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Man as predator : qualitative behaviour of a continuous deterministic model of a fishery system Huberman, Gur 1976

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MAN AS PREDATOR: QUALITATIVE BEHAVIOUR OF A CONTINUOUS DETERMINISTIC MODEL OF A FISHERY SYSTEM by Gur Huberman B.Sc, Tel Aviv University, 1975 .  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 (c*)  Gur Huberman, 1976.  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  the U n i v e r s i t y of B r i t i s h  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  of  the requirements  Columbia,  I agree  r e f e r e n c e and  t h a t p e r m i s s i o n for e x t e n s i v e copying o f  this  of  representatives.  this  thesis for  It  financial  is understood that copying or gain s h a l l not  written permission.  Depa rtment The  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  f\^V  m>  that  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s  for  or  publication  be allowed without my  i(a)  Man  as P r e d a t o r :  Q u a l i t a t i v e Behaviour of a  Continuous D e t e r m i n i s t i c  Model of a F i s h e r y  System  ABSTRACT  A g l o b a l p o r t r a i t of the phase p l a n e i s o b t a i n e d f o r any v a l u e s o f the parameters. recovered.  The  first  3 d i f f e r e n t s t r u c t u r e s of the phase p l a n e  p r e d i c t s an  e v e n t u a l c o l l a p s e of the  second p r e d i c t s an u n s t a b l e l i m i t c y c l e and s o l u t i o n s which s t a r t  catch.  catch  last  r a t e , and  The  b o u n d a r i e s of these domains are  d i f f e r e n t i a l equation f o r a saddle-to-saddle  phase p l a n e .  structure  predicts  the o t h e r  found by  one  the  solving  separatrix in  T h i s procedure u t i l i z e s r e g u l a r p e r t u r b a t i o n  The  of  Each s t r u c t u r e corresponds to a d i f f e r e n t domain i n  parameter space. relevant  with high  The  are  fishery.  an e v e n t u a l s t a b i l i t y  i n s i d e the l i m i t c y c l e .  2 p o s s i b l e s t a b l e e q u i l i b r i a , one w i t h no  acceptable  methods.  the  the  i(b)  TABLE OF CONTENTS  PAGE  INTRODUCTION I. II. III. IV. V. VI.  "'  2  THE MODEL  6  PRELIMINARIES  ' .  9  THE TRAJECTORIES AFFILIATED WITH THE SADDLE POINTS . . .  17  A SOLUTION FOR  21  b  .'  THE GLOBAL PORTRAIT OF THE PHASE PLANE  31  INTERPRETATION  38  APPENDIX A:  POSSIBLE VALUES OF  APPENDIX B:  THE NUMERICAL  SCHEME  R  AND  Q . . . . . .  .  39 41  ii  FIGURES  PAGE F i g u r e 1.  The e q u i l i b r i a i n the phase p l a n e  F i g u r e 2.  A p r e l i m i n a r y d e s c r i p t i o n o f the motions i n the phase p l a n e  15  F i g u r e 3.  T  2  18  F i g u r e 4.  T  £  F i g u r e 5.  The phase p l a n e p o r t r a i t when  F i g u r e 6.  The phase p l a n e p o r t r a i t  F i g u r e 7.  D i v i s i o n o f the phase plane by  F i g u r e 8.  The phase p l a n e p o r t r a i t  F i g u r e 9.  The number o f r o o t s o f  and > T  T  3  3  19  3  values f o r R  and  b <  32  f o r x^ < b < b  34  •  for b > b g(x)  Q . . .  35 .  36  as a f u n c t i o n o f v a r i o u s 40  iii ACKNOWLEDGEMENTS I wish to express my appreciation to my supervisor, Dr. D. Ludwig, for his original suggestion of my thesis topic as well as for his encouragement and guidance during the preparation of this thesis.  I would also  like to thank Dr. F. Wan for many helpful discussions considering the final draft of this work.  LEAF 1 OMITTED IN PAGE NUMBERING.  2  INTRODUCTION  Commercial e x p l o i t a t i o n o f animate r e s o u r c e s oldest occupations,  a l r e a d y mentioned i n Genesis.  In r e c e n t y e a r s an  a c c e l e r a t i n g d e c l i n e i n the p r o d u c t i v i t y o f important Clark  [1] mentions the great whale f i s h e r i e s ,  Peruvian  anchovy  i s one o f man's  f i s h e r i e s was o b s e r v e d .  Grand Banks f i s h e r i e s  and the  fishery.  A model o f the dynamics o f animal p o p u l a t i o n and human e f f o r t t o harvest  i t i s analysed  i n t h i s work.  For a non-harvested p o p u l a t i o n predation.  i t assumes l o g i s t i c  and d e t e r m i n i s t i c .  growth p e r t u r b e d by  T h i s i m p l i e s t h a t t h e r e a r e two p o s s i b l e e q u i l i b r i a f o r the  population: population  The model i s continuous  a v e r y low one and a h i g h one.  