as common. The latter phenomena are only found to occur in 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 formu- lations 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 r ) -plane indicate the domains of attraction for the homomorphic and polymorphic attractors. The possibility of extinction is also high for most parameter sets. Cr i ter ia for determining the existence and stabil ity of polymorphic equil ibria 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. The 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 stabil ity properties of polymorphic equil ibria, 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 in 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 140 6.2 Background One of the few studies which considers the more general setting of non-monotone density- dependent fitness functions is by Selgrade and Namkoong [110]. They prove a number of results concerning the existence and stability of polymorphic equil ibria for both con- tinuous 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 equil ibr ium. For the case of heterozygote inferiority the equi l ibr ium is unstable. For the discrete model (which is the one consid- ered in this chapter), if the heterozygote is superior and - 2 < JVff < 0 (JV=population density, u)=mean fitness) at the polymorphic equil ibr ium then this equi l ibr ium is stable. Higher order attractors are not studied in [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 The fitness functions used in this chapter are given by u>ij = kijNtexpln^l-kijNt)] j = 1,2 (6.1) and wlj = kijNtexpln^l - kijNt)Nt] i,j = 1,2. (6.2) Expressions of the form (6.1) can be found in [109] and also in [110] for the analogous continuous model . These functions have a single hump (see figure 6.1) and are known as cl imax fitness functions in some applications [109]. Apart f rom the results for polymorphic equil ibria obtained by Selgrade and Namkoong [110], not much work has been done on discrete models such as (5.2) in which both geno- types have cl imax fitnesses. In order to simplify the analysis I look at an additive model Chapter 6. Population Genetics Model II 141 in which the fitness parameters for the heterozygote are averages of the corresponding parameters for the homozygotes. That is, This reduces the number of parameters for consideration. 6.4 Model analysis 6.4.1 Approach I begin by studying the existence and relative frequency of occurrence of stable equil ibria for the additive model with 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 As in the previous chapter bifurcation diagrams and plots of the (p, AQ-plane are the main tools for communicating results. In some situations the fitness functions are plotted so that the relative fitnesses for various population densities can be seen. This is analogous to calculating relative carrying capacities in 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 , & 2 2 and r 2 2 (h2 and r i 2 are averages of the cor- responding homozygote parameters). Since there were no prior results wi th which to begin, I used M A P L E 1 [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 t ime, I fixed r 2 2 and fc22 (that is, I fixed the fitness function W22) and investigated the (hi, rn) -parameter space. Therefore, the results wi l l indicate J A n y other ma thema t i ca 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 [125], could have been used. and r i 2 k •12 hi + k22 2 ru + r 2 2 2 (6.2). Chapter 6. Population Genetics Model II 142 1.2- 1— 0.8- 0.6- 0 .4 - 0 .2 - 0- \ I I I To o N Figure 6.1: w22 with r 2 2 = 0.8 and & 2 2 = 0.6. the effects of varying the relative positions and magnitudes of the fitness functions. It should be kept in mind that altering ru and ku wi l l affect both wu and w\2. 6.4.2 ( & n , r n ) - p a r a m e t e r space After using A U T O to vary the parameters one by one to locate a stable polymorphism, I chose r 2 2 = 0.8 and k22 = 0.6 as starting values, which gave the w22 fitness function shown in figure 6.1. The a im is to divide the (ku, r n ) - p a r a m e t e r space into regions corre- sponding to stable fixed points at p = 0,p = 1 (homomorphic equilibria) or 0 < p < 1 (polymorphic equil ibria). The 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 wi th r n = 0.7. Only 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 143 0 i 1 1 1 1 1 0 0.4 0.8 1.2 1.6 2 hi Figure 6.2: A one-parameter bifurcation d iagram obtained by vary ing k\\ w i t h r n fixed at 0.7 (r22 = 0.8 and fc22 = 0.6). Branches marked w i t h a * correspond to interior values of p, namely 0 < p < 1. L P marks the l i m i t points and B P the t ranscr i t ical bifurcation points . There are many bifurcations in figure 6.2. Most are transcritical bifurcations (see section A.2.25) but there are also two l imit points (see section A.2.13). Diagrams of the (p, AQ-plane for a number of different values of hi are shown in figure 6.3 to help clarify the changes in 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 equil ibria 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 imit points and period-doubling bifurcations in two param- eters. As can be seen from figure 6.2 we wi l l also need to know how the positions of the transcritical bifurcations vary with r n if we want to delimit regions of stable behaviour Chapter 6. Population Genetics Model II 144 O P 1 0 P 1 0 P 1 Figure 6.3: D iag rams of the (p, 7V)-plane (obtained using D S T O O L ) for a number 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 n - v a l u e s (such as 0 .3 ,0 .5 ,0 .7 , . . . ) . For each value I obtained starting points (that is, equi l ibr ium points) for A U T O using D S T O O L 2 . I then used A U T O to generate a one-parameter bifurcation diagram by varying ku in both directions, and recorded the A; x l-values corresponding to the various bifurcat ions—both transcritical and l imit 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) 2 I t is impor t an t that a number of different fixed points are located for each parameter set to ensure that unconnected cont inuat ion branches are not missed. Chapter 6. Population Genetics Model II 145 and (d) show the same diagram with shaded regions corresponding to stable equi l ibr ia 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 iag rams of the {ku, rn ) -pa ramete r space, (a) T h e basic d iagram showing a number of bifurcat ion curves, (b) T h e regions corresponding to stable po lymorph i sms are shaded, (c) T h e regions corresponding to stable equ i l ib r ia at p = 0, N > 0 are shaded, (d) T h e regions corresponding to stable equ i l ib 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 r ) -plane corresponding to the re- gions marked A to P are shown in figures 6.5 and 6.6. These figures are schematic, representations of diagrams obtained using D S T O O L . The 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 equil ibria, figure 6.4 shows where we can expect simultaneous homomorphic and polymorphic attractors. In fact, there are no regions where the only attractor is in the interior. Thus, there is always the possibility that one of the alleles wi l l be excluded. The relative sizes of the domains of attraction of the homomorphic and polymorphic equil ibr ia are shown in figures 6.5 and 6.6. These diagrams also highlight the ever-present possibility of extinction. 6.4.3 Criteria for polymorphisms It would be helpful to be able to predict when a stable polymorphic equi l ibr ium is l ikely to occur. One possibility is to investigate the fitness functions corresponding to the different regions in 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 U , W\2 and W22 can be plotted on the same pair of axes. F ixed points at p = 0 and p = 1 occur at those values of N where w22 = 1 and u> n = 1 respectively. The stabil ity 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, Ar)-plane corresponding to the regions A to H in figure 6.4( Chapter 6. Population Genetics Model II Figure 6.6: D iagrams of the (p, A f ) -plane corresponding to the regions N to P in figure 6 Chapter 6. Population Genetics Model II 149 F i g u r e 6.7: Examples of the fitness functions for parameter values corresponding to (a) a sta- ble polymorphism (rn = 0.7, ku = 2.0, r2i = 0.8, £22 = 0.6) and (b) an unstable polymorphism (rn = 0.4, jb n = 2.0, r 2 2 = 0.8, k22 = 0.6). Chapter 6. Population Genetics Model II 150 al l parameter combinations which gave similar configurations of the fitness functions correspond to the existence of polymorphisms. This supports the conclusion in [110] that heterozygote inferiority or superiority is necessary but not sufficient for a polymorphism 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{2 + (1 -pt)w*22, and wt = ptw\ + (1 - pt)wl, where w\ is the marginal fitness of allele A{ and wl is the mean population fitness at t ime t. A n interior equi l ibr ium (0 < p < 1, N > 0) requires Pt+i = Pt and Nt+i = Nt. From equations (5.2) in the previous chapter we can see that the above equations wi l l be satisfied if w* = 1 (6.3) and w\ — w*. (6-4) For 0 < p < 1 the latter condition can only be satisfied if w[ = w2. (6.5) We 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 in the fitness func- tions. We 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 in figure 6.8. Chapter 6. Population Genetics Model II 151 3 0 1 1 1 0 0.5 1 P Figure 6.8: Curves given by w\ — w2 ( thin dotted line) and w = 1 (thick sol id line) for parameter values corresponding to a stable po lymorph i sm ( r n = 0.7, k\\ = 2.0, r 2 2 = 0.8, fc22 = 0.6). From 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 equi l ibr ium. Thus, we can predict the existence of polymorphic equil ibria. 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 equil ibria, we can pre- dict the existence and the stability properties of polymorphic equil ibria. The relative positions of the fitness functions, such as in figure 6.7, give most of the required stabil ity 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 sit- uation it is probably easier and quicker to use D S T O O L to find and classify the fixed Chapter 6. Population Genetics Model II 152 points. In the next section I turn to higher order dynamics. As in the previous chapter we would like to know whether stable period-2 (and higher period) polymorphisms are possible with this additive model since the maintenance of genetic diversity need not be restricted to the existence of polymorphic equil ibria. 6.4.4 Stable per iod -2 po lymorphisms Attempts to find a stable period-2 polymorphism using D S T O O L and A U T O proved to be t ime-consuming. However, I was finally successful in locating a period-2 sink for r n = 1.3, &n = 0.5, r 2 2 = 7.5 and k22 = 4.57. The fitness functions and (p, AQ-plane 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 smal l . A lso, figure 6.9(a) shows that the fitness function w22 has an unrealistically high max imum. Neither situation is particularly satisfying. The rij values determine the heights of the fitness function max ima. We would l ike to know whether altering one of the other parameter values would allow r22 to be lowered while st i l l maintaining the interior period-2 attractor. A U T O can be used to generate two-parameter diagrams for this purpose. The first step is to create a one-parameter bifurcation diagram by varying r22. The relevant curves are shown in figure 6.10. The period-doubling bifurcation HB* is the point at which stable period-2 orbits are init iated. 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 k22 as the second parameter are shown in figures 6.11(a), (b) and (c) respectively. As can be seen from these diagrams, r22 needs to be greater than 6.4 for a stable period-2 polymorphism. This is st i l l rather large and not particularly satisfactory. Chapter 6. Population Genetics Model II 153 (a) 2 - W22 1— (b) T N Figure 6.9: (a) Fi tness functions and (b) (p, 7V)-plane for r n = 1.3, kn = 0.5, r22 = 7.5 and k22 = 4.57. There is no reason why we should be confined to the fitness functions given in equa- tions (6.1). The 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). The steeper the slopes of the fitness functions (the right-hand slope in particular) , the less stable the dynamics corresponding to that homozygote. This is similar to the previ- ous chapter where higher growth rates, which caused exponential fitnesses with steeper Chapter 6. Population Genetics Model II 154 0.75 Figure 6.10: A pa r t i a l one-parameter bifurcat ion d iagram obtained by vary ing 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 mode l was used so that the period-2 orbi ts could be continued. However, using this mode l the per iod-doubl ing bi furcat ion H B * is label led as a b i furcat ion point by A U T O which means that it cannot be continued i n two parameters. T h e o r ig ina l m o d e l needs to be used for such a cont inuat ion. slopes, resulted in more complex dynamics. We expect higher period interior orbits to occur in 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 polymorphism for m = 0.2, fen = 5.0, r 2 2 = 0.3 and fe22 = 0.4. Both D S T O O L and A U T O were used in the search. The fitness functions and (p, AQ-plane corresponding to these values can be found in figure 6.12. As can be seen from figure 6.12(a), the fitness functions are much more reasonable than before. Again we can get some idea of the size of the region in parameter space for which Chapter 6. Population Genetics Model II 155 Figure 6.11: Two-parameter continuations of the per iod-doubl ing bifurcat ion H B * i n figure 6.10 ob- ta ined by va ry ing (a) ru, (b) & n and (c) k22 i n add i t ion to r 2 2 - T h e shaded regions indicate wh ich side of the bi furcat ion continuat ions corresponds to periodic behaviour. Chapter 6. Population Genetics Model II 156 Figure 6.12: (a) Fitness functions and (b) (p, iV)-plane for ru = 0.2, ku = 5.0, r 2 2 = 0.3 and k22 = 0.4. Chapter 6. Population Genetics Model II 157 (a) 2 N 1.5 - (b) 2 N 1.5 i 1 ? H B _ B P I • 1 ^ ^ H B 2 i ' • •. 4 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 & 2 2 ) and examine the (ku, rn) -parameter space as was done earlier. The first step is to use A U T O to create a one-parameter bifurcation diagram by varying ku- D S T O O L was used to generate starting points for A U T O . Using the first iterate of the model equations resulted in figure 6.13(a). Only the period-doubling bifurcation indicating the change from a stable equi l ibr ium 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. This 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. This 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 period- doubling (at ku = 4.226) changes as ru is varied in 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 in figure 6.13(b)). Both curves are plotted on the same Chapter 6. Population Genetics Model II 158 0.1 - 0 I 1 1 1 1 0 2 4 6 8 10 hi Figure 6.14: A bifurcat ion d iagram showing the two-parameter cont inuat ion of the per iod-doubl ing at & n = 4.226 and the subsequent per iod-doubl ing at ku = 5.358 in figure 6.13. T h e shaded region approximates the region of stable po lymorph i sms of per iod greater than 1. set of axes in 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) within 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 in 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 much simpler behaviour near the p = 0 boundary. A n important question is whether a fitness function configuration of the form shown in figure 6.16 could result in period-2 (or Chapter 6. Population Genetics Model II 159 3.5 Figure 6.15: A n example of an interior chaotic at tractor obta ined for r n = 0.18, ku = 5.0,r22 = 0.3 and &22 = 0.4. higher period) stable polymorphisms. In such a situation both homozygotes have similar fitness properties but with one of the homozygotes slightly out-competing the other at each population density. The region in ( & n , r n ) - p a r a m e t e r space corresponding to such fitness configurations is plotted together with figure 6.14 to produce figure 6.17. Clearly, in this case the two regions do not overlap. Thus in this range it is not possible to have addit iv i ty (in the sense just described for the fitness functions) and stable polymorphic behaviour. So far w22 has been fixed to have the shape shown in figure 6.12(a). The slopes of this function are fairly gentle. Suppose we replace W22 by the function shown in figure 6.18. We expect such an alteration to reduce the stabil ity 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 polymorphism. Figure 6.19 was obtained using the same procedure as for figure 6.17. Again 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 161 Figure 6.19'. T h e new two-parameter space showing the region of higher order stable p o l y m o r p h i c behaviour (vert ical lines) and the region corresponding to fitness funct ion configurations of the type shown i n figure 6.18 (horizontal 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 with results in the previous chapter for period-4 orbits. 6.5 Conclusion The population genetics model studied in this chapter is more complicated than that of the previous chapter due to the form of the fitness functions. As a result any mathematical 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 with one-humped fitness func- tions, 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 equil ibria from mean fitnesses was demonstrated and the relationships between interior and boundary stable equil ibria were shown using two- parameter bifurcation diagrams and corresponding diagrams of the (p, A r ) -plane. The latter diagrams also highlight the high possibility of extinction for most parameter com- binations. Other one-humped fitness functions may lead to different conclusions than those obtained in this study. The techniques outlined in this chapter could be used for such investigations. Chapter 7 Spruce Budworm Mode l 7.1 Introduction In this chapter I concentrate on a discrete model of a defoliating insect system. The 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 Ludwig [22]. In it the budworm, the branch surface area of the trees and their foliage are al l 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 . The model by Clark and Ludwig [22] is fairly complicated and includes a number of processes such as dispersal, predation, food l imitat ion, and parasitism. Section 7.3 gives a description of the equations. Discrete models of this complexity have not been analysed in detail before. Section 7.4 contains the model analysis. D S T O O L is the main package used. Because the system is discrete and because of its complexity, continuation packages such as A U T O are of l imited value. This is discussed in more detail in section 7.4.1. For the analysis I chose to focus on one aspect of the model rather than attempt an exhaustive parameter study. The 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, mult ip le stable states and 163 Chapter 7. Spruce Budworm Model 164 periodic outbreak behaviour are identified. Whi le Clark and Ludwig [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 in the model is responsible for the observed behaviour and which ones have a lesser effect. Knowing 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 main process responsible for outbreak cycles is small larval dispersal. This agrees with the findings of Clark [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 main influence of predation is at fairly low budworm densities which means that it affects the t ime period between outbreaks. 7.2 Background The eastern spruce budworm, Choristoneura fumiferana C lem. (Lepidoptera: Tortr ic i - dae) is found throughout the Canadian Marit imes and northern New England as well as westward and northward through middle Canada up to the boreal forest (McNamee [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 in the eastern regions, resulting in budworm outbreaks of epidemic propor- tions. Damage to the preferred host trees, balsam fir (Abies balsamea) and white spruce (Picea glauca), is extensive and can approach 100% in dense, mature stands [19]. These Chapter 7. Spruce Budworm Model 165 outbreaks are documented as far back as the 1700's with some of the worst ones occur- r ing in the Canadian province of New Brunswick. Intensive insecticide spraying began in this area in 1952 in an attempt to protect the foliage and, thus, l imi t tree mortal i ty [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]). The budworm itself is a univoltine insect which means that there is one budworm gen- eration 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]). This stage causes the most defoliation and large larval feeding levels influence both fecundity and adult dispersal. The large larvae are also subject to b i rd predation and parasitism and are the target for insecticide spraying. As mentioned earlier, dispersal also affects the dynamics. The small larvae spin silk threads and are transported aerially by wind. 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. This model has been used as a research tool by forest managers and scientists in New Brunswick (Clark and Hol l ing [21]). However, a ful l 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, Ludwig et al. [74] developed a simplified model consisting of a system of three ordinary differential equations which they studied qualitatively. A l though they obtained some interesting results and demonstrated the potential of the model to exhibit outbreak behaviour, Clark and Ludwig [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 aim in ! mind , Clark and Ludwig [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 in [65] but st i l l 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 in a unit of branch surface area. It is an average value representing conditions on the whole site and is measured in '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 addit ion to being consumed by budworm larvae, the foliage provides oviposition sites for adult moths [19]. A l though balsam fir retains its foliage for about eight years, it is sufficient to consider only two classes ('new' or present year foliage and 'o ld ' 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 in i t ia l foliage density then, following Jones, the effect of larval constimption on new foliage can be described by: Fb Remaining new foliage = e~A—— (7.1) KF Chapter 7. Spruce Budworm Model 167 where A = d0Lb—^. Here Lb represents the in i t ia l budworm larval density and d0 is the m a x i m u m foliage consumption rate for an individual larva during the feeding season. Equat ion (7.1) is a standard competit ion function (see appendix D for an explanation) and it is used to represent the competit ion 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~B—^——-Ft, (7-2) Kp where B = cx{A - 1 + e~A). ( c i = 0.357 is a constant.) The 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 + (KF - l)e~B]Fb. (7.3) KF 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 Fe (the foliage density after one year) cannot exceed Kp, then we obtain Fe = ?\ , (7.4) where the denominator introduces density-dependence. This completes the foliage dy- namics. Chapter 7. Spruce Budworm Model 168 7.3.2 B r a n c h surface area Another feature of trees that is important to budworm is the surface area of branches. This 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 simplif ication to a single age class. Severe defoliation by budworm may k i l l branches. This is modelled by setting S1 = [ l - ds(l - (7-5) where Sb is the in i t ia l branch surface area density and d$ is an average death rate. S\ represents the l iv ing branch surface area which remains after defoliation. Because of the quadratic term (1 — ) 2 , the difference between Sb and S± is only significant if there is 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 max imum branch surface area density. 7.3.3 B u d w o r m The preceding equations are only slightly more complicated than the ones in Ludwig et al. [74]. However, those for the budworm dynamics are much more complex. Most of the following equations are based on Jones's model [65]. Following his approach, an in i t ia l large larval density of Lb larvae per unit of branch surface area (per tsf) is assumed. Parasit ism 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 sig- nificantly before the outbreak collapse has begun [19]. According to Jones, the rate of parasit ism is a decreasing function of larval density, with a m a x i m u m of 4 0 % at low bud- worm densities that decreases exponentially. The number (density) of larvae surviving parasit ism is given by Li = (1 - qmaxe-c)Lb, (7.7) 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: L> = (̂ fer)L- <7-8) where kr, — 0.425 is a proportionality constant. Predation by birds is l imited to the large larvae. As in Ludwig et al. [74] this process is modelled by a Hol l ing type III functional response. Thus, L3 = e~DL2, (7.9) where " Sb(PseLtFb2 + Lj)' and pmax is the max imum predation rate and p s a ^ is a half-saturation rate. The ex- 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/Sb. The predators search foliage and they switch to alternate prey if the ratio of larvae to foliage, L2/Fb, is too small . If pmax/Sb is smal l , then predator consumption is Chapter 7. Spruce Budworm Model 170 L 2 - L 3 = ( l - e - D ) L 2 « DL2 = P m a x L " (7 10) which indicates that p s a ^ is the half-saturation value of the ratio L\jF2. C lark and Ludwig [22] chose the more complicated exponential form (7.9) over (7.10) in order to take into account competit ion among predators. The survival of pupae is correlated with the survival of large larvae [65] and is given by L4 = (AP + BAL3 (7.11) where Av = 0.473 and BP = 0.828 are regression constants obtained by Jones [65]. This expression gives the density of adult moths. According to Jones, female fecundity depends on their weight. He calculates this weight, W, as W = AF1K A ' + AF2(KF - 1){ A ] + BF (7.12) where AFI = 34.1, Ap2 = 24.9 and BF = —3.4. This formula expresses the differing nutr it ional values of new and old foliage. Fecundity is proportional to the cube root of W [65] which results in the following equation for egg density: L5 = {ExW1'3 - E2)A„L4 (7.13) where E\ = 165.64 and E2 = 328.52 are regression constants obtained by Jones [65] and Asr 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 nutr i t ion 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 moth wi l l deposit some of her eggs on the site under consideration and wi l l remove some to other locations [65]. Following Clark [19], it is assumed that dispersal always leads to death. Thus, in this model, adult dispersal serves to increase egg mortality. Unl ike normal mortality, however, this removal of eggs from the population depends on female adult density in the following way: A fPm Le = (l~ jfj^)L. (7.14) where ASTL\ & = —. A-thr is the ratio of female adult density to a threshold density, Athr- If the parameter ra is large then Em w i l l be large if E > 1 and small if E < 1. This means that a fraction Adisp 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 in determining the survival of small larvae. The reason is that small larvae disperse twice (using silken threads to give them buoyancy in the wind) and the success of their dispersal depends upon landing on suitable foliage. It is assumed that the scaled probabil ity of successful dispersal is given by G = H{2-H) (7.15) where H = K - F n - The parameter n is analogous to ra in equation (7.14). If n is smal l , H w i l l only vary slightly as F\ decreases below its max imum, Kp. But if n is large, changes in 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 in 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 Le = dSL^G2L6 (7.16) ft 5 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 Le represents the new large larval density. This completes the description of the three-dimensional model developed by Clark and Ludwig [22]. A summary of the equations can be found in appendix D. Table 7.1 gives the standard parameter values. As with all models there are a number of simplify ing 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 to be evaluated, but this is outside the scope of the present study. 173 P a r a m e t e r Descr ip t ion Va lue Foliage KF maximum foliage density 3.8 do maximum foliage consumption rate/larva 0.0074 rF foliage growth rate 1.5 Branch surface area ds branch surface area death rate 0.75 rs branch surface area growth rate 1.15 KS maximum branch surface area density 24 000 Budworm <lmax maximum parasitism rate 0.4 Pmax maximum predation rate 23 000 Psat half-saturation value (predation) 0.085 A\ST adult sex ratio (females/total) 0.46 fraction of females dispersing 0.5 m steepness of dispersal transition 4.0 Athr female threshold density (dispersal) 5.0 n effect of foliage density on larval dispersal 1.0 dsL small larval survival rate 0.28 Table 7.1: Table of standard parameter values. 7.4 M o d e l analysis 7.4.1 Preliminaries Tradit ional methods of qualitative analysis, such as isocline analyses, are not possible in 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. This is t ime-consuming and often misleading as the solutions depend on the in i t ia l conditions and on complex joint distributions of parameters [19]. Since many ecological models of Chapter 7. Spruce Budworm Model 174 practical interest are similar in complexity to the one under discussion, it is desirable that methods be found for understanding the behaviour of these models. Wi thout an understanding of the range of behaviour that a model can exhibit , it may be difficult to explain results obtained from computer simulations or t ime series analyses of the model [82]. A l though A U T O can perform continuations of equi l ibr ium 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 period- doubling from an equil ibr ium 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 imited as none of these higher period cycles would be detected. Also, 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 l imitations of continuation software in the context of complex discrete models, it is st i l l possible to obtain the desired qualitative information. However, bifur- cation diagrams have to be obtained 'manually ' rather than through automated continu- ation programs. This can be done using D S T O O L as wi l l be described shortly. The 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. No further discretisation or approximation is required as in the case of differential equations. As in 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 Chapter 7. Spruce Budworm 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- This ordering is arbitrary. The a im is to analyse the effects of varying the values of these parameters and to divide the (dsL, A i / i r ) -parameter space into regions of different qualitative behaviour. Following this I study the effects of different processes on model behaviour to determine which one(s) is(are) responsible for the outbreak cycles. Final ly , I study the two processes predation and small larval dispersal in more detail. Because there are three state variables in 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 in al l the D S T O O L calculations, branch surface area, S, is st i l l allowed to vary—the results are just projected into the (Foliage,Budworm)-plane. This differs from Clark and Ludwig's study where S was constant in their analysis and means that the following analysis applies to the ful l three-dimensional model whereas that in [22] is restricted to a simpler two-dimensional system of equations. Chapter 7. Spruce Budworm Model 176 7.4.2 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 in the S E T T I N G S - S E L E C T E D window and fixed al l the other parameters at their respective values given in table 7.1. Using D S T O O L I found and recorded the equi l ibr ium points corresponding to this parameter set as well as their local stabilities. This 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 in i t ia l points, or entering in i t ia 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 in 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) in the two-dimensional plane. After recording the results I increased dsL to 0.10 and repeated the above procedure. I continued in this way (incrementing dsL by 0.05 each time) unti l I reached dsL = 1-0. In regions where qualitative changes occurred (such as an equi l ibr ium point changing stabi l - ity 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 t ime-consuming and I did not think it necessary for a qualitative study. The results of the above parameter study are summarised in the one-parameter b i - furcation diagram in figure 7.2. (Similar diagrams for F and S can also be drawn.) This diagram was obtained by plott ing the budworm densities of the equi l ibr ium points for each value of dsh and then connecting these points—sol id lines for stable equil ibria and dotted lines for unstable equil ibria. M a x i m a and min ima of the cycles that were found were then p lot ted—sol id dots for stable cycles and open circles for unstable cycles. Chapter 7. Spruce Budworm Model 177 250 200 B u d w o r m ^ g g la rva l density ( larvae/tsf) 100 50 0 1 • 1 i i - — o u t b r e a k • • m a x i m a - . o o • • • • — 1outbreak * • • . minima • o 0 I . . . ol 0 0 0.2 0.4 0.6 0.8 dsL Figure 7.2: One-parameter bifurcat ion d iagram of b u d w o r m larva l density versus dsL- We can see from figure 7.2 that for low values of d$L there is a single stable equi l ibr ium value. A diagram of the (Foliage,Budworm)-plane for the particular value d$L = 0.2 is shown in figure 7.3(a). A l l orbits spiral in to the equi l ibr ium point (denoted by the triangle). As dsL increases a bifurcation occurs, and for dsL > 0.28 there is the possibility of outbreak cycles. Figure 7.3(b) shows the (Foliage,Budworm)-plane for d$L = 0.35. In this case, depending on the in i t ia l values of the state variables, either stable equi l ibr ium or outbreak behaviour can occur. This is further exemplified in 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 from these plots, the max imum of the outbreak cycle varies but the period remains fixed at 15 years. (Strictly speaking the period may be some larger mult iple 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 max imum varies slightly in consecutive outbreaks. Cycles of very long period or totally aperiodic motion wi l l both appear to be almost periodic (or chaotic) in a practical biological setting [84].) Since both the equi l ibr ium and the outbreak cycle are stable phenomena, there must be some k ind of boundary Chapter 7. Spruce Budworm Model 178 (a) 350 (c) 350 / ] / \ •i ' (b) 350 (d) 350 Figure 7.3: D iag rams of b u d w o r m larva l density versus foliage for (a) dsL = 0.2, (b) dsL — 0.35, (c) d$h — 0.8 and (d) dsL = 0.9. T h e dots indicate densities i n consecutive years. Chapter 7. Spruce Budworm Model (a) 350 179 B (b) 1/4" u Time Time (c) 25000 100 Time F i g u r e 7.4: Time plots of (a) budworm larval density, (b) foliage density and (c) branch surface area density versus t ime for dsL — 0.35. In each case two trajectories are shown for a t ime period of 100 years. As can be seen, i n i t i a l values of the three state variables affect the result ing behaviour of the system. The two s tar t ing points, A and B, are the same in all three graphs. Chapter 7. Spruce Budworm Model 180 del imit ing their domains of attraction (see section A.2.5) . This is not easy to locate in three dimensions and may not be a smooth surface. The dashed line in figure 7.3(b) gives a rough approximation to a two-dimensional projection of part of this boundary (the boundary also varies with S) and is only included to indicate that the domain of attraction for the outbreak cycle is much larger than that for the equi l ibr ium point. For 0.28 < d$L < 0.68 the behaviour remains the same qualitatively. Near dsL — 0.68 the stable equi l ibr ium undergoes a bifurcation resulting in an unstable equi l ibr ium (de- noted by a plus sign in 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 in figure 7.3(b) and that the points corresponding to these outbreaks appear to f i l l 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 in the magnitudes of consecutive points in the cycles. However, the period is fixed for a given value of dsL- Returning 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 in figure 7.3(d). For any in i t ia l values the system sti l l oscillates but the amplitudes of these oscillations get larger and larger unti l the budworm finally becomes extinct. Having classified the dynamical behaviour of the system for different values of dsL, I w i l l now vary Athr, the threshold value for female adult dispersal. 