H a r v e s t i n g may d r i v e the  to t h e low e q u i l i b r i u m , where no h a r v e s t i n g i s w o r t h w h i l e . For a h a r v e s t e d  p o p u l a t i o n we s u b t r a c t the h a r v e s t  growth, and o b t a i n the e q u a t i o n  from the n a t u r a l  f o r the dynamics o f the p o p u l a t i o n .  For  the human e f f o r t we assume t h a t the r a t e o f change i n the e f f o r t i s p r o p o r t i o n a l to the n e t income. population hunting  i s a common p r o p e r t y .  This r e f l e c t s  the f a c t  t h a t the hunted  Everybody has f r e e access  ( o r f i s h i n g ) and t h e r e f o r e the t o t a l human e f f o r t  t o commercial  increase i s  p r o p o r t i o n a l to the t o t a l n e t income ( n e g a t i v e net income and n e g a t i v e i n c r e a s e a r e not e x c l u d e d ) .  The n e t income i s the d i f f e r e n c e between the  t o t a l revenue ( i . e . the h a r v e s t ) The model i s p r e s e n t e d  and t o t a l c o s t . i n Section I.  S e c t i o n I I i s a p r e l i m i n a r y d i s c u s s i o n of the equations in Section I.  They a r e s c a l e d and brought t o the form  obtained  3  dx ,. — = x g(x) - xy ,  dt  ay(x - b) ,  g(x) = R ( l - ^)  ^ 1 + x  The quantities x and y are proportional to the population density and the human e f f o r t respectively, and hence x,y >_ 0. i = 1, 2-, 3 . E ^ = (x^,0) E„ 2  and  g(x) has three p o s i t i v e zeros x  The possible e q u i l i b r i a of the system are  i = 1, 2, 3, ;  E ^ = (b,g(b)).  E » are saddle points and 3  E, 4  E Q = (0,0);  E ^ i s asymptotically stable,  is  ^  L I  unstable  if (b) • dx  _ • > 0  We  de denote by  X  q  the point where  (x) = 0,  and then give a preliminary  description of the motions i n the phase plane.  They are summarized i n  the following f i g u r e :  Figure 1.  The e q u i l i b r i a i n the phase plane.  4  Section II concludes with an i n d i c a t i o n of the main problem of the work:  the completion of the phase plane p o r t r a i t .  this i s the information about E  2  T^  and the one which leaves  T  2  E^,  and  T^ - the separatrix which goes to  respectively.  are defined and the problem a r i s e s :  to-saddle separatrix This  b  saddle-to-saddle  The main clue for  In Section III these  for which value of b i s a saddle-  obtained, i . e . when do the  i s found i n Section IV.  T^  intersect?  Approximate solutions for the  separatrix are obtained, and they imply a unique value  of b (denoted b) .  Of course, b depends on R, Q, a.  This  b  i s found  with two a l t e r n a t i v e assumptions: 3/2 (i)  a«Q  ( i i ) a*Q^^  i s large, or i s small.  1  The i n t e r e s t i n g thing i s that always  b > x^ .  This enables us to draw  only three d i s t i n c t phase plane p o r t r a i t s and eliminate other p o s s i b i l i t i e s . These p o r t r a i t s - as given i n Section V - are: (i)  For  b < XQ  (ii)  For  XQ < b < b  a l l the t r a j e c t o r i e s converge to  .  an asymptotically unstable l i m i t cycle appears.  Its existence i s proved by the Poincare-Bendixon theorem. which start inside the l i m i t cycle converge to outside the l i m i t cycle converge to  Those which s t a r t  E^ .  ( i i i ) For  b > b,  while  i s the a t t r a c t o r for o r b i t s which start above  .E^  E^  E^ .  A l l the solutions  i s the a t t r a c t o r for o r b i t s which start under T  2  T  2  .  From these 3 s t r u c t u r a l l y d i f f e r e n t p o r t r a i t s we see that given an i n i t i a l point (x(0), y(0)), the solution may What happens w i l l depend upon the parameters.  or may  not converge to E^ .  In certain cases, a modification c  5  the parameters can a v o i d a c o l l a p s e o f the p o p u l a t i o n The a n a l y t i c a p p r o x i m a t i o n s were accompanied n u m e r i c a l scheme.  (and the h a r v e s t ) .  and v e r i f i e d by a  I t i s d e s c r i b e d i n the appendix.  6  I."  THE MODEL  This model is a deterministic continuous model which describes an animal population subject to human harvest. A system of two coupled ordinary differential equations is introduced. One equation describes population changes while the other describes changes in human effort to catch the animals.  The  equations are simple and therefore one should not expect them to f i t reality in every detail.  