7.4.3 The 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 in the previous section in order to obtain one-parameter bifurcation diagrams. In this case Athr is not restricted to lie between 0 and 1 since Athr 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 imited to outbreaks so we would like to investigate values of Athr between 0 and, say, 20. I used an increment of 0.5 for this study, decreasing this to 0.1 in regions of qualitative change. The resulting bifurcation diagram for dsi = 0.45 is shown in figure 7.5. In this case 120 B u d w o r m la rva l 80 density ( larvae/tsf) 40 h F i g u r e 7.5: One-parameter bifurcat ion d iagram of b u d w o r m la rva l density versus Athr for dsL = 0.45. there are two Hopf bifurcations resulting in two regions, L\ and L2, where periodic orbits 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 al l Athr values. For clarity of the smaller amplitude orbits, only the min ima of the outbreaks are shown in figure 7.5 (these are just above zero). The max ima vary between 355 larvae/tsf at Athr = 0.01 and 480 larvae/tsf at Athr = 14. For larger Athr values the behaviour corresponds to that at Athr = 14 but with greater outbreak amplitudes. The diagrams for the other values of dsL are qualitatively s imi la r—on ly the values of Athr at which the Chapter 7. Spruce Budworm Model 182 0 0.2 0.4 0.6 0.8 1 dsL Figure 7.6: Two-parameter bifurcat ion d iagram of Athr versus dsL- bifurcations occur and the amplitudes of the cycles are different. For d$L = 0.2 outbreaks only occur for Athr > 10.5. Having obtained an idea of the qualitative behaviour which corresponds to varying Athr, we can begin constructing a two-parameter bifurcation diagram in the (dsL, Athr)- 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. The values of Athr at which each of the above phenomena occur can be found through incrementing dsh by 0.1 (or 0.05 in regions where significant changes occur). This results in the two-parameter bifurcation diagram shown in figure 7.6. The solid lines indicate where Hopf bifurcations occur and the small dotted lines indicate the outer l imits for Chapter 7. Spruce Budworm Model 183 stable cycl ing behaviour corresponding to these bifurcations. The thick dotted line sepa- rating 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. The other thick dotted lines indicate the boundaries of the regions F, G and H where budworm extinction occurs and region I where two equi l ibr ium states are possible. This single diagram summarises nine qualitatively different types of behaviour that can be obtained by varying dsL and Athr- The nine regions are marked A - I . Diagrams of the (Foliage,Budworm)-plane corresponding to each region are shown in figure 7.7. Str ict ly 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 in i t ia l values spiral in 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).) Trajectories starting near this point spiral out towards a stable periodic orbit. Trajectories from other in i t ia l points st i l l spiral inwards but in this case they approach the periodic orbit instead of the equi l ibr ium point. Region C is similar to region A in that the equi l ibr ium points are again a spiral sink and a saddle. However, in this region outbreaks are also possible. Since both the sink and the outbreak cycle are attracting, there must be a basin boundary div id ing their domains of attraction. A rough approximation to part of this boundary is denoted by the dashed line. Again , the position of the boundary wi l l vary with S. It appears that in most cases the domain of attraction for the outbreak cycle is much larger than that for Chapter 7. Spruce Budworm Model 184 Foliage Foliage Foliage Foliage Foliage Foliage Foliage Foliage Foliage Figure 7.7: D iag rams of budworm larval density versus foliage for the regions marked A-I in figure 7.6. Tr iangles denote sinks (usually spiral sinks in this model) and plus signs denote unstable saddles ( equ i l i b r ium points having at least one stable and one unstable eigenvalue—the th i rd eigenvalue may 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. The only difference is that the spiral sink has been replaced by a stable periodic orbit. As Athr is varied (and we move from region D to E ) , the periodic orbit becomes unstable and we are left wi th al l the trajectories approaching the outbreak cycle. As we move into region F even the outbreak cycle becomes unstable. The system may oscillate a few times exhibit ing 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 equi l ibr ium. The 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 inks—one corresponding to no budworm and the other to a positive budworm density. The ini t ia l values of F, S and L (budworm larval density) determine which sink is approached. 7.4.4 B io log ica l interpretat ion The first point to note is that, from an experimental viewpoint, it may be difficult to distinguish between the equi l ibr ium behaviour associated with regions A and C and the cycl ing behaviour of regions B and D, respectively, due to random variation and measurement errors. This is true even in 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. When 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 in 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. The 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. The 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. In contrast regions F and G correspond to high survival rates. These have the effect of destabilising the system and causing wi ld oscillations. This destabilisation as a survival rate is increased is a phenomenon characteristic of many ecological models [84]. Region I corresponds to two stable equi l ibr ia—one at very low budworm densities and the other at higher population densities. Although the possibility of mult ip le 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 mult iple stable states with significant area. From 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. The 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 Model 187 I i , 1 . 1 d ' •, i e - - a I) ' ' • ' '•: - : • c \ - - ; Reg ion of mul t ip le stable states \ f . • • d ' . e " .. . • i • • i i i 0 0.2 0.4 0.6 0.8 1 dsL a. E x t i n c t i o n b . Sp i ra l s ink/s table l i m i t cycle behaviour (endemic popula t ion) c. Sp i ra l s ink/s table l i m i t cycle + outbreak behaviour d. O n l y outbreak behaviour e. W i l d oscil lat ions leading to ext inc t ion f. S m a l l region of stable behaviour + same as region e Figure 7.8: S impl i f ied two-parameter bifurcat ion d iag ram of Athr versus dsL- stable state is possible but for the rest of the outbreak region no alternative is possible. A l though 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 corre- sponding to the equil ibria and cycling behaviour as well as the amplitudes of the cycles. The corresponding one-parameter bifurcation diagrams for foliage and branch surface area density show that the equi l ibr ium values for these quantities decrease as dsL and/or Athr are increased. Thus they show the severity of the defoliation corresponding to dif- ferent dispersal rates. A l though the state space diagrams in the preceding analysis were only for the (Foliage, Budworm)-plane, they could also have been projected into the (Surface Area,Budworm)- Chapter 7. Spruce Budworm Model 188 or (Foliage,Surface Area)-plane, since the branch surface area, S, varies in al l the situa- tions studied. The dynamics are similar in all the planes—just the shapes of the cycles and the positions of the equi l ibr ium points are different. 7.4.5 W h a t causes outbreak cycles? A p p r o a c h In the preceding sections parameter values corresponding to low (0-2 larvae/tsf) and medium (30-50 larvae/tsf) equi l ibr ium 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 equil ibria and outbreaks) have been observed in the field [90]. However, the model under discussion is a complex one with many component processes. We 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 in the current model is to start with a basic model and add indiv idual processes one at a t ime (such as parasitism or the effect of larval weight on fecundity) to see what effect this has on model behaviour. Clark and Ludwig [22] included a number of switches in their model so that the different processes can be turned on and off. In order to remove parasit ism, equation (7.7) can be replaced by The effect of food on large larval survival (equation (7.8)) can be replaced by a constant survival rate, namely L\ — Lb- (7.17) L2 - kLLi. (7.18) Chapter 7. Spruce Budworm Model 189 As for parasit ism, predation (equation (7.9)) can be removed by setting L3 = L2. (7.19) Instead of correlating pupal survival with that of large larvae, equation (7.11) can be replaced by L4 = BPL3. (7.20) The 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 L5 = BSeLA (7.21) where Bje = 96 was the chosen average fecundity per female moth [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 Le = dSLL&. (7.22) In order to obtain an idea of the effects of these processes on model behaviour, I turned off al l the switches init ial ly and then added each process to the model in 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 al l the switches off, the budworm density increases exponentially causing foliage density (and hence branch surface area) to decline to zero. The 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.) Even with very high predation rates, predation alone cannot produce cycl ical behaviour. The other processes in the model, namely parasit ism, 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. Smal 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 in the original model . (For brevity this simplified model wi l l be referred to as the dispersal model.) The two-parameter bifurcation diagram in (dsL, A t - i r ) -parameter space shown in 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 main phenomena are st i l l present, which supports the c laim that dispersal is responsible for the qualitative behaviour of the model . The 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 t ime plots for the simplified model (as is done in the process of obtaining figure 7.9), it is easily seen that the outbreaks for this model have Chapter 7. Spruce Budworm Model 191 0 I — L L - J • 1 1 1 1 0 0.2 0.4 0.6 0.8 1 dsL Figure 7.9: Two-parameter bifurcat ion d iagram of Athr versus dsL for the s impl i f ied mode l wh ich only includes dispersal . Regions are marked 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 with max ima 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 model , then we can make another important deduction. The processes which have been left out of this simpler dispersal model are important for biological realism. As stated above, their effects are quantitative rather than qualitative. They have a moderating effect on the system dynamics and prevent densities becoming too high or too low for reasonable parameter values. Each process may have a relatively small influence but together they exert considerable control over the system. Chapter 7. Spruce Budworm Model 192 The observation that small larval dispersal is responsible for the outbreaks contrasts wi th McNamee's [89] explanation that cycling is the result of movement between bud- worm equil ibr ia at low and high budworm densities. However, the result agrees with 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 l inked to dispersal, is the determining factor in the occurrence of outbreaks. In their study of the larch budmoth, Baltensweiler and Fischl in [12] suggest that the cycles in 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 in good condition with high branch surface area and foliage densities. The 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 in an epidemic collapse. F ischl in and Baltensweiler [37] come to similar conclusions in 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. McNamee et al. [90] recognise the importance of forest biomass on outbreak behaviour but maintain that the outbreaks are movements between high and low equi l ibr ium bud- worm 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. The state space and bifurcation analyses show that the equil ibria 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 with 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 equi l ibr ium manifolds, the method used by McNamee et al. [90], is not particularly informative (cf. chapter 4). 7.4.6 The 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. M e t h o d A technique suggested by Clark and Ludwig [22] involves beginning with al l the processes turned off (as described earlier) and then turning them on one at a t ime and determining budworm recruitment values over one year for a wide range of in i t ia 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 with the processes being turned on in different orders so that the results can be checked. A n example is shown in figure 7.10. Discussion From 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 3- 1 2 u I p Figure 7.10: Isoclines of recruitment versus b u d w o r m density when 1) a l l processes are switched off, 2) the dependence of large l a rva l surv iva l on food is added, 3) the effect of feeding his tory on fecundity is also added, 4) s m a l l l a rva l dispersal is added, 5) paras i t i sm is added, 6) predat ion is added, and 7) adul t dispersal is added g iv ing rise to the ful l mode l . then the effect is not very pronounced. Feeding history has a small effect on fecundity at al l budworm densities and has the most effect at high densities. Parasit ism 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. Adul 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 medium 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. The extent of their influence wi l l therefore affect the length of t ime between outbreaks. 7.4.7 The effects of predation C o m p a r i n g predat ion 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 budworm 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 wi l l be referred to as the predation model for simplicity (although it also includes dispersal). The two parameters affecting predation are pmax, the m a x i m u m predation rate, and psat, a half-saturation value. Increasing pmax or decreasing psat both lead to increased predation (see equation (7.9)). W i t h these two parameters at their nominal values given in table 7.1, the new two-parameter bifurcation diagram is shown in 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). However, the upper Hopf bifurcation curve is lowered by predation. Thus predation decreases the size of the region ( A , B , C , D and G) where a Chapter 7. Spruce Budworm Model 196 14 r 12 • 10 • 8 • 6 • 4 • 2 • 0 • 0 Figure 7.11: Two-parameter bifurcat ion d iagram of Athr versus dsL for the predat ion mode l wh ich includes dispersal as well as predat ion. Regions are marked according to figure 7.6. stable endemic state is possible. The 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 in figure 7.9. Thus predation seems to have a destabilising effect from this point of view. The 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 in the dispersal model. This is due to the appearance of a sink corresponding to zero budworm density in 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 rom figures 7.9 and 7.11 it appears that the main 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 mult ip le stable states is smaller (due to the shift in the upper Hopf bifurcation curve). However, the qualitative dynamics have not been significantly altered. Compar ing figures 7.6 and 7.11 we can deduce the effects of adding the remaining processes (other than dispersal and predation) which are included in the ful l 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 t h r values) by adding these processes since both Hopf bifurcation curves are lower in figure 7.6 than in figure 7.11. In other words, in the ful l model endemic equil ibria occur for higher rates of female adult dispersal than in the predation model since lower dispersal thresholds imply that there is more dispersal (see equation (7.14)). The region of extinction for low dsL values (region H) is larger in 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 ful l model. For Athr < 12, the onset of outbreaks occurs for higher dsL values in the ful l model than in the preda- t ion model . This emphasises the above observation that the additional processes in the original model have a stabilising effect on the system provided that adult dispersal is sufficiently high, that is, Athr is sufficiently low (below 12 larvae/tsf in this case). The role of predation In order to study the effects of predation in more detail , one-parameter bifurcation dia- grams can be generated by varying pmax or psat. I used the original model which includes al l the processes since its output is more biologically meaningful. The method used to obtain figures 7.2 and 7.5 was used again and the resulting one-parameter bifurcation Chapter 7. Spruce Budworm Model 198 diagram for pmax, corresponding to dSL = 0.4 and Athr = 5, is shown in figure 7.12. To simplify the scale pmax is given in multiples of 23 000 which is the nominal value for pmax given in table 7.1. 100 80 Budworm gg larval density (larvae/tsf) 40 20 0 0 1 2 3 4 Pmax ~ F ~ . c a Figure 7.12: One-parameter bifurcation diagram for p m a x with dsL = 0.4 and Athr = 5. For Pmax less than half its nominal value there is a single equi l ibr ium state, cor- responding to endemic budworm densities (see figure 7.12). No outbreaks occur in this range although the system oscillates as it approaches the equi l ibr ium. Outbreaks are pos- sible in the range of pmax values denoted by a. M a x i m a for these cycles are around 270 larvae/tsf for pmax — 0.6 x 23 000 and increase as pmax increases. A t pmax = 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 Athr- Around pmax = 2.3 x 23 000 the endemic equi l ibr ium bifurcates to produce a nine year periodic orbit of small ampli tude 1 . 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 Chapter 7. Spruce Budworm Model 199 of parameter values denoted by 6. In region a, in i t ia l values for the budworm and forest variable determine whether the outbreak cycle or an endemic equi l ibr ium state (stable equi l ibr ium point or small amplitude periodic orbit) is attained. For pmax > 3.3 x 23 000 (region c) there is a single equil ibr ium corresponding to low budworm densities. This 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 in the field. (Recall that the nominal or standard value for pmax is 1.0 x 23 000.) The recruitment curves generated earlier (figure 7.10) using the standard parameter values in 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 budworm densities cannot escape from the low numbers where predation is prevalent. A diagram similar to figure 7.12 can be obtained by decreasing psat. F rom the d i - agrams of the (Foliage,Budworm)-plane and the t ime plots generated in doing these parameter studies, an important observation can be made—vary ing the predation pa- rameters 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 pmax = 3.4 x 23 000. These increased periods do not have much effect on the t ime 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 ig - 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 pract ical impor tance i n themselves. W e are more interested i n the qual i ta t ive change from an equ i l ib r ium to cycles. Chapter 7. Spruce Budworm Model 200 (a) 350 Time 100 (b) 350 Time Figure 7.13: T i m e plots of b u d w o r m larva l density for (a) Pmax — 0.6 x 23 000 and (b) p m a x = 3.4 x 23 000. Outbreaks last 7 or 8 years in bo th cases but the t ime between outbreaks is longer i n (b). B o t h figures are plots of the behaviour after the i n i t i a l transients have died away. Chapter 7. Spruce Budworm Model 201 outbreak (such as its t ime 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 t ime periods between outbreaks. As a final test of the effects of predation relative to small larval dispersal, I constructed a two-parameter bifurcation diagram of pmax versus dsi using the ful l model (which includes al l processes). The results are shown in figure 7.14. Clearly, the higher dgL (and •4 3 P m a x (x23000) 2 1 0 0 0.2 0.4 0.6 0.8 1 dsL A . O n l y lower sink (budworm: 0-2 larvae/ tsf) B . T w o sinks (lower and interior) C . Interior sink (budworm: 30-50 larvae/ tsf) D . Interior sink + outbreaks E . Interior saddle + outbreaks F . Interior saddle + stable l i m i t cycle G. Interior saddle + stable l i m i t cycle + outbreaks Figure 7.14: Two-parameter bifurcat ion d iagram of p m a x versus dsL- hence the greater small larval dispersal success) the greater the chance of outbreaks. Only for low dsL and high pmax 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 equi l ibr ium only exists for Chapter 7. Spruce Budworm Model 202 dSL < 0.12. This 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 C o n c l u s i o n This chapter has focussed on analysing the budworm-forest model developed by Jones [65] and Clark and Ludwig [22] using the techniques of dynamical systems theory. Proce- dures for obtaining one- and two-parameter bifurcation diagrams using diagrams in 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 f / i r ) -parameter space was obtained. Whi le Clark and Ludwig [22] found parameter combinations corresponding to behaviour in a number of these regions, namely A , B, C, D and H, this study found some addit ional possibilities for model behaviour and is much more comprehensive as the resulting bifur- cation diagrams summarise the behaviour for al l possible combinations of dsL and Athr- The 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 in 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 in the model and that predation has an added destabilising effect. The main 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 ime plots was crucial in the analysis. C h a p t e r 8 C o n c l u s i o n 8.1 M a i n r e s u l t s A variety of models, both continuous and discrete, theoretical and practical , have been analysed in the preceding chapters. The same basic techniques have been used in 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 rst , 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 with tradit ional techniques. For the sheep-hyrax-lynx model in chapter 3 bifurcation diagrams were found to give more information than a tradit ional sensitivity analysis, and for the ratio-dependent model in 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 dynamical systems techniques can be helpful in constructing more plausible models. The techniques can also identify which parameters or processes are crucial for determining the behaviour of the model. This is i l lustrated in chapters 3 and 7. The 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 in chapters 2 203 Chapter 8. Conclusion 204 and 5 where the numerical results were compared with previously obtained theoretical results. In chapter 2 the numerical results were in fact more accurate than Bazykin 's approximate analytic results [14]. The computer packages also allow more complicated models to be studied than is possible by hand. A l l the models i l lustrate this point. As a result, previously unobtainable insights can be discovered. In addit ion, 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. Al though the dynamical systems techniques can be applied in a variety of situations, they are obviously not suitable for al l types of ecological models. 8.2 Limitations Systems of difference equations or ordinary differential equations can be studied but not systems of part ial differential equations. Also, the models must not be t ime or space dependent. These l imitations are serious, however it is generally the case that models which are fundamentally different in structure require different methods of solution. For certain types of partial differential equations and time-dependent models it is possible to overcome these l imitations by transforming the equations so that they fal l into the required categories, but the mathematics required to do this is not t r iv ia l . A l though the computer packages allow the dynamical systems techniques to be ap- plied 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 with which the computer packages can be used should not be taken as an argument for bui lding Chapter 8. Conclusion 205 complicated models. Simple models are sti l l most l ikely to give us insight into system behaviour because of our own l imits in understanding. As anyone who has used a computer wi 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 in previous chapters and others are recorded in appendix B. It is usually a good idea to check the results using an alternative technique or software package. Al though a variety of models have been studied in 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]). However, the dynamics and bifurcation structure are considerably more complex. Including such an example in this thesis would have confused rather than clarified my intent and would perhaps have discouraged rather than encouraged readers who were considering trying the techniques. 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Introduction to Theoretical Ecology, New York: Harper and Row Appendix A Dynamical systems theory A . l Introduction This appendix explains the dynamical systems terminology used in my thesis. Section A . 2 is a glossary of the basic concepts such as equil ibr ium point, domain of attraction and bifurcation point. Extensive use is made of diagrams in order to introduce the concepts as simply and intuit ively 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 in any introductory text on dynamical systems theory. A few examples which I found particularly readable include [57, 111, 128]. A.2 Basic concepts This section is ordered alphabetically. A l l the examples given are for continuous systems of equations. Discrete systems are discussed in section A .3 .5 . W i t h i n each subsection in this glossary, italics is used to highlight terms which are explained in 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. The 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 equil ibria and dotted curves are used for locally unstable equil ibria. M a x i m a and m i n i m a of l imit cycles are indicated using c i rc les—sol id ones for stable cycles and open circles for unstable cycles. See, for example, the figures in sections A.2.10, A .2 .13, A.2.16, A.2.18, A.2.25. It is important to note that bifurcation diagrams summarise the behaviour associated with 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 in the diagram corresponds to an equi l ibr ium point or a periodic orbit ( l imit cycle). The 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 = p2 respectively in figure A . 17(a). 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 in 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 in figure A .19. 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 in 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 equil ibr ium to oscillatory behaviour. Examples can be found in sections A.2 .10, A .2 .13, A.2.16, A.2.18, A.2.25. A.2.3 Chaos Chaos is difficult to define but intuit ively it refers to the (apparently) irregular and un- predictable behaviour which many nonlinear mathematical models (systems of equations) exhibit [11]. If a system is chaotic then ini t ia l values which are very close together may lead to vastly different behaviour as t ime progresses. However, this behaviour is st i l l bounded by a region in 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 equi l ibr ium point (or l imi t cycle or bifurcation point) change as a parameter (or parameters) is altered. Together a number of these branches make up a bifurcation diagram. A.2.5 Domain of attraction Suppose the system in which we are interested has a stable equilibrium point (see section A.2.14). Then the collection of all in i t ia l state variable values from which the system tends towards this equi l ibr ium as time progresses is the domain (or basin) of attraction of the equi l ibr ium point. The equil ibr ium point is called an 'attractor' . A n y stable phenomenon, such as a stable limit cycle, also has a domain of attraction and is referred to as an attractor. Appendix A. Dynamical systems theory 219 For example, in figure A . l a population density of 10 is locally stable. For init ial population values which lie between 5 and 20 the population tends towards a density of 10 as t ime progresses. For init ial values below 5 the population tends to extinction and for in i t ia l values greater than 20 it increases steadily. The range of values between 5 and 20 is the domain of attraction for the equil ibrium point at 10. Part (a) of figure A . l shows t ime plots corresponding to various init ial points and part (b) is a one-dimensional phase portrait of the situation. (a) Popu la t i on density 20- T i m e (b) A—• q •—* • — - 0 5 1 0 2 0 Popu la t i on density Figure A . l : (a) T i m e plots showing the domain of a t t ract ion of an equ i l ib r ium point . A popu la t ion density of 10 is stable and the range of in i t i a l popula t ion values between 5 and 20 constitutes its doma in of a t t rac t ion . T h e values 5 and 20 are unstable equ i l ib r ium points, (b) A one-dimensional phase por t ra i t of the s i tua t ion in (a). T h e arrows indicate the direct ion of change corresponding to i n i t i a l points 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 equil ibria. 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. The curve separating these regions is called a separatrix. Curves with arrows indicate how the population densities vary over t ime beginning at various in i t ia l points. Figure A . 2 : Phase por t ra i t showing the domains of a t t ract ion of two equ i l i b r i um points in two d imen- sions. A and C are stable equi l ibr ia . The arrows indicate how the popula t ion densities vary over t ime beginning at various i n i t i a l points. A.2.6 Equi l ibr ium point If the values of the state variables representing an ecological system do not change as t ime progresses then we say that the system is at an equil ibr ium point. Other commonly used terminology is singular point or fixed point. See also section A.2.14. A.2.7 Hard loss of stability In the case of a Hopf bifurcation this occurs when there is a sudden change from stable equi l ibr ium behaviour to stable limit cycles of large amplitude. A n example is shown in figure A . 3 . When the parameter fi is increased beyond the Hopf bifurcation at //*, the system suddenly jumps to l imit cycles of large amplitude instead of starting off with Popu la t i on 2 A Popu la t i on 1 Appendix A. Dynamical systems theory 221 (popula t ion density) \ HB o o o e o Mi P-* Figure A.3: H a r d loss of s tabi l i ty (adapted from [111], P-74). T h i s phenomenon gives rise to sudden changes between stable equ i l ib r ium behaviour and limit cycles of large ampl i tude . small l imit cycles which grow in size as p increases. The latter phenomenon is shown in figure A.6(a) and is called soft loss of stability. Also, as p is decreased, there is a jump from large amplitude cycles to a zero amplitude equil ibrium point but this takes place at pi which is less than p.*. For pi < p < p* there are two stable att ractors—an equilibrium point and a limit cycle. This is a kind of hysteresis phenomenon. Examples of hard loss of stabil ity arise in the analysis of the ratio-dependent model in chapter 4. Equ i l ib r ium states can also undergo a hard loss of stability (for example, in the vicinity of a pitchfork bifurcation). However, such phenomena are not encountered in the main 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 equil ibrium point intersects a stable manifold of the other Appendix A. Dynamical systems theory 222 equi l ibr ium point then the system is said to have a heteroclinic orbit. A n example is shown in figure A .4 . For any init ial point on this orbit the system wil 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. equi l ibr ium 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 H o m o c l i n i c orbit This is similar to a heteroclinic orbit except that in this case the unstable and stable manifolds of the same equil ibrium 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. The 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 homocl in ic orbi t . A n unstable and a stable mani fo ld of the saddle point intersect. A.2.10 H o p f b i f u r c a t i o n A Hopf bifurcation 1 (HB) is a bifurcation point at which an equilibrium point alters stability and a limit cycle (period orbit) is init iated. 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 imit 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 equil ibr ium point or sink. After the Hopf bifurcation is encountered at p = p*, A becomes a source and a stable l imi t cycle is ini t iated. The corresponding dynamics at p = p2 are shown in figure A.6(b)( i i ) . Notice that the amplitude of the l imit cycle increases as p increases (see figure A.6(a)) . This is called soft loss of stability. In figure A .6 a stable l imit cycle surrounds an unstable equil ibr ium point. It is also possible for an unstable l imit cycle to encircle a locally stable equi l ibr ium 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 name H o p f bifurcation is usually used, A r n o l d [7] points out that this is inaccurate. B o t h Poincare and Andronov studied this bifurcation prior to Hopf. W i g g i n s [124] refers to the Poincare- A n d r o n o v - H o p f bifurcat ion. Appendix A. Dynamical systems theory 224 Figure A . 6 : (a) A bifurcation d iagram of a Hopf bifurcation ( H B ) , (b)(i) a phase por t ra i t corresponding to fi = fii at which there is a stable equ i l ib r ium point and (b)(ii) a phase por t ra i t corresponding to n = /J2 at which there is an unstable equ i l ib r ium point surrounded by a stable l i m i t cycle. Appendix A. Dynamical systems theory 225 (popula t ion density) o° Figure A . 7 : B i fu rca t ion d iagram of a Hopf bifurcat ion where unstable per iodic orbits surround a stable e q u i l i b r i u m point . The first examples of Hopf bifurcations occur in 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 in figure A . 8 . This results in multiple equil ibria corresponding to a single parameter value. For [i < u*^ and [i > u.*2 there is a single equilibrium point, which is stable in this example. For parameter values n*A < fj, < pi*2 there are three equil ibr ium points—two stable and one unstable. The unstable equil ibrium point divides the domains of attraction of the stable equil ibria. For init ial population densities above the dotted line in the range //*! < \i < u*2 the population tends towards C. For ini t ia l values below the dotted line the population is attracted towards A . The way in 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 wi l l continue to tend towards C for all p such that Appendix A. Dynamical systems theory 226 (popula t ion density) • L P LPC Mi M* M2 M2 p Figure A .8 : Bifurcat ion d iagram of hysteresis. p < p*2 since our init ial point is above the boundary B of the domain of attraction of C. However, as p increases beyond p*2 a catastrophe or sudden change occurs and the system tends towards A instead. What is more, if-// is now decreased we do not return to the equi l ibr ium at C. The system continues to tend towards A as we are below the dividing point B. This occurs unti l p \ is passed. Then the system jumps up towards C again. The situation that has been described is known as hysteresis. Occurrences of this phenomenon arise in chapter 4. In nature there are many unpredictable influences on a system which means that there wi l l be fluctuations around any equil ibrium. Notice that the domain of attraction of C is smaller the closer p is to p*2. This makes the system much more susceptible to crashing towards A as p*2 is approached [128]. A . 2 . 