They do not refer to any particular animal  population, but mainly fish populations were in my mind during the work on this paper. The basic equation for the dynamics of the population i s  4^ = NG(u) dx Where u  i s the population density,  -H .  (1.1)  T is time,  NG  is natural growth  rate (i.e. growth rate of the population with the absence of human harvest) and  H  is the harvest.  This equation has been employed in Clark and  Munro [2] and Smith [9]. The equation for a population which is not subject to human catch i s j g  2 = NG(u) = r u ( l - ^ ) - B u a +u U  u  2  2  ;  ..  (1.2)  The term  r u(l - — ) i s the right hand side of the logistic u , equation, often used by biologists (cf. McNaughton-Wolf [3]). It describes U  .  a growth which is exponential i n i t i a l l y and then decays due to the finiteness  7  of the environment resources. density.  K  i s the maximal possible population  u  It is determined by factors such as limited food supply or  space.  2 u The term  -8  due to predation.  describes.an effect on the growth rate,  2  a  +u  This particular choice of the predation term represents  a type III S-shaped functional response (cf. Holling [4]). According to Rolling, the effect of predation saturates at fairly low population densities, i.e. there is an upper limit to the rate of > a mortality due to predation. This implies-that — is small. K  feature of predation is a  Another  U  decrease in the effectiveness of predation at  very low densities. This is attributed to searching and learning on 2 the predator's part. Finally, we have to remark that -B "—J a +u i  s  n  o  t  2  the only way to represent a type III S-shaped response.  This particular  form is chosen because of mathematical convenience. In order to incorporate the effect of human harvest we follow Clark-Munro [2] who give  H(E,u) - the human harvest - the form: l 2 H(E,u) = y u E Y  where  E  Y  ,  i s the human effort and Y. > 0  (1.3) For reasons of convenience  1 —  we set Y = Y = 1 • x  (1.4)  2  Combining (1.1), (1.2), (1.3), and (1.4) we obtain  dT  =  r  u  u(1  - r> -  e  u  V^-T a +u  -  Y  U  E  •  tt-*)  8  For the economic  p a r t we assume t h a t  c • Y ,  where  Y  i s the n e t income y i e l d .  (1.6)  Following Schaeffer  [5] we assume  t h a t the t o t a l c o s t i s p r o p o r t i o n a l t o t h e e f f o r t .  C = p • E  .  (1.7)  T h i s i s a common assumption among economists ( c f . C l a r k Munro [ 2 ] ) . In g e n e r a l ,  p = p(E),  but h e r e  p  i s assumed t o be independent o f  E.  The t o t a l p r o d u c t i o n has a l r e a d y been g i v e n by (1.3), (1.4) and when we combine t h i s w i t h  g =  Combining  (1.7) and s u b s t i t u t e i n (1.6) we o b t a i n  c YE(u - * > ' • •  (1.5) and (1.8) we get the system  u  —  = c  Y  E(u -  a  + u  •  The main aim o f t h i s work i s to study  system  (1.9).  (1-8)  9  II.  This section  PRELIMINARIES  i s devoted  to a s i m p l i f i c a t i o n of the problem and  to the d e r i v a t i o n of some s t r a i g h t forward r e s u l t s . done by s c a l i n g the v a r i a b l e s  and b r i n g i n g  Simplification i s  (1.9) t o the form  x' = x g(x) - xy , y' = a y ( x - b) ,  where  X g(x) = R ( l - —) Q  found and then used stability. motions complete II.1  '  1 + x  .  Zeros and a maximum of  i n a discussion  The s e c t i o n  i n the f i r s t  X  o f the phase p l a n e .  d e s c r i p t i o n o f the  The f o l l o w i n g  this description.  Scaling  We i n t r o d u c e the f o l l o w i n g parameters ar  K  c a = - • Y '  b -  -B- , ya  are  on the e q u i l i b r i a and t h e i r asymptotic  concludes w i t h a p r e l i m i n a r y  quadrant  g(x)  2  a ,  and q u a n t i t i e s :  sections  10  u x = — , a v - ^  t  =  F  — T  a  In terms of these we have x' = x g(x) - xy » y' = ay(x - b) where  ' = 4r  J g(x) = R(l - £)  d t  Since  K u  —j 1+ x  Q  (II.1.1)  , •  2  is the density where predation saturation occurs and K is the total capacity, Q = — is large. This w i l l be used later, a  II.2  a  g(x) : i t s zeros and maximum g(x)  may have either one or three zeros.  only in the latter case.  We are interested  This implies certain restrictions on  Q . They are discussed in the Appendix. We employ algebra to obtain:  g  (x) =  R(I - h *  ~— = 1 + x  P ( x )  Q(l + x )  where P(x) = x To evaluate x^(i=l,2,3)  3  - Qx + (1 + |)x - Q .  