1 2 L i m i t cycle L i m i t cycle behaviour occurs when a state variable (such as a population density) os- cillates 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 imit ing behaviour once the in i t ia l transients have died away. A l imit cycle is also called a periodic orbit and is often associated with a Hopf bifurcation. L imi t cycles may be locally stable or unstable (see section A.3.4 for more details). (a) P o p u l a t i o n density (b) predator T i m e prey Figure A . 9 : (a) T i m e plot and (b) phase por t ra i t of stable l i m i t cycle behaviour . A.2.13 Limit point A l imit 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 imit point (LP) . For p < p* there are no equi l ibr ium points at which both populations are nonzero, p* is thus the l imi t ing value of p for which equil ibr ium 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 equil ibrium point and B is a saddle point. The ini t ia l values of xi and x2 determine the subsequent behaviour of the system. If the init ial point is in the domain of attraction of A (to the right of point B in figure A.10(b)) , then the system wi l l approach A . If the init ial point lies on the other side of B, however, it wi l l be repelled away from B in the opposite direction to A . Notice how the size of the domain Appendix A. Dynamical systems theory 228 (a) . 0-2- (popula t ion density) (b ) x2 (predator) •B LI / i ! Ax? 1 0-2 (prey) Figure A .10: (a)Bifurcat ion d iagram showing a l i m i t point ( L P ) and (b)a phase por t ra i t corresponding to p = H\. For fi < fi* there are no equ i l ib r ium points at which both popula t ions are nonzero. of attraction of A , in terms of the state variable x\, decreases as p decreases towards p* (see figure A.10(a)). The 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 equi l ibr ium, then the equil ibr ium point is said to be locally stable and is called an 'at- tractor' . Otherwise it is said to be unstable and is a 'repeller'. Locally stable equi l ibr ium 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 t ime, Appendix A. Dynamical systems theory 229 a parameter is kept constant as time progresses. 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 equi l ibr ium may cause the population to start cycling. A . 2 . 1 6 Period-doubling bifurcation A period-doubling bifurcation occurs when a limit cycle undergoes a bifurcation and there is an exchange of stabil ity to cycles having double the period. The situation is depicted graphically in figure A . l l . Part (a) shows a bifurcation diagram wi th period-doubling bifurcations at Ai and A 2 , and part (b) shows the behaviour over t ime for different values of the parameter A. Chapter 4 contains the first examples of this phenomenon in the main body of the thesis. A . 2 . 1 7 Phase portrait Suppose our system has two state variables, say a prey (x i ) and a predator ( £ 2 ) . We can represent the behaviour of both populations in a single diagram called a phase portrait . A n example is shown on the square base of the diagram in figure A . 12. In this example both x\ and x2 exhibit oscillations of decreasing amplitude as they approach the stable equilibrium point. This translates into an inward spiral in the (xx, x 2 ) -phase space. 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) • o o o o • • • * • # 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) Xl A < A i i) H i i i ) A / ^ < A < Ax T i m e A i < A < A 2 T i m e Figure A . 1 1 : (a) Per iod-doubl ing bifurcations at \ \ and A 2 (adapted from [111], p.259). (b) Behav iour over t ime for the state variable for (i) X < \JJ (stable equ i l ib r ium) , (ii) A# < A < A i ( l i m i t cycle) and ( i i i ) Xi < A < A 2 (period-2 cycle—each cycle consists of a b ig hump and a sma l l hump) . A.2 .18 P i tchfork bifurcat ion 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 equi l ibr ium points on the other side. A n example is shown in figure A . 13. Part (a) shows a bifurcation diagram and part (b) gives phase portraitsior parameter values on either side of the bifurcation point. In the example shown in figure A .13 , A is a stable equil ibr ium point for p < p*. For n > fi* A is a source and the other two equil ibrium points, B and C, are locally stable. These stability properties vary from situation to situation but the symmetry is Appendix A. Dynamical systems theory 231 Figure A. 12: Der iva t ion of a phase plane showing the time-dependent behaviour of two variables, xi and X2 (adapted from [61], p-3). D a m p e d oscil lat ions over t ime give rise to an inward spira l in 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 ind of threshold point as it separates the domains of attraction of B and C. The ini t ia l values of Xi and x2 determine whether the system tends towards B or C (see figure A.13(b)(i i)) . No examples of this type of bifurcation occur in this thesis but the above description is included for completeness. A . 2 . 1 9 Qual i tat ive behaviour When 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) pop- ulation densities for each instant in time. For example, different types of qualitative Appendix A. Dynamical systems theory 232 (a) « 2 o i a 2 CL3 (prey) (prey) Figure A .13 : (a)Bifurcat ion d iagram of a pitchfork bifurcat ion, (b)(i) a phase por t ra i t corresponding to ix = Hi and (b)(ii) a phase por t ra i t corresponding to LI — p 2 - In (b)(ii) the unstable manifolds from A d iv ide up the domains of a t t ract ion 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 sec- tion A.2.5) . Appendix A. Dynamical systems theory 233 A . 2 . 2 0 S a d d l e p o i n t A saddle point is an equilibrium point which attracts in certain directions and repels in others. In figure A.14 the equil ibrium point has an unstable manifold (see page 241) and a stable manifold as indicated by the dashed lines. Initial points ly ing on these manifolds are repelled from or attracted towards the equil ibrium point respectively. Other in i t ia l points may first be attracted and then repelled as shown by the solid lines. 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 . 2 1 S i n k A sink is a locally stable equilibrium point. A sink may be either a stable node or a spiral attractor. Phase portraits and t ime plots corresponding to these two possibilities are shown in figures A.15(a) and A.15(b) respectively. The t ime plots begin at the point marked with a * in the phase portraits. predator prey Appendix A. Dynamical systems theory 234 (a) i) f H) Xi 0 T i m e (prey) Figure A .15: (a)(i) Phase portrai t of a stable node and (a)(ii) t ime plots starting at point * in (a)(i) . (b)(i) Phase por t ra i t of a spiral attractor and (b)(ii) t ime plots starting at poin t * i n (b)( i) . A . 2 . 2 2 Soft loss of stabi l i ty For a Hopf bifurcation this occurs when there is a continuous change from stable equilib- rium 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 systems theory 235 A . 2 . 2 3 Source A source is an equilibrium point which is locally unstable. Any disturbance to the system wil l 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. Both are examples of sources. (a) i) | ") a . X i 0 T i m e (prey) F i g u r e A .16: (a)(i) Phase por t ra i t of an unstable node and (a)(ii) t ime plots s tar t ing at poin t * in (a)(i) . (b)(i) Phase por t ra i t of a spiral repeller and (b)(ii) t ime plots s tar t ing at point * in b( i ) . A.2 .24 State variable Suppose we are interested in a system consisting of plants, herbivores and predators. Then the 'state' of the system can be described by the relative biomasses or densities of Appendix A. Dynamical systems theory 236 these populations. The 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 transcrit ical bifurcation point two equilibrium points coincide and exchange stabil - ities. A n example is shown in figure A.17. There are two equil ibr ium points, A and B, (a) (popula t ion density) a>- A - B- fJ-i (b) i) X2 (predator) ii) X2 (predator) Xi (prey) Figure A . 17 : (a) A bifurcation d iagram of a t ranscr i t ical bifurcat ion, (b)(i) a phase por t ra i t corre- sponding to fi = fi\ and (b)(ii) a phase por t ra i t corresponding to / i = [in. T h e stabi l i t ies of the two equ i l i b r i um points interchange for parameter values on either side of the bifurcat ion point . 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 The situation is reversed for B. Figures A.17(b)(i) and A.17(b)(ii) show possible phase portraits in two dimensions for p = pi and p = p2 respectively. Note that the bifurcation diagram in figure A.17(a) only indicates the positions of the equi l ibr ium points in 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 This 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 wi l l assume that the model under study consists of a system of m ordinary differential equations of the form: where x is a vector of m state variables and the dot denotes differentiation wi th respect to t ime, that is, x = For example, if we.were studying a predator-prey model then we would have m = 2 and equation ( A . l ) in expanded form would be where / i and f2 are the components of f representing the dynamics of xi (prey density, say) and x2 (predator density), respectively. The results for systems of difference equations (discrete models) of the form: x = f (x) ( A . l ) X\ = fi{xi,x2) X2 = f2{xi,X2) X I—> f(x) (A.2) or x t + 1 = f(xO, t = 0 , l , 2 , . . . (A.3) Appendix A. Dynamical systems theory 238 are very similar. Section A .3 .5 highlights some of the differences. A.3.2 Equi l ibr ium points and local stability A n equi l ibr ium 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 t ime zero, it would remain there for al l t ime. However, in nature it is very unlikely that a system wi l l remain exactly at an equi l ibr ium 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 from x* as t ime progresses. That is, we would like to determine the local stabil ity behaviour near x*. We can sometimes do this by using a linearised analysis 2 . We begin by perturbing the system slightly from x*. That is, we replace x by x* + u in equation ( A . l ) where u is a small perturbation. (We use a small perturbation since we are investigating the local behaviour near the equi l ibr ium point.) Our new vector of state variables is u since x* is fixed. Expanding f in a Taylor series about x* and neglecting nonlinear terms in u (since u is small) we obtain the linearised system ii - A u where A is the matr ix of first order part ial derivatives of f evaluated at the equi l ibr ium 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 stabil ity properties of x*. If al l the 2 M o r e detai led in t roduct ions to linear analysis can be found in [34, 128]. Appendix A. Dynamical systems theory 239 eigenvalues of A have non-zero real parts 3 then x* is said to be a hyperbolic equi l ibr ium point of ( A . l ) . If any eigenvalue has a zero real part then x* is said to be a nonhyperbolic equi l ibr ium point. The local stabil ity behaviour near hyperbolic equi l ibr ium points is relatively easy to determine. Nonhyperbolic equi l ibr ium 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 equi l ibr ium 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 asymptotical ly stable equil ibr ium point of ( A . l ) (that is, trajectories starting near x* move towards x* as t ime progresses). If 3̂ 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 in (A . l ) ) even more information can be obtained about the behaviour near In this case there are two (since m — 2) eigenvalues, Ai and A 2 , of A . They satisfy the equation 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). The various possibilities are summarised in figure A . 18. Alternatively, the results can be summarised as follows: 3 A complex number, A, has a real part , 9£A, and a complex or imag inary part , 9 A . For a br ief in t roduc t ion to the relevant theory of complex numbers see the appendix i n [128]. A 2 Appendix A. Dynamical systems theory 240 3 Saddle point Tr 2 = 4 D e t Tr ^ Saddle point Figure A .18: A summary of the local s tabi l i ty behaviour near an equ i l i b r ium point , x*, of ( A . l ) when m = 2 (adapted from [34], p.190). A i , A 2 real • Ai < 0, A 2 < 0 => x* is a stable node (region 1 in figure (A.18)) • Ai > 0, A 2 > 0 => x* is an unstable node (region 2 in figure (A.18)) • Ai > 0 ,A 2 < 0 (or vice versa) x* is a saddle point (region 3). A i , A 2 complex • 9£Ai < 0, 9?A2 < 0 => stable spiral or focus (region 4) • SftAi > 0,!RA 2 > 0 => unstable spiral or focus (region 5). Appendix A. Dynamical systems theory 241 Saddle points and sources are both unstable equi l ibr ium points (see section A.2.14) but a saddle point differs from a source in that solutions may be attracted towards it for a while before being repelled, depending on the ini t ia l values of the variables (see region 3 in 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 in phase space such that in i t ia l points on these curves are attracted towards the saddle point (for in i t ia l points on the stable manifold) or repelled away from the saddle point (for points on the unstable manifold) (see region 3 in figure A . 18). Nonhyperbolic equilibrium points As mentioned earlier it is the nonhyperbolic equi l ibr ium points that are associated wi th bifurcations, that is, with changes in the qualitative behaviour of the system of equations. There wi l l be a threshold value at which the change in behaviour occurs—the 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 in sections A.2.10, A .2 .13, A.2.16, A.2.18, A.2.25. For nonhyperbolic equi l ibr ium points the principle of linearised stabil ity used above does not apply and other methods need to be used. Two 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. Normal form theory uses systematic coordinate changes to transform this reduced system of equations into a 'normal form'. The behaviour corresponding to a number of normal forms has already been classified by various mathematicians and can be found in most dynamical systems texts. The abovementioned examples have al l been Appendix A. Dynamical systems theory 242 classified using normal form theory. Another method due to Liapunov is described in Wiggins [124]. One-parameter local bifurcations Suppose that the system of equations ( A . l ) has the form x = f(x,/x) (A.4) where fj, is a parameter 4 and suppose that the equi l ibr ium point x* undergoes a bifurcation at pL — (j,*. (We assume init ial ly that there is only one zero eigenvalue or one pair of complex conjugate eigenvalues with zero real par ts—the 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 one- parameter local bifurcation. Examples of bifurcation points having one zero eigenvalue include l imi t point (see section A.2.13), pitchfork (see section A.2.18), and transcrit ical (see section A.2.25) bifurcations. A Hopf bifurcation (see section A.2.10) is also a one- parameter bifurcation but it has one pair of complex conjugate eigenvalues whose real parts are zero. Two-parameter local bifurcations Suppose we allow two parameters in our model to vary, that is, x = f(x,//,A) (A.5) where fj, and A are parameters. W i t h two parameters more complex behavioural patterns such as hysteresis, which is described in section A .2 .11, are possible. The extent of the region of overlap in figure A . 8 (that is, the difference fi*2 — f i \ ) may vary with a second parameter, A, as shown in the fwo-parameter bifurcation diagram 4 T y p i c a l l y a system of equations has more than one parameter but we only need to consider one of these exp l i c i t ly at the present t ime. W e assume that the values of any other parameters are f ixed. Appendix A. Dynamical systems theory 243 in figure A.19(a). Part (b) of this figure shows one-parameter bifurcation diagrams (a) Ml fJ-2 '_P3 M4 M M iii) M — i — i — Pi M3 M Mi M4 M Figure A . 19: (a)Two-parameter bifurcation d iagram showing a cusp point and the posi t ions of the two l i m i t points associated w i t h the hysteresis as both fi and A are varied. (b)One-parameter bifurcat ion d iagrams corresponding to different, fixed values of A in part (a) and w i t h fi as the bifurcat ion parameter, (i) A = A i , (ii) A = A 2 and (i i i) A = A 3 . These one-parameter bifurcation diagrams correspond to the hor izonta l dashed lines in part (a). corresponding to different values of A (that is, corresponding to the horizontal dashed lines in part (a)). A t A = Ai the equil ibrium point does not undergo any bifurcations in behaviour. For A = A 2 the two l imit points are close together and for A = A 3 they are further apart. The point (p, A) = (M,A*) is called a cusp point. A t this point the two l imit points coincide. The curves in figure A.19(a) thus show how the positions of the l imit points (bifurcation points) vary with p and A. Compare this with 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 G l o b a l b i f u r c a t i o n s So far we have been looking at the local dynamics associated with bifurcations of equi- l ib r ium points and l imit 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 equil ibr ium points and let fi be the bifurcation parameter. Phase portraits of two possible degenerate situations that can arise are shown in figure A .20. In (b) x2 Xi Xi Figure A .20: (a)Phase por t ra i t of a saddle connection or heteroclinic orbi t . (b)Phase por t ra i t of a saddle loop or homoclinic orbi t . part (a) of this figure a heteroclinic orbit joins two saddle points. That 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 equi l ibr ium point. The dynamics associated with the second equi l ibr ium point vary depending on the model equations. The situations in figure A.20 are degenerate. That is, they only exist for a particular value of f i . A lmost any small perturbation wi l l disrupt the coincidence of the stable and Appendix A. Dynamical systems theory 245 (a) i) x2 X2 i i i ) X2 7 T P < P Xl P — P Xl P > P Xl (b) i) X2 i i) i i i ) ^2 p < n Xl p - p Xl p> p Xl Figure A . 2 1 : (a)Phase portrai ts for parameter values near a saddle connection or heterocl inic orb i t . (b)Phase por t ra i ts for parameter values near a saddle loop or homocl in ic orbi t , (i) p < p*, (ii) /j, = fi* and ( i i i ) p > p*. (p*\s the point at which the heteroclinic or homocl in ic orbi t occurs.) unstable manifolds. In figure A.21 we see what happens to the stable and unstable man- ifolds when p is perturbed from the bifurcation point, p*. The reader may be wondering what happens near the second equil ibrium point in part .