we set  2  .  t  R and  11  = 0  P(x)  Hence x ^  .  .  - | x ^  + 1 - ± ( x ^ +x j ^ )  1  Therefore  f o r 0 < eg £ R < -j  i + A -4 Xl=^-P  w  *  have :  e  -rOCQ" ) ,  x  Since  x  =  2  3 x  /  R  1  + 0(Q )  R  _ 1  = - Q + — ^ + R • x_  = 0 , we o b t a i n  I  x-  3  3  x^  = Q - \ + 0(Q ) . _ 1  3  An approximate v a l u e of  X Q i s found as f o l l o w s  2 SLS. = _  _  Thus A  In  x  Q  ~ x  , and  1  (i  +  2  )  2  <*o> - ° •  t  a  x  Q  d x  2 R Hence x_ = 0  +  R  A _ 8R  + Q R /  2  -  Q  2  « J\  - f /l+OCQ- ) 1  general: (x - x ) ( x - x ) = e ; 1  2  x  ^ x  1  2  e 1  x  "2 X  1  x, + 1  £  — - x,  x - x„  1  2  X  l " X2  .  2,  12  II.3  Equilibria and their asymptotic stability From the equations: x' = x g(x) - xy , y' = ay(x - b) ,  we see that the equilibria in the f i r s t quadrant are: E  Q  = (0,0)  E, = ( x r 0 )  , ,  E„ = (x 2 ,0) , E„ = (x 3 ,0) , E, = (b,g(b)) E, occurs only i f 0 < b < x, or x 4 — — 1 1 y  0  < b < x_. The restriction b > x„ means — 3 3  —  that even i f the biomass were at i t s maximal possible value, i t would not be worthwhile to make an effort to harvest i t . The restriction b < x means that 2  i t is worthwhile to make a harvesting effort in densities below x - the 2  collapse threshold. Neither case is r e a l i s t i c . on the case  x  2  We shall concentrate only  < b < x^ •  In order to compute the asymptotic stability of the equilibria, we f i r s t compute the variational matrix of the system. x g (x) + g(x) - y x  -x  M(x,y;b) = ay  Let  >L  a(x - b)  be' M(x,y;b) evaluated at E.^ . Then we obtain  13  M,  R  0  0  -ab  -x. a(x  X  M  2 ' x 2 g  2 =  ( x  x  3  a(x  • g (x ) x  M = 3  0  4 -  a(x  b g (b)  -b  a g(b)  0  From the assumptions on  - b)  2  -x.  3  x  M  -x.  )  0  - b)  1  3  - b)  g(x) and on b  (x < b < x ) 2  3  we  can draw the following conclusions: (1)  E Q is a saddle point which attracts in the y repels in the x  direction.  (2)  E^  i s a stable equilibrium (because  (3)  E  is a saddle point (because  2  repels in the x (4)  E  3  (5)  E^  § ( j^ x  S ^ 2^  >  ^» £  S ( 3^  <  0;  x  0;  <  x  x  x  <  x^ < b) that  direction.  i s a saddle point (because  attracts in the x  direction and  x  x  x  3  > b) which  direction.  i s asymptotically stable (unstable) according to:  14 If  g(b) < 0, then  If  ^  g(b) > 0, then  is asymptotically stable. E^ is asymptotically unstable.  Graphically, this means that i f b > x^ then stable; i f b < x^ then II.4  E^ Is asymptotically  E^ is asymptotically unstable.  Phase plane description We consider again the system x' = x g(x) - xy , y' = ay(x - b) .  Along the axes: Along the y axis  x' = 0 ; y' < 0 .  Along the x axis  y' = 0 ; sgn(x') = sgn g(x) '.  In the interior of the f i r s t quadrant we have 5 distinct regi dy ay(x — b) Considering -f- = — ,—r we can immediately t e l l the dx x g(x) - xy direction of the motion in each case.. to discuss.  and y < g(x)  then 4^~ dx  1.  When  x < x, 1  2.  When  x < x < b and y < g(x)  3.  When b < -x < x. and y < g(x) 3  then  4.  When  x <b  then  5.  When  x >b and y > g(x) then <0. At the boundaries of these regions we have  dy 2  and y > g(x)  0 •  <  then 4^ dx dy  < 0. >  0  •  > 0 i.  15  (1) y (2)  =  g(x) implies x'=0;  x = b implies y'=0;  y  1  > 0  x' > 0  if  x > b  and  if  y < g(x)  y' < 0 and  if x < b .  x' < 0  i f y > g(x)  The motions i n the phase plane can be i l l u s t r a t e d by the f o l l o w i n g f i g u r e :  Figure 2.  A preliminary d e s c r i p t i o n of the motions i n the phase plane.  16  At this stage we are ready to indicate the main problem of the work - the global picture of the phase plane.  Given i n i t i a l  x  and y,  we would like to know where the solution of ( I I . 1.1) goes as t -»• . 00  Clearly, when E^  is asymptotically stable there are two possible answers  and E^ . But even when E^ is asymptotically unstable there is no  reason to believe that a l l the solutions converge to E^ . The answer depends not only upon the i n i t i a l data but also on the particular value of the parameters.  In the next sections the asymptotic behaviour  of the solutions w i l l be discussed in the various domains of the parameter space.  17  III.  THE TRAJECTORIES AFFILIATED WITH THE SADDLE POINTS  A possible way to deal with the problem raised at the end of the last section is a division of the f i r s t quadrant into domains bounded by solution trajectories.  