(b) of this figure. Some examples of possible phase portraits are shown in figure A.22. 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 in chapter 2 (section 2.4). Appendix A. Dynamical systems theory 246 Figure A .22: Phase portrai ts near a saddle loop or homocl in ic orbi t showing possible behaviour near the second equ i l i b r i um point . Appendix A. Dynamical systems theory 247 A . 3 . 4 Per iod ic orbits L o c a l s tabi l i ty So far we have discussed equi l ibr ium points and the stabil ity behaviour associated with them. The concept of a Hopf bifurcation introduced the idea of periodic orbits or l imi t cycles. Cycles have been studied in many biological settings (for example, the spruce budworm [74], nerve action potentials [58], glycolysis [47], cellular slime mold [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 main results are presented here. More detailed discussions can be found in [49] and [124]. In general 5 , a Poincare section S is an (m — l ) -dimensional hypersurface chosen so that al l trajectories of ( A . l ) a) intersect the hypersurface transversally, and b) cross the hypersurface in the same direction. In particular, the l imit cycle passes through the hypersurface transversally at a particular point, q*. Figure A.23 shows a periodic orbit in 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 l imit cycle and ip(t;z) is a solution of ( A . l ) starting at z (that is, satisfying the in i t ia l condition x(0) = z) , then q* = <p(T;q*). Let q be a point on S and let T v (q) be the t ime taken for a trajectory ,x(t; q) to first return to S. Then the Poincare map or first return map P(q) is defined by P(q) = <p(Tv(q);q). This is i l lustrated in figure A .23 . Note that P(q*) = q* and Tv(q*) = T. 5 T h e fo l lowing is summar ised f rom [111]. Appendix A. Dynamical systems theory 248 Figure A .23 : Schematic representation of a Poincare section and a l i m i t cycle in three dimensions (from [111], p.244). q*is a point on S that lies on the l i m i t cycle, q is another point on S and P ( q ) is its poin t of first return to S. In order to determine the stability of the periodic orbit, we need only investigate the behaviour of P near its equil ibrium point q*. That is, we need to determine whether this equi l ibr ium point is attracting or repelling. As in section A.3.2 we linearise P about the equi l ibr ium point q*. In this case A = a P J ^ \ Stabil ity is again related to the eigenvalues of A but the conditions are slightly different as we are dealing with 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 matr ix The matr ix M always has an eigenvalue equal to 1. It can be shown that the remaining Floquet multipliers. Analogous to the discussion on equi l ibr ium points of ( A . l ) , the M = d<p{T;q*) dz (m — 1) eigenvalues are the eigenvalues of A . The eigenvalues of M are called Appendix A. Dynamical systems theory 249 stabil ity of a periodic orbit can be determined by calculating the Floquet multipl iers. These are easier to find than the eigenvalues of A. In general, in the vicinity of a Hopf bifurcation, unstable periodic orbits encircle stable equil ibrium points and stable periodic orbits encircle unstable equil ibrium points however there are some other possibilities. A few cases are shown in figure A.24. Both the min ima and m a x i m a of the periodic orbits Figure A.24: E x a m p l e s of H o p f bifurcations having soft loss of s tabi l i ty (adapted from [111], p-72). are shown. Figure A.24 gives examples of soft loss of stability or soft generation of l imi t cycles (see section A.2.22). Section A.2.7 gives an example of hard loss of stabil ity or hard generation of l imit cycles. Bifurcations of periodic orbits The 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 f rom 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 stabil ity of the periodic orbits to orbits having double the period. The situation is depicted in figure A . 1 1 . Period-doubl ing occurs in many situations such as chemical reactions, nerve models, the Navier-Stokes equations and ecological models involving three trophic levels. (Period- doubling does not occur in 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 stabil ity 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. Aga in m > 3 is required. Details of this can be found in [111, 124]. From a practical point of view it is probably more informative to return to generating numerical solutions of a model over t ime when these complicated phenomena are encountered in 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 in section A .2 .3 . Further mathematical details can be found in [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 in [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 251 removes the chaotic behaviour. However, in [82, 84] it is demonstrated that chaos is prevalent in 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) The 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 equil ibr ium point occurs when f (x) = x or x i + i = x«. A linear stabil ity analysis can again be done for hyperbolic equi l ibr ium points. For maps a hyperbolic equi l ibr ium point is one for which none of the eigenvalues of the matr ix of part ial derivatives has unit modulus (that is, no eigenvalues have a magnitude of 1). Aga in it is the nonhyperbolic equi l ibr ium points which result in interesting bifurcations. If the linearised matr ix has a single eigenvalue equal to 1 then a l imi t point, trans- cr it ical 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 dia- grams in state space consist of discrete points rather than continuous curves. Examples corresponding to a spiral sink are shown in figure A .25. Consecutive points are labelled in 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 equi l ibr ium point undergoing a, period-doubling bifurcation to become a stable period-2 orbit as fx is increased. This period-2 orbit undergoes a further period-doubling to produce a stable period-4 orbit. Figure A.26(b) gives examples of diagrams in 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. Consider the second iterate, f2, of the map (A.3): f 2 (x f ) = f ( f ( x 0 ) = f ( x , + 1 ) - x t + 2 . In general, the kth iterate of the map is given by f f c(x t) = xt+Jfe. Suppose there exists a value of x , x , such that f f c(x) = X Appendix A. Dynamical systems theory 253 (a) Xi - « O O 0 c oo • •••••• • • B O O o o o - f i—I—I—I Mi Mi* M2 M2 M3 M (b) X2 1 • 9 , «4 s . A 3« ii) ^2 1. 2* 4». * • • \ lll) 1 z 2 1 2 » 3' 5* 7' 11* 4 6 10 • . • ,14 12 V .9 +- .13 Figure A .26: (a)One-parameter bifurcation d iagram showing per iod-doubl ing bifurcations at // = p\ and p = p*2 for a discrete system. (b)State space diagrams showing the dynamics at (i) \x — \x\ (stable equ i l i b r ium) , (ii) p = \in (stable period-2 orbit) and (i i i) u = fi3 (stable period-4 orb i t ) . but f j ( x ) ^ x f o r j = l , 2 , . . . , k - l . Then x is called an equil ibrium or fixed point of period k. This means that the system has a cycle or periodic orbit whose period is equal to k t ime units. Suppose k = 2 and we have a one-dimensional system. Then / 2(x) = /(/(*)) = x but f(x) + X. Appendix A. Dynamical systems theory 254 Hence, if we start at x then after the first t ime unit we are at f(x) but after the second t ime unit we have returned to x. This is shown in figure A.27. A state space diagram i i i \ i \ — i — - 0 1 2 3 4 5 6 7 T i m e Figure A.27: T i m e plot of a period -2 orbi t for a discrete system. for such a situation in two dimensions (that is, for two state variables) is shown in figure A.26(b)( i i ) . In this example both xi and x2 wi l l have t ime plots resembling figure A.27. 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. This bifurcation corresponds to a pair of eigenvalues of modulus 1. Instead of a periodic orbit, however, an invariant circle is init iated at this bifurcation point. Whi le 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 t ime plot. The stabil ity 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 with 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 stil l lie on the circle. If the map is allowed to iterate for a long t ime in the latter case, then what is eventually observed in state space wil l Appendix A. Dynamical systems theory 255 (a) X2 (b) X l 1» »9 4t • 6 5« 6' .8 5« «9 2 • • 4 .3 T i m e Figure A .28: (a) State space diagram showing an invariant circle, (b) T i m e plot of the s i tua t ion in (a) in terms of x\. 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 bui l t - in t ime delays in the feedback relation- ships [128]. Even one-dimensional maps can exhibit chaotic behaviour. May [82] shows how the behaviour of the discrete analogue of the logistic equation changes from stable equi l ibr ium 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 Stabi l i t y of bifurcations under perturbat ions The 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 imi t point bifurca- tions, Hopf bifurcations and hysteresis phenomena are stable under small perturbations but transcrit ical 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 equi l ibr ium points and periodic orbits) but can affect the occurrence of the latter two. Figure A.29 illustrates the destruction of transcritical and pitchfork bifurcations graphically. It turns out that all bifurcations of one parameter families of equations which have an equi l ibr ium point with a single zero eigenvalue (or single eigenvalue of modulus 1 for maps) can be perturbed to l imi t point bifurcations [49]. Figure A.29: Possible results of per turbing t ranscr i t ical and 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]. The 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. Dynamical systems theory 257 A.3.7 Mult iple degeneracy A l l the cases discussed so far have assumed simple zero eigenvalues or a single com- plex conjugate pair of eigenvalues with zero real part (or the analogous situations for maps). Higher order singularities, or mult iple degeneracies do, of course, occur but^the behavioural dynamics associated with them can be difficult to interpret and there is st i l l much research being done in 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 in a region surrounding these complex points rather than struggling with the details of the bifurcation structure. For those interested some of these higher order degeneracies are investigated in [124]. A . 4 Conclusion ( The introduction to dynamical systems theory given in this appendix has been mainly intuit ive and not mathematical ly rigorous. Although the concepts are fairly simple, the mathematics involved in studying a particular system of equations can be quite formidable. Appendix B describes how computers can be useful in this regard. Appendix B Numerical details B . l Introduction This 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 wi l l be mentioned later, enable particular solu- tions 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 equi l ibr ium 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 mathe- matics had to be done by hand. The process can be repeated for different parameters in order to obtain a more comprehensive picture of the qualitative behaviour associated with different regions in the parameter space. Section B.2 introduces some of the techniques used by continuation programs for continuing equi l ibr ium points and determining the stability of the solution branches (see section A.2.4) . Methods for detecting bifurcations along these branches are also described wi th particular reference to A U T O 1 . The section is intended as a brief overview. More H w i l l use the abbrevia t ion A U T O to refer to A U T 0 8 6 , A T J T 0 9 4 and Interactive A U T O as the lat ter is i n essence jus t a graphica l interface for A U T 0 8 6 and A U T 0 9 4 is an updated version of A U T 0 8 6 together w i t h a graphica l interface. Even when using X P P A U T I w i l l refer to A U T O when t a lk ing about bifurcat ion diagrams since X P P A U T generates these diagrams through a graphica l interface w i t h A U T 0 8 6 . In cases where I need to dis t inguish between the packages, I w i l l use their fu l l names. 258 Appendix B. Numerical details 259 detailed discussions as well as a comprehensive list of references can be found in 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 ime plots, phase portraits and bifurcation diagrams. Final ly , section B.5 gives a few pointers and warnings regarding the use of some of the packages. These are based on my experiences gained through analysing the models in the main body of the thesis. B .2 Theory B.2.1 Continuation methods As has already been mentioned, continuation or path-following methods generate a chain of solutions (equil ibrium points, periodic orbits or bifurcation points) as a parameter is varied. A typical path-following method is the predictor-corrector method. This 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 im to improve this approximation and bring it back to the actual curve [4]. Typical ly Newton or gradient type methods are used in this step [4]. Some form of parameterisation of the curve is required for these steps. The obvious choice is the control parameter (that is, the parameter being varied) as it has physical significance. However, this leads to difficulties at l imit points (see section A.2.13) [111]. A n alternative is to choose another variable which involves adding another equation to the system. This 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 in [30, 67]. The 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 t ime. In some cases the objective is jusf to follow the curve as rapidly and safely as possible unt i l a crit ical 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. A U T O fulfills both criteria. Stepsizes are changed automatically in the program depending on the speed of convergence of Newton's method (that is, depending on the number of iterations required to fulfi l l 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 in which choices of predictor, corrector, parameteri - sation 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 stabil ity of Newton correctors, more stable higher order predictors based on polynomial interpolation (instead of on the direction of the tangent to the curve as with an Euler predictor) are often advantageous [4]. No continuation method can guarantee that all possible solutions wi l l be found in a given example [111]. Isolated branches (branches which are not attached to other branches v ia bifurcation points) are very difficult to detect. It is suggested in the A U T 0 8 6 manual [30] and by Seydel [111] that t ime integration of the governing system of equations using random in i t ia 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 equi l ibr ium points which lie on these isolated branches. B .2 .2 Detection of bifurcations As 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 addit ional equations which characterise the bifurcation point. Indirect methods on the other hand utilise data obtained during a continuation together with a test function. The latter are generally recommended in practical computations (and are used in A U T O ) as direct methods involve solving much larger systems of equations and have higher storage re- quirements. Indirect methods do have more difficulty achieving high accuracy but when discretisation errors are present and when bifurcation points are unstable to perturba- tions, 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 evalu- ated 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 max imum of all real parts a ; of the eigen- values of the Jacobian matr ix A (see section A.3.2) . However, this choice may not be smooth and it does not signal bifurcations in which only unstable branches coalesce [111]. This problem can be overcome by choosing r = with |afc| = m i n . . . , |am|}. This choice also detects Hopf bifurcations. In general, the accuracy of T depends on the accuracy with 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 in 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 in 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 in the A U T 0 8 6 manual [30]. B.2.3 Stability Stationary branches The stabil ity of stationary branches (that is, of continuations of equi l ibr ium points) is de- termined from the real parts of the eigenvalues of the Jacobian matr ix A. The calculation of eigenvalues (for example, v ia Q R factorisation or L U decomposition methods [4, 111]) can be very t ime-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 in 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 matr ix [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 stabil ity of periodic orbits. These multipliers are eigenvalues of a particular matr ix and, thus, an eigenvalue solver is again required. Since Floquet multipliers can be very large or very small in value, there may be a loss of accuracy when evaluating them numerically. This 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 stabil ity determination) breaks down. In X P P A U T and A U T 0 9 4 the routine for calculating Floquet multipl iers has been improved and is more accurate than that in A U T 0 8 6 , and hence in Interactive A U T O . Floquet multipl iers 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 264 B.3.1 A U T O 8 6 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 wi l l describe its capabilities and l imitations since most of them 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 imited bifurcation analysis of algebraic systems of the form f ( x , A ) = 0 , x , f e 7 e m (B.6) where A denotes one or more free parameters, and of systems of ordinary differential equations of the form x = f ( x ,A ) , x , f e 7 t m (B.7) It can also do certain continuation and evolution computations for the diffusive system x t = D x u u + f ( x , A ) , x , f e 7 l m (B.8) where x = x ( u , £ ) and D denotes a diagonal matr ix of diffusion constants. 