A solution which starts in such a domain i s  destined to remain there because two solutions cannot intersect. For the  same reason there are only two possible boundaries of this type:  periodic solutions and orbits which connect c r i t i c a l points.  E^, E  E^ are already known to be connected by one orbit, namely the x There exists another unique orbit which goes to E which goes from  2  and  axis.  and another one  2  E^ . Motivated by this^one i s led to investigate  properties of these trajectories. Since the right hand side of the system (II.1.1) i s twice continuously differentiable we may use the following theorem, which i s a slightly modified version of a theorem given by Coddington-Levinson [6]. Theorem. Consider the system r  d dt  X  \  ' f (x,y)' n  f (x,y)  >y .  2  for which the following conditions hold:  (i)  ( 0'^0^  (ii)  2 f ,f e C  x  2  *  S a  sadd  -'-  e  P°i ' nt:  in the neighbourhood of  (x ,y ) .  Then there exist exactly two orbits tending to  0  Q  (xQ,y^) as t -*• °°  The angle between these two orbits i s 180°, and any orbit starting sufficiently near either of these orbits in the neighbourhood of ( o> o^ x  v  18  tends away from them as t -*• . 00  A corollary of the theorem is that i f ( i ) , ( i i ) hold, there exist exactly two orbits tending to  (XQ,VQ)  as t  . The angle  between the orbits i s 180°, and any orbit starting sufficiently close to either of these orbits and to At  E = (x ,0) 2  on the x-axis.  2  We define  (x ,y ) tend away from them as t •+ -<*> . 0 0  we already know that the latter orbits l i e T  as the orbit of the former type which lies  2  above the x-axis.  X  Figure 3. T  T along that  T T  2  2  2  must l i e above  near  g(x) near  E , and x > x 2  does not tend to E At  E  Q  we define  and T  2  2  2  on T  3  E 2  . Therefore  2  . Indeed, otherwise  for T T  2  2  under  x' > 0  g(x) imply  lies in region 4 near  T^ - the trajectory leaving  E^ . In a  E . 2  19  similar way to  T  2  we can show that  lies in region 5 near T the line (if  T  2  2  line  E^  or i t converges to T  2  crosses "higher" than  and T  3  as  t ->-<»,  or both happen  in a converging fashion).  lies in region 5 near  If both  and  . Therefore, either i t crosses  or i t converges to E^  spirals around  x = b  g(x)  ,.  lies in region 4 near  x = b  T^  lies above  T  3  and  Figure 4.  E^ . Hence, either i t crosses the  E^  as  cross T  > T  3  T  2  t -> °° or both.  x = b  T  2  > T  3  in the opposite case  2  > T  we say  3  if T  2  20  If one T. does not cross the line x = b, then T. > T 0 J h At least one of them must cross x = b . Otherwise both converge to E^, J  X  but a simple non saddle equilibrium cannot attract and repel at the same time. There i s another possibility yet: both x = b  at the same point.  Then - by the uniqueness of the i n i t i a l value  problem - they are identical. separatrix.  1^ ~ ^3*  This trajectory connects  anc  * this is a saddle-to-saddle and E^, so i t can serve us  in the way described at the beginning of this section. to find the value of b  and T^ cross  Hence the motivation  for which such a separatrix is obtained. From \  here on we shall refer to this as the solution for b, and denote i t by  b.  21  IV. A SADDLE TO SADDLE SEPARATRIX  IV.1  Introduction In this section we shall concentrate on the equation dy_ dx  We shall find the  b  =  ay(x - M x g(x;Q) - xy  (IV.1.1) .  for which exists a solution y(x ) =0  y  i = 2, 3  which satisfies (IV.1.1)  ( j  i.e., there is a saddle-to-saddle separatrix. Since (IV.1.1) is too complicated to integrate exactly, we shall use perturbation method. The crucial parameter was discovered in two steps: i s . Assuming (i) Clearly, the larger a i s , the larger |dx| "large  a"  we obtained an approximate solution and then compared i t with  numerical results.  It appeared that the approximation was good also for  fairly small values of (ii)  Observing that  a  such as x„ =  -2 10- . is where structural changes occur*  we scaled: In fact, i t w i l l be shown later that a Hopf bifurcation takes place near b = x^ .  22  x (IV.1.3) B = —  Now the new form of (IV.1.1) i s : dY du Here  Y(u) = y  ai/Q  =  (u - B) • Y u(G - Y)  (IV.1.4) u/Q  G(u;Q) = R(l - -± ) /Q  1 + Qu  We tried a parameter of the form expansions for the saddle-to-saddle For small Y  (s)  P = aQ  m  and the following  separatrix and B:  P ,  ( u ; Q ) = Y £ ( U ; Q ) + PYJ (u;Q) + P Y^ (u;Q) + . g(s)  =  B  For large  (s)  ( Q )  2  s)  s)  +  p B  (s)  ( Q )  +  p  2 (s) B  ( Q ) +  S)  ^  (IV.1.5)  ^  P ,  Y (u;Q) = Y< (u;Q) + P ^ ^ ( u j Q ) + p " Y ^ (u;Q) + ... , U)  0  2  )  (IV.1.6) B  ( i )  ** Following this,  = B< (Q) + A>  i u.