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 imit points (saddle-node bifurcations) and continue these in two parameters. Appendix B. Numerical details 265 • do al l 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 in more parameters. For the ordinary differential equations (B.7) A U T O can: • compute branches of stable or unstable periodic solutions and Floquet mult ipl iers. Starting data for the computation of periodic orbits are generated automatical ly at Hopf bifurcation points. • locate l imi t points, transcritical and pitchfork bifurcations, period-doubling bifurca- tions and bifurcations to tori along branches of periodic solutions. Branch switching is possible at transcrit ical, pitchfork and period-doubling bifurcations. • continue Hopf bifurcation points, l imit points and orbits of fixed period in two parameters. • compute curves of solutions to (B.7) on the interval [0,1] subject to general nonlinear boundary or integral conditions. • locate l imi t points and bifurcation points for such boundary value problems. Branch switching is possible at bifurcation points. Curves of l imit points can be computed in 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 ho- mogeneous 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. The wave speed c is fixed but the wave length L wi l l normally vary. • trace out branches of waves of fixed wave length L in two parameters. If L is large, then one gets a branch of approximate solitary wave solutions. • do t ime 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. The 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. Also, the adaptive mesh guards to some extent against computing spurious solutions. When spurious solutions do occur they are often easy to recognise by the jagged appearance of the solution branch. Some general l imitations of A U T 0 8 6 are listed below. Limitations The following difficulties are noted in the manual [30]: • degenerate (multiple) bifurcations cannot be detected in general. A lso , 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. This 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 crit ical eigenvalue pair. A n often occurring case is a Hopf bifurcation close to a l imit 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 with 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 imit points, period-doubling bifurcations or bifurca- tions 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. As noted above A U T O may have some difficulty with identifying bifurcations on periodic orbits. Problems arise when Floquet multipliers are close to the unit circle. Also, unstable orbits can be difficult to locate, especially when they are close to a stable orbit or equi l ibr ium. Accuracy may be increased by decreasing dsmin or increasing ntst, the number of mesh points used in the discretisation of the periodic orbits. Some other l imitations of A U T O which have resulted from doing bifurcation analyses on a few models are as follows: • transcrit ical and pitchfork bifurcations cannot be continued in two parameters be- cause the determinant of the Jacobian matr ix A is zero at these bifurcation points. Other continuation and bifurcation packages wi l l have the same l imitat ion. A possi- ble solution is to do a number of one-parameter continuations with different (fixed) Appendix B. Numerical details 268 values of the second parameter and then to plot the bifurcation points obtained from each continuation in two-parameter space. A n approximate curve can be drawn through these points. This is done in chapter 2. • error messages from A U T O can be misleading (as with 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 with 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 in progress. The corresponding eigenvalues are shown simultaneously in a separate graphics window. The advantages and disadvantages listed below relate mainly to the suitabil ity of this package for analysing ecological models and are included for comparison wi th the other packages. Since the packages were not set up specifically for the models that are analysed in this thesis, these comments are not necessarily criticisms. Advantages • This package has good on-screen graphics. Different colours are used to indicate the number of unstable eigenvalues corresponding to a particular branch. The locations of the eigenvalues relative to the real and imaginary axes and the unit circle are also shown. Bifurcation diagrams can be viewed in three dimensions if desired and the mouse can be used to shift diagrams and to zoom in and out. Appendix B. Numerical details 269 Disadvantages • Bifurcat ion diagrams cannot be printed out or saved as postscript files for later printing. • Each t ime the model equations are altered the driving routine needs to be recom- piled. • The package only runs on Personal Iris and Iris 4D workstations (instead of on all systems supporting X-windows as for the other packages). • Interactive control of the graphics output is l imited. Once a one-parameter bifur- cation diagram has been generated it is not possible to view it with a different variable on the y-axis. The 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 ia anonymous ftp from ftp.math.pitt .edu and is in the directory pub /hardware. Capabilities The A U T O interface allows most of the capabilities of A U T 0 8 6 to be enjoyed. How- ever, the a im of making A U T O easier to use has led to some restrictions which wi l l be mentioned below. This 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 pos- sible to obtain nullclines, arrows indicating the direction of flow, as well as the positions and stabilities of equi l ibr ium points (singular points). Other capabilities include curve-fitting of data, spreadsheet type data manipulat ion and generation of histograms. Advantages • T i m e plots, phase portraits and bifurcation diagrams can all be generated using the same computer package. • The A U T O interface is easy to use—the number of constants that need to be altered has been reduced. • Hardcopies of bifurcation diagrams can be obtained through saving them as postscript files. The data can also be saved so that it can be read into other graphics packages. • The package has good on-screen graphics. The development of a bifurcation d i - agram can be seen while a calculation is in progress. The 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 one- and two-parameter bifurcation diagrams viewed are kept in memory. Once a one- parameter 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 multipl ier 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. • The driving program is easy to write for most systems and is compiled automatical ly when the X P P A U T command is given. • W h e n calculating t ime plots and phase portraits, data information for auxil iary variables (variables which are subcomponents or composite functions of the state variables) can also be observed. • The package runs on any X-windows system as well as on L I N U X . • There is a comprehensive tutorial on the World Wide Web to help the user become acquainted with the package. The address is: ftp://mthbard. math, pitt.edu/pub/bardware/xpptut/start. h tml . D i s a d v a n t a g e s • W h e n generating phase portraits only one equi l ibr ium point is located at a t ime and the in i t ia l point often has to be quite close by. • The 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 imi t points cannot be turned off in this version of A U T O . This may cause difficulties when calculat- ing periodic orbits as at a l imit point of a periodic orbit two Floquet mult ipl iers equal 1 and this affects the convergence properties of the continuation algorithm. Decreasing dsmin and dsmax may help. • There is l imited control of the appearance of printouts. The stabil ity nature of a particular branch in 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 indiv idual 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 lso, the data for a diagram can be saved separately and then read into another graphics package. When two-parameter bifurcation diagrams are printed out, data from the previously generated one-parameter diagram is superimposed. B.3.4 A U T 0 9 4 Capabilities This package is an updated version of and graphical interface for A U T 0 8 6 . The 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. • Model equations can be changed interactively. • The package runs on any X-windows system. • It is possible to start a continuation from a numerically calculated periodic orbit. • The routine for calculating Floquet multipliers has been improved. Disadvantages • Printouts of bifurcation diagrams cannot be obtained. Appendix B. Numerical details 273 • The 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. The way in which bifurcation points are labelled also makes the diagrams difficult to read. The disadvantages were the main reason why I did not make use of A U T 0 9 4 . B .3.5 D S T O O L This package can be obtained v ia anonymous ftp from macomb.tn.cornell.edu and is in the pub/dstool directory. Capabilities This package generates t ime plots and two-dimensional phase portraits for both discrete and continuous systems of equations. It calculates equil ibr ium points and their stabilities as well as stable and unstable manifolds for saddle points. There is provision for exten- sions to the package—three-dimensional graphics as well as continuation and bifurcation routines may be incorporated in the near future. Advantages • A mouse can be used for specifying starting points for t ime plots and phase por- traits. This is very convenient and speeds up the generation of these diagrams considerably. • The 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. • The package runs on any X-windows system as well as on L I N U X . Disadvantages • Three subroutines need to be modified each t ime a new model is entered. • The package sometimes crashes when calculating equi l ibr ium 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 in [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 bifur- cation analysis of equi l ibr ium points, l imit 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 much 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 dynamical 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 equi l ibr ium points and for detecting l imit points. As 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 st i l l needed in 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 t ime plots, phase portraits and bifurcation diagrams using some of the packages I have mentioned. B.4 U s i n g the packages Although 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. Many of the manuals have introductory examples for the reader to work through and these are invaluable for getting acquainted with the capabilities of the package. There is a comprehensive tutorial for using X P P A U T on the Wor ld Wide Web. The address is: f tp://mthbard. math, pitt.edu/pub/bardware/xpptut/start. h tml . 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 . The user s imply chooses time as the variable to be plotted on the x-axis and one of the state variables for the y-axis. After entering in i t ia l values for the state variables and the t ime 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 in i t ia l values or parameter values. These steps can al l be done interactively. Both packages have a number of choices for the numerical algorithm that is used to calculate solutions. M i n i m u m and max imum 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 in i t ia l point, then only the in i t ia l t ime and the value of the state variable on the y-axis wi l l be altered automatically. Values for the other state variables wi l l remain unchanged unless new values are typed in . When analysing a discrete system using D S T O O L , changing the t ime increment to 1 in 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 in the 'settings' window that correspond to the max imum 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 though three- dimensional portraits are possible in X P P A U T , I w i l l only discuss two-dimensional por- traits 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 plott ing on the x-axis and another for the y-axis. For systems of dimension greater than two the solution trajectories wi l l be projected into this plane. Init ial values can be typed in manually or the mouse may be used. Only the values of the state variables shown on the axes wi l l be altered when using the mouse. The user can again choose between the various numerical methods for calculating solution trajectories. In addit ion to solution trajectories the positions of equi l ibr ium points can be shown in phase diagrams. When asked to find equil ibr ium 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 stabil ity of these points. X P P A U T tends to locate one singular point at a t ime and often the in i t ia 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 'f ind' option in the 'fixed point' window a few times generally locates all the relevant points. If desired, the user can choose the in i t ia l point from which to begin a calculation instead of using the Monte 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 un- stable manifolds associated with saddle points in the plane. These help del imit 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 in the phase portrait. Equ i l ib r ium 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. This choice must be specified in the driving program for Interactive A U T O but can be done interactively in the other two packages. In addit ion 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 imit 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 in the A U T 0 8 6 manual [30] as well as in 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 in others. In order to begin generating a bifurcation diagram a fixed point (equil ibrium point) , corresponding to a particular parameter set, is required. Such a point can be determined analytical ly (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 transcrit ical, 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. The second iterate of the model must be used for this purpose. Since a (period-1) Appendix B. Numerical details 279 equi l ibr ium point of the original model is also an equi l ibr ium point of the second iterate of the model , both the period-1 and the period-2 equi l ibr ium point branches wi 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. Equ i l ib r ium 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. This approach is used in chapter 7. Two-parameter bifurcation diagrams Once a l imi t point or Hopf bifurcation point has been detected in a one-parameter con- t inuation, it can be continued in 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 in Interactive A U T O and A U T 0 9 4 or choosing the two-parameter option in X P P A U T . L i m i t cycles of fixed period can also be continued in two parameters. This 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 2 . The resulting two-parameter curve gives the required approximation to the curve of homoclinic or heteroclinic bifurcation points. This technique is used in section 2.4. When continuing 2 A po in t sat isfying the U S Z R function w i l l be given a label wh ich can be used as a restart value for the two-parameter cont inuat ion. Appendix B. Numerical details 280 these curves in two parameters, detection of l imit points should be turned off as it may lead to spurious bifurcation points. This is done automatically in X P P A U T . Any bifurcation points should also be checked against the numerical output to see whether an eigenvalue or Floquet mult ipl ier has, in fact, changed sign. B.5 Pointers and warnings As with al l computer packages, the ones that I have discussed do not work exactly as one might wish in a given situation. Each 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 my experiences with the packages. Many of these wi l l only make sense once the examples in the main 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 wi l l not respond to input. I have not been able to locate the cause of this but quitt ing 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 in 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). The 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 wi l l generate bifurcation diagrams faster if the accuracy is lower. In many cases the results are st i l l sufficiently accurate, however some bifurcation points may need to be checked with 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 lso , if there is a sudden jump in a continuation, or a curve becomes very jagged, then the continuation should be repeated using greater accuracy. • The choice of step size depends on the extent of the parameter range over which changes in 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 within fairly small parameter ranges. Scaling the equations and parameters (as is done in chapters 3 and 4) circumvents this problem. • Bifurcation diagrams can become very complicated even when studying simple models. However, not al l continuation branches are always of interest. For example, some may correspond to negative or zero values of the state variables. A lso , 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 generat- ing 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 in X P P A U T where the bifurcation dia- grams and phase portraits can be generated within the same package. T ime plots and phase portraits can be checked by choosing different numerical methods to calculate solutions. Both D S T O O L and X P P A U T offer a variety of methods. B y looking at the eigenvalues corresponding to an equi l ibr ium point one can check that the stabil ity 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 with certainty. The general behaviour and the types of changes that can occur are of greater practical interest. For example, l imi t cycles of small amplitude wi l l be indistinguishable from equi l ibr ium points due to the natural variation of field data. Also, a system wi l l easily be perturbed from a stable node with a small domain of attraction and, hence, such a phenomenon may be of minor interest. It is also important to look at t ime plots corresponding to fixed parameter combinations to see how quickly a system approaches the l imi t ing behaviour indicated in a bifurcation diagram. If a system takes a long t ime to attain its equi l ibr ium configuration then the transient dynamics may be more important than the final equi l ibr ium 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 in the relevant chapters. Clearly, the more one utilises the techniques and packages that have been introduced, the more familiar one becomes with them, and the more creative one can be in their use. The above comments wi 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 in one of its components. Often there is a time lag between the ini t ia l change and the response of the system. For example, a change in hyrax population density wi l l only affect the size of the hyrax population the following season. It takes time for an increase in population density to affect the reproductive success of the hyrax and the survival of their offspring. In order to represent this in the model, averaged or smoothed versions of certain quantities are used in calculating growth rates. Three quantities are averaged in this mode l—hy rax density (Hp), prey abundance (Ap) , and the grazing mult ipl ier (GM)- For example, when sheep have been grazing more than their usual amount their condition is expected to improve resulting in a decline in juvenile mortal ity and a rise in fecundity (Swart and Hearne [116]). However, this wi l l only occur after a prolonged increase in the average amount of pasture consumed. Hence, sheep fecundity and juvenile mortal i ty are functions of the grazing multipl ier average, GM, instead of GM- A first order distributed delay (see MacDonald [77] or May [81]) is used to calculate GM- That is, <1GM _ GM — GM dt tdei where t^ei is the average delay t ime. Examples of this type of delay equation can be found in Forrester [39] and a detailed explanation of the mathematics underlying the above differential equation can be found in MacDonald [77]. The 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 in dimension does not pose a problem when solving the system numerically. Further discussions on delays in biological systems and their effects on system be- haviour can be found in [78, 81]. For the sheep-hyrax-lynx model the delays were not found to affect stabil ity 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 and Vi = — Si dvi 1 dvi dt Si dt = fi(siVi, • • • , SiVi, . . . , SmVm). Si Dropping the bars for convenience we get ^1 - Lf.