=-^-  ?-h[ \Q) + l  P" B< (Q) + ... . 2  £)  X  j = 0, 1, 2, 3 ;  B=  b —  23  (s) B Q (Q) appears to be 0 ( 1 ) as Q -> » and m w i l l be chosen ( s) so that also is asymptotic. IV.2  = 0 ( 1 ) as Q -»• °° . This w i l l indicate that ( I V . 1 . 5 )  B^  m = 3/2  It appears that  An approximate solution for small  i s the appropriate  m.  P  We substitute ( I V . 1 . 5 ) in ( I V . 1 . 4 ) and obtain:  dY<  dY<  s)  -r^-—+ P - r ^ -  du  . dY<  S)  + P  du  ——  s)  (IV.2.1)  + ...  du  a Q^(Y< + PY< s)  + P Y^  s)  2  S)  + ...)(u - B<  - PB<> - P B<  s)  S  2  u(G(u) - Y< - PY< - P Y^ - ...) At this point we have to decide i f aQ i s "small" or "large". This i s k s)  s)  2  S)  z  necessary i n order to simplify the equation. is small.  (I) (ii)  m = 3/2 .  We assume this, and w i l l show later that Assuming that  (IV.2.1)  Certainly, i f m _> \, aQ  aQ  i s small we can balance the two sides of  2  by:  Y^ (u;Q) = 0 , or  setting  s)  setting Y^ (u;Q) = G(u;Q) . s)  (s) We know that  Y  (u;Q) lies above G(u;Q), so we reject Y< (u;Q) = G(u;Q) . s)  (i) and set (IV.2.2)  Now the equation has the form: (s) dG  du  +  p  ^  l  +  du  (s) P  2 ^ 2  +  (IV.2.3)  du  Q ( G + PY^ V m  }  + P Y^  u(Y<  2  s)  +...)(«- B<  S)  + P Y <  2  )  +P Y^ 2  S)  s)  - PB<  + ...)  s)  - ...)  S)  24  Formally, we expand the right hand side: (s) (s) 2 (s) ,<s) ^ „2 (s) (G + P Y ^ + P Y^' + ...)(u - B j - P B - PB D /  v  W/  ...)  1  u(Y  Q^CG  + PY<  Cs) +  +P Y^  s)  2  P  Y  (  S  )  (S)  1  +PM I  ,(s). _2 ,(s)  (1-P  S )  +  u-Y  + ...)  S)  (s)  t  f  ,(s)  ,(s)  ,(s)  -B  W 0  -PB^  2  S)  - ...)  - ...)  S)  ,(s)  ,(s)  (s)  P B^  Y  ' ,(s)  s)  PB  s)  (s) 2 3  (s)  ,)(  Q/*-G(u - B< ) m  2  + ...)(«- B<  S)  Y  ^u — Y, (G + PY1  + P Y^  ,(s  uY ,<s)  _ 2 P  F  (  Y  ( 8 )  (s) 2 u Y, U  (  U  l u  v  ,(s)  ,(s)  ,<s)  ,(s)  .  B  ( 8 )  *0  _  )  ;  Y  Y  (S)  l  B  (8) _  l  ,(S)  Br'G - - ^ y  (Y{ (u - B^') s )  21 G(u - B ) + Q  This expansion is valid only i f — 1  is bounded.  Now we substitute this in (IV.2.3) and formally equate the coefficients of the respective powers of P . For  P° we have  25 ^  dG du  B< ) s)  u Y (s) C>  Y^ (u;Q)  G(u -  • G(u;q) ' [u - B^ (Q)]  m  S)  (IV.2.4)  s)  u - f  (u;Q)  (s) This Y (u;Q) i s continuous except - maybe - at u = dG — (u ;Q) = 0 . To avoid a singularity there we set du u.  U Q ( Q ) ,  N  where  n  B  For dY  (s) 1  P  du~~ " (Y Y  (s) 2  " (s) „ rY  ~U~Y7  ( S )  )  =  (IV.2.5)  = V°-> '  S)  we have  ,  _  o  [ Y  r  I  (  U  R  "  (s)s O  B  }  ,(s)  (s) "  B  G  I  _2_ f„ r  "  (s) 1  R  " 0  G ( u  B  ( M S  } J  2  '  U , ( Y  1  }  d  Y  l  _(s) (a) ~ l l y  G  +  G  B  u - B (s)  Y  (IV.2  Now to avoid singularity we set  Q "S- i <yQ> m  B^ (Q) = S)  Y  s)  GTu^Q)  From (IV.2.6) we see that  i du~ d  Y  (IV.2.7)  r ( U  0  ; Q )  fq")  (s)  Y^  (u) = H(u) • Y^ (u) where H(u) Y< (u) a bounded function on [u.,u,] . Therefore —r~\ H(u) is bounded Y| (u) there. This indicates that the geometric expansion was valid. s)  =  1  S;  To get a more explicit form of (IV.2.7) we use l'Hopital's rule to obtain to obtain:  (s, Y^ (u^jQ)  *V  and  — —  (u^;Q) and substitute in (IV.2.7)  26 G(u ;Q)[2 G"(u ;Q) + u G'"(u ;Q)] °~ ^ ~  =  ( s ) B l  2 UQ (G"(u ))  1  . Q  ,  (IV.2.8)  J  0  where  ( )' -  ^  We recall that u  Q  ~p+  =  vR  o(l) ,  G(u ;Q) = R + o(l) , Q  3/2 G"(u ;Q) = - 2 R _ Q G'"(u  +  0(Q  -3/2  ^  }  2  ;Q) -  0(Q- ) . 3/2  Therefore 3/2-m B  i (Q) - B ! 8 )  = a  +  o ( 1 )  •  This gives us two things (i)  m  has to be |  in order that  B^  is required to make the expansion (IV.1.5) (ii)  B^ (Q) > 0, S)  and therefore  B  (s)  s)  (Q) = 0(1),  Q  « . This  asymptotic. (Q) > u  Q  for large  Q . Th  w i l l serve us later. The continuation of the expansion i s similar to what was done for the 0(P) terms. (s) S)  For 0(P ), n  (s)  n > 1 we have  « * ' i -^- ±) ' ' " 0 ^ " V l G + (u - B ) U  Y  d Y  y  6  [U  B  S  Q  1  (s) (s) (s)._2 3 n-1' (s) ' ( s ) 1 1 y  H H  , (s) 1 0' Y  ( Y  Y  Y  Y  Y  (s) n-1 (s) 1  )  Y  ( +  W  Y  _2 . -Qzi) , n-1' Y ' "' ' ' Y  27  2 — ,  n-1 —^—  Y  where  H^,!^  vanish at  are polynomials in Yg,  Y^  Y  which  (s) . From this we calculate the expressions for Y^ ,  u^i  B . Y w i l l automatically satisfy Y (u.) =0, 0 = 2, 3, and n n ' n l . (s) (s) B , is determined so that Y (u) has no singularities in [u„, u ] . n-l n z j ( s )  ( s )  (s)  }  IV.3  An approximate solution for large  For large  P  we expect  Let  z(u) =  y  (I)  (SL), y  U  P .  (u)  to be large, so we scale i t :  v  (IV.3.1)  J  p  Then we assume the following expansions: z ( u ; B ) = (u;B^ ) + p" Z (u;B^ ,B^ ) a)  £)  1  )  Zo  )  1  p- Z (u;B^,Bf ,B^ ) 2  +  )  )  2  +  (IV.3.2) B  U )  o  B  < «  >  +  P" B^  .  2  +  +  We denote the solution which corresponds to T^ (IV.3.1), (IV.3.2) , (IV.1.3) dZ^ du  d —± du  ( 1 )  + P  1  Z l  in  by  Z^  . Substituting  (IV.1.1) yields:  ( Z ^+ + ...= — —  rh^  +  ...)(u - B<» -  - Qu(Z^  l}  + P Z^ 1  l )  p"V*> -...)'  + ... - P G) 1  (IV.3.3) Using the geometric series expansion we obtain:  28  •  , (i) dZ (i) n -1 i du + P —du -— + 7  d  B  }  (IV.3.4)  -Qu  B U)  -l  -UK - Q  ( U  Z  (u - B  )  Q  + ...  u This expansion is valid only i f later that  —  G  i  s bounded in  —  is bounded.  U)  [ ' Q U  B  ^  2  A N D  G (3)~  It w i l l be shown bounded  i s  i n  0  [B^,u ] 3  Along with (IV.3.4) we must satify  Z  j  i  )  (  u  i  }  =  1 =  3 = 0, 1,  °  (IV.3.5)  1, 2  Formally, we equate the coefficients of equal powers of P  and solve term  by term. Hence  Z  0  ± ) ( u ; B  0  O )  =  ^  [ B  0  u~ i  1 }l n  Q  Now we can show that — ^ y we have  ( u  - i u  )  ]  2  1  (  U  }  A )  2  B„U) n  ?9 ^ 0<  Z  u = u  2  > u) 2  2  -Qu Therefore  (IV.3.6)  B< ] . At is bounded in [U ,BQ  G(u) " Z G(u) ° — = 7—r< » , (we assume lim —-prr = j u->u Zj (u) u - BQ g  i = 2, 3  U  )  j  2  z  0  (2)  is bounded for u e [u„,6] for some 6 > o . — r — > o du  (a)  for u E [u ,Bg ] and therefore  dZ  (2)  and  (( A ) -  ZQ '-(u^) = 0 implies Z^ ' (u) >_ e for u e [<5,Bg i - bounded for u e [6,B^^] . The final conclusion s  V)  (2~)  29  is that  o i^ Z  Z^  treat  0 3  *  bounded ^  S  o r  u  E  t 2>**o^] • u  ^  n  a  similar way we can  ( u )  (u)  to show that  — Z  Now  to find  BQ  0  *  s  bounded in  I Q ^ > 3^ • B  u  ( u )  we equate  (Bg  ;BQ )  Q  Z  0  '0  '  and obtain the unique solution of this algebraic equation (A) 3 " 2 o = i 1 0 In u^ - In u^ U  B  U  (IV.3.7)  The next thing to be shown is that numerically and for large u  3  - u  Q  Q ->- »  3 - = In ^- + 0(1), In -^T 2 = 2 U  i„S_  3  u  Q -> «, where  x  9  = 0(1)  x  B,U ) _ 0  Therefore  > U Q . This was done  we have  = /Q + 0(1)  2  B^  Vq  x  2  While  uo = 0(1) u  n  Hence  u  0  for  ,  Q ->  Q  In order to continue this procedure one has to equate the higher order terms.  d Z  l ) n  du Where  H  _  R(G  Qu  In general, from (IV.3.4) we shall have:  >  Z  0  X ) >  Z  1  X ) >  •••» n - 1 Q QuZ (u)  is a polynomial such that  Z  ;  B  £ ) >  ••" "n-l*  Q  — 0 Z  is continuous in  [u„,u ]  30  Since  Z ^(u.) =0  i = 2, 3 we shall have  (l  B<°  ru  z«>(„> -  Qs I  Z^Cs); B ° ,  H(G(s), Z ^ C s ) ,  (  B<*>)  Q  ds  Q sZ^ (s) }  i—  i = 2, 3 (2)  Now we require  (v) =  obtain the expression for u.  1  n  Q  (3)  B n  (v) for some v e [ ^ j U ] and  from ds = 0 H s • Z (s)  S  Q  u„  IV.4  Summary Approximate solutions for the saddle-to-saddle separatrix were  obtained in this section. One solution i s based on the assumption that 3/2 aQ  3/2 aQ  i s a small parameter and the other assumes that  A unique value of b  i s large.  for which the existence of a saddle-to-saddle  separatrix is possible was recovered in each case.  In the former case  "(s) (s) b = X Q + Pb^ + ... . The expression i  this  b has the expansion As) (s) l B^ = — — i s given in (IV.2.8). b  ~  0  +  P  b  l  +  •••  W h e r e  b  0  In the latter case we have - ln x  3  - In x  2  '  The important conclusion is that in both cases,  b > X Q . This  w i l l serve us as the main tool in the next section.  *  In fact at this stage we already know that  (2) (3) Z^ (u) = Z_. (u) 0 <^ j < n  31  V.  THE GLOBAL PORTRAIT OF THE PHASE PLANE  At this stage we aire ready to answer the question which was asked at the end of Section I I . We use the ideas suggested at the beginning of Section III. When T  2  < T^  (which starts at E ) must go to E . As  y  3  1  a result we have a domain bounded by solution trajectories.  