( \ . — Ji\S^V\, . . . , SiVi, . . . , SmVm I dt Si as required. The state variables have been replaced by s,-u,- and the differential equation for Vi is divided through by s,- as stated in 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 indiv idual larva to depend on the availabil ity of new foliage per larva, that is, to depend on FbL^F (Fi/Kp represents the amount of foliage that is new foliage). The larvae prefer new foliage to old and thus, in most circumstances, wi l l eat al l the new foliage before moving on to old foliage. Assuming 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. We can model this competit ion (see Starfield and Bleloch [113]) by including the factor L d0Lb \ V Fb/KF) where do is the m a x i m u m foliage consumption rate per larva. This 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 in intraspecific competit ion. The new equation is n . .. n Fb ( doKpLh\ new foliage consumption/larva = — — — 1 — . KpLb V Fb 286 Appendix D. Mathematical details for the budworm-forest model 287 This 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 l i tt le new foliage available per larva. This gives Ff, ( d O K F L h new foliage consumption/larva = — — — I 1 — e F>> KFLI, V The total amount of new foliage consumed is then Fb / „ d O K F L b new foliage consumption and this gives KFLh 1 - e Fb remaining new foliage = —r ;T~~(1 ~ e A ) KF KF -A Fb where _ d^KpLb D.2 Summary of model equations The subscript b denotes the in i t ia 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-l)e-B}Fb, KF Appendix D. Mathematical details for the budworm-forest model A B — doLt KF Cl(A-l + e-A), cj = 0.357. Branch surface area (S) where Budworm (L) where Se rsSi Si = [1 - ds(l - ^)2}Sb. •Tb Le — dsh-rrG2 L& Ks G = H = Le = u = H(2-H), Fi -n. A P m I 1 - 1 , P m . )LS> E = 1 + Er< AsrL\ Athr (EiW1/s - E2)AsrL4, Ei = 165.64 and E2 = 328.52, W — Api (1 + A F 2 ( / ^ F - 1) (1 A F I = 34.1, AF2 = 24.9 and BF = - 3 . 4 , Appendix D. Mathematical details for the budworm-forest mo> L4 = (Ap + Bp-^-)L3, Ap = 0.473 and Bp = 0.828, L3 = e~DL2, Sb(PsatFb2 + LIY kL = 0.425, Li = (1 - qmaxe~C)Lb, 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. The entire calling program does not need to be rewritten each t ime. 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 # Mode 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) # Init ial 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 model—XPPAUT # Sheep, hyrax, lynx and pasture model with pasture l imit ing 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*sjmn- max(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)/del l dpaa/dt=(pa-paa) / del2 dgma/dt=(gm(pai)-gma)/del3 dtr/dt=(-tr - f (mutwool-culling-cssu*ssu+ssu*ssuval)/trmax) / tau # # State variables and ini t ia 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 ixed variables—these are quantities which are used repeatedly # in the model equations. Mak ing 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 ixed variables needed to define revenue (mutton and wool sales and cost # of cull ing). 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* # # Aux i l ia ry 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 in the data window # we need to have the following statements: fhut=hut f lut=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,l jdn=0.5,lfdn=0.13,lmdn=0.13,ljmn=1.0,lcn=0.0 292 param sfn=0 .75,sjdn=0.1,sfdn=0.02,smdn=0.02,sjmn=0.5 param sjpn=90.0,sjcn=0.09,sfcn=0.28,smcn=0.28 param sjmv=55.0,sfmv=79.0,smmv=75.0,sjwv=5.0,sfwv=7.0,smwv=9.0 param ccl=30.0,cch=1.0,cssu=2.0,ssuval=200.0,pmax=1.0e8 param hjmax=500000,hfmax=500000,hmmax=500000,ljmax=500,lfmax=500 param lmmax=500,sjmax=100000,sfmax=100000,smmax=100000 param dell=0.9,del2=0.6,del3=0.9,tau=0.05,trmax=1.0e7 # # User functions—these are the multipl ier functions. a r g E ( A , B ) = ( A / B ) - l a rgC(A ,B )= ln (a rgE (A ,B )/ (A - l ) ) argr(A,B,slope)=slope*A/((A- l )*argC(A,B)) argf (A,B,s lope,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) l jdm(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 model—XPPAUT # Mode l equations. d M l / d t = g a m l * ( l - e x p ( - o m e g a l / M l ) - M l A b l ) * M l - ( 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 # Init ial values. M l (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 Populat ion genetics model with exponential fitness functions, c P R O G R A M A U T O c I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) c c N O T E : parameters l iw and lw are V E R Y I M P O R T A N T . They set aside c space for A U T O in the work arrays IW and W . In Interactive A U T O c they must be "hardwired" into the code. If you begin to have problems c with large continuations (e.g. periodic solutions using a big N T S T ) , try c setting l iw 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 IW(l iw) ,W(lw) . dimension ipar(50),rpar(50),icp(20) character*10 params(20) character*50 name c cal l dfinit c c N D I M (number of state variables) ipar( 1)=2 c IPS (+1 for ode's, -1 for maps) ipar( 2 )= - l c IRS ipar( 3)=0 c I L P ipar( 4)=0 c N T S T ipar( 5)=15 c N C O L ipar( 6)=4 c I A D 295 ipar( 7 =3 ISP ipar( 8̂ =1 I S W ipar( 9̂ =1 I P L T ipar(10 )=0 N B C i p a r ( l T =0 N I N T ipar(12} =0 I A D S ipar(13 =1 N M X ipar(14^ =100 N U Z R ipar(15^ =0 N P R ipar(16} =50 M X B F ipar(17) =5 IID ipar(18) =2 I T M X ipar(19) =8 I T N W ipar(20) =5 N W T N ipar(21) =3 J A C ipar(22) =0 ICP( i ) icp( l ) = 1 icp(2) = 2 D O 1 1= =1,2 ipar(30+i)=icp( 1 C O N T I N U E DS rpar( l )= :0.0001d0 D S M I N rpar(2)= :0.000002d0 D S M A X rpar(3)= :0.01d0 RLO rpar(4)= =0.0 R L 1 rpar(5) = 4 .0 AO rpar(6) = -10.0 A l rpar(7) = 250.0 E P S S rpar(8) = = l .d -6 E P S L ( i ) , i =1,20 rpar(9) = = l .d -6 E P S U 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 S U B R O U T I N E VECFLD(ndim,u , icp ,par , i jac , f , t ) c c This subroutine evaluates the right hand side of the first order system c c input parameters : c nd im - dimension of u and f. c u - vector containing u. c par - array of parameters in the differential equations. c icp - par( icp(l ) ) is the in i t ia 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 cal l . c t - current t ime. c c value to be returned : c f - f(u,par) the right hand side of the ode. c impl ic 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 ) eN = u(2) c Fitness functions w l l = e x p ( a l l - b l l * e N ) w l 2 = exp(al2-bl2*eN) w22 = exp(a22-b22*eN) 298 c Mean fitnesses w l m a r g = P * w l l + ( 1 - P ) * w l 2 w2marg = P*wl2+(1-P)*w22 fitmean = P*wlmarg - f ( l -P )*w2marg c c D I F F E R E N C E E Q U A T I O N S 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 , i jac ,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 , i jac ,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(nd im,u ,par ) c c in 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 init ial ized here or else in init . c 299 c n d i m - dimension of the system of equations, c u - vector of dimension ndim. c upon return u should contain a steady state solution c corresponding to the values assigned to par. c par - array of parameters in the differential equations. c impl ic i t double precision (a-h,o-z) c dimension u(ndim),par(30) c c init ial ize 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 init ial ize 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 in the bifurcation window. This subroutine is called c when the ' S P ' is toggled O N (issue the command successively c to turn the toggle from O N to O F F , and vice versa). c c input values: c nd im - dimension of u. c u - vector containing coordinates of current solution. c isw - the number of parameters being used in the current c continuation. 300 icp - par( icp(l ) ) is the in i t ia 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 in unit 7 (file fort.7). return values: pt - array whose 1st, 2nd and 3rd elements are plotted the x, y and z axes, respectively. impl ic 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 pt ( 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 in the projection defined in the subroutine sproj. input value: nd im - dimension of u. 301 c IABS( isw) - the number of parameters being used in the current c continuation. c icp - par( icp(l ) ) is the in i t ia 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 with the x, y, and c z axes names, respectively. c integer*4 nd im, isw, icp(20) character* 10 axes(3) c if (isw.eq.l) then axes(l) = ' icp( l ) ' axes(2) = ' N ' axes(3) = ' ' else if (isw.eq.2) then axes(l) = ' icp( 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, l intog, labtog, ldsplt, @ leigen, lfltog, lsavpt, @ lgraph, lvideo, lsproj, nproj , @ ndmplt , delay, sclbif, scldis, @ sclev, filext ) c logical ldebug, l intog, 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 l intog: 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 Graph window c leigen: T => open eigenvalue plott ing window c lfltog: T => il luminates points temporarily as they are plotted c lgraph: T => use graphics (F => can then run jobs in the background) c lvideo: T => reposition windows in botton 1/4 of screen for videotaping c lsproj: T => plot special projection denned in subroutine sproj c in the bifurcation window c lsavpt: T => eigenvalues saved in fort. 11 ('svaut *' moves this to m.*) c D O N O T A L T E R the following lines. ldebug = .false. l intog = .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 = par( icp( l ) ) , u ( l ) , u(2); c when isw = 2 => default plot axes x,y,z = par( icp( l ) ) , par(icp(2)), u ( l ) c The 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 ndmplt = 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—DSTOOL. # include <model_headers.h> /* Note: The symbols /* and */ denote the beginning and end of comments respectively. */ / * - ; Required function used to define the vector field or map. The values of the vector field mapping at point x with parameter values p are returned in 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 wl l ,wl2 ,w22,margwl ,margw2,meanf i t ; /* Parameters whose values can be changed interactively. */ a l l = p[0]; a l 2 = p[ l j ; a22 = p[2]; b l l = p[3]; b l 2 = p[4]; b22 = p[5]; /* S T A T E V A R I A B L E S */ P = x [0] ; /* Frequency of allele A l */ N = x[l] ; /* Population density */ /* F I T N E S S F U N C T I O N S * / w l l = e x p ( a l l - b l l * N ) ; w l 2 = exp(a l2 -b l2*N) ; w22 = exp(a22-b22*N); /* 305 M A R G I N A L F I T N E S S F U N C T I O N S * / margwl = P * w l l + ( 1 - P ) * w l 2 ; margw2 = P*wl2+(1-P)*w22; meanfit = P*margwl-|-(1-P)*margw2; /* D I F F E R E N C E E Q U A T I O N S * / f[0] = P*margwl/meanfit ; f[l] = meanfit*N; } /* E n d of model equations. */ / * - ; ; — Opt ional function used to define the Jacobian m at point x with parameters p. The matr ix m is pre-allocated (by the routine dmatr ix) ; A t exit, m[i][j] is to be the partial derivative of the i 'th component of f wi th respect to the j ' t h component of x. = */ /* int user_jac(m,x,p) double **m, *x, *p; { } */ /* ; ; Optional function used to define the inverse or approximate inverse y at the point x with parameter values p. The array y is pre-allocated. : */ r int user_inv(y,x,p) double *y,*x,*p; { } */ /* Optional function used to define aux functions f of the variables x and parameters p. The array f is pre-allocated. T ime 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 : You 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 in this segment */ int n_varb=2; /* d im of phase space */ static char *variable_names[]={"P","N".}; /* list of phase varb names */ static double variables[]={0.,0.}; /* default varb in i t ia 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 max for display */ int n_param=6; /* d im of parameter space */ static char *parameter_names[ ]= { "a l l " , "a l2" , "a22" , "b l l " , "b l2" , "b22" } ; /* list of param names */ - static double parameters[] = {2.1,1.9,1.1,1.0,0.904,0.56}; /* in i t ia l parameter values */ static double parameter_min[]={0.,0.,0.,0.,0.,0.}; /* default param m i n for display */ static double parameter_max[]={2.,2.,2.,3.,3.,3.}; /* default param max 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 do- main */ static double period_end[]={l., l .} ; /* if P E R I O D I C , end of fundamental do- main */ 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 (*def_name)()=genetics_def; /* the eqns of motion */ int (*jac_name)()=NULL; /* the jacobian (deriv w.r.t. space) */ int (*aux_funcjtiame)()=NULL; /* the auxil iary functions */ int (*inv_name)()=NULL; /* the inverse or approx inverse */ int (*dfdt_name)()=NULL; /* the deriv w.r.t t ime */ 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: The symbols /* and */ denote the beginning and end of comments respectively. */ / * - ; Required function used to define the vector field or map. The values of the vector field mapping at point x with parameter values p are returned in 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 exfrac ,exthr ,A,B, fo lnew,fo lo ld , fo l tot ,C ,Sl ,BLl ; 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]; /* S T A T E V A R I A B L E S */ F O = x[0]; /* Foliage */ S = x[l] ; /* Branch surface area */ B L = x[2]; /* Budworm density */ /* F L A G S 309 */ npara = 1; nfood = 1; npred = 1 ; nsurv = 1 ; nhist = 1; mdisp = 1; ndisp = 1; /* C O N S T A N T S * / fgrow = 1.5; fmax = 3.8; smax = 2.4E4; sgrow = 1.15; sharp = 4; dsearch = 1.0; /* P R E L I M I N A R Y E Q U A T I O N S 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; /* Budworm dynamics * / /* The if -then statements in this section are for the switches. */ C = 0.003*BL; if (npara = = 1) 310 B L 1 = ( l -0.4*exp(-C))*BL; else B L 1 = B L ; 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) { tempi = 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) { tempi = exp(sharp*log(El)); B L 6 = ( l - (exf rac*templ )/ ( l+templ ) )*BL5; } 311 else B L 6 = B L 5 ; } else B L 6 = B L 5 ; 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; /* D I F F E R E N C E E Q U A T I O N S * / 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 with parameters p. The matr ix m is pre-allocated (by the routine dmatr ix) ; A t exit, m[i][j] is to be the partial derivative of the i 'th component of f wi th respect to the j ' t h component of x. */ /* int user_jac(m,x,p) double **m, *x, *p; { } */ / * - ; ; ; — Opt ional 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; { } */ / * - ; Optional function used to define aux functions f of the variables x and parameters p. The array f is pre-allocated. T ime 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 : You 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 budwormJn i tQ { /* define the dynamical system in this segment */ int n_varb=3; /* d im of phase space */ static char *variable_names[]={"F","S","B"}; /* list of phase varb names */ static double variables[]={0.,1.68E4,0.}; /* default varb in i t ia 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 max for dis- play */ static char *indep_varb_name=="time"; /* name of indep variable */ double indep_varb_min=0.; /* default indep varb min for display */ 313 double indep_varb_max=1000.; /* default indep varb max for display */ int n_param=7; /* d im 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}; /* in i t ia l pa- rameter 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 max 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 min for display */ static double funct_max[]={3.0}; /* 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 , 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 (*def_name)()=budworm_def; /* the eqns of motion */ int (*jac_name)()=NULL; /* the jacobian (deriv w.r.t. space) */ int (*aux_func_name)()=budworm^aux; /* the auxil iary functions */ int (*inv_name)()=NULL; /* the inverse or approx inverse */ int (*dfdt_name)()=NULL; /* the deriv w.r.t t ime */ 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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