The boundaries  are the x-axis and T^, and any solution which starts inside this domain cannot leave i t s boundaries. When  T > T^, T 2  2  serves as a part of the boundary, and similar  conclusions w i l l be drawn. Throughout this section we assume shown for "large  P" and "small  b > X Q . This has been  P" . For intermediate values of P i t  was verified by a numerical scheme. The details of the numerical scheme are given in the appendix. V.l  T >T 3  2  For  In this case x  2  x < b <_ x 2  Q  or x  Q  <b <b.  < b < X Q there i s only one asymptotically stable  equilibrium in the phase plane, namely  E^ . E^ is on the boundary of  K - the compact set bounded by ' T^ and the x-axis.  Therefore every  solution (except those starting exactly at the equilibria or on T ) 2  w i l l converge to E^ . This conclusion holds for solutions which start inside  K or outside i t in the f i r s t  quadrant.  /  Figure 5.  The phase plane portrait when b < X Q  .  Another possiblity, s t i l l in case (1), is that  X Q< b < b .  Following the formulation of Coddington-Levinson [6] we define as the negative limit set of * T 2 ' Theorem. L(T ) 2  Proof.  T, 2  and  T  2  LCTJ)  as a negative semi-orbit of  is a limit cycle.  We use Poincare-Benedixon theorem as given by  Coddington-Levinson.  There i t refers to positive semi-orbits but i t can  be applied also to negative semi-orbits, which is our case. L(T )<r K 2  are  E  j  The singular points in  K  . , i = l,2,3,4.  E^, E^ £ L(T ) 2  to  which is a bounded set.  E E.j as  2  £ L(T ) t -> 2  E^ { L(T ) 2  because they are asymptotically stable. because there are only two trajectories which converge and they are on the x-axis. because  E^ = K T ^ )  by definition of  *i.e. T~ is obtained by starting somewhere on trajectory as time goes backwards.  T^  and there  T , and following the solution 2  33  is only one trajectory in the f i r s t quadrant which converges to  as  t -*- -°° . The conclusion is that  L(T )  contains only regular points  2  and hence either (i)  T  is a periodic orbit, or 2  (ii)  L(T ) (i)  periodic. T  2  is a periodic orbit (a limit cycle).  2  is excluded because i f T  But we know that  T  2  -* E  2  as  is periodic then  2  t  <*> and  4 E  is not periodic, and hence a limit cycle exists inside E^  unstable.  is asymptotically  T 2  2  is  . Therefore K .  stable so the limit cycle is asymptotically  Every solution which starts in the domain bounded by the limit  cycle w i l l remain there, and converge to  E^ .  Outside the closed set  bounded by the limit cycle, a l l the solutions converge to The existence of a limit cycle for also by noticing that at  b = x^  b  near  E^ .  X Q can be shown  a Hopf bifurcation takes place.  We  shall show this by using the Hopf bifurcation theorem as given by HowardKoppel [7], Theorem.  For  b  near  x  Q  the system (II.1.1) has a one  parameter family of solutions which l i e in the neighbourhood of  E^,  and  there are no other periodic solutions wholly in this neighbourhood. Proof. where  Near  b = x  Q  the eigenvalues at  E^  are  X^(b) ± iX (b) 2  34 b g (b)  /4  ag(b) - b g (b)  A (b) = 2  At  b = x, Q  8 (b) = 0 x  and therefore b g  We also have  X  db  x  A^b)  = 0;  A (b) 4 0 . 2  ( b )  < 0  b=x 0  Another condition which is automatically satisfied is that the other eigenvalues are bounded away from the imaginary axis. We see that a l l the conditions of the theorem hold and hence the conclusion. As we can see, a structural change takes place as bigger than  b  becomes  X Q . One feature of this change is the appearance of a  limit cycle. By numerical methods i t was observed that i t s amplitude is small near place until  b = X Q and increases as T  0  = T  0  b  gets larger. This increase takes  and the limit cycle disappears.  1  - Figure '6.  The phase plane portrait for  X Q< b < b  35  is the only asymptotically stable eqilibrium in E^  is the only asymptotically stable equilibrium in  solution which starts in starts in K„  w i l l converge to  w i l l converge to  E. .  E^  and  • Therefore a  and a solution which  36  Figure 8.  The phase plane portrait for  b > b .  Summary. We pointed at 3 structurally different portraits of the phase plane.in the interior of the f i r s t quadrant. The f i r s t portrait was obtained for start at an equilibrium or on converge to  E  T^  b < X Q . Solutions which  stay there, while a l l the other solutions  . .  The second portrait was obtained for b e (Xg,b) . An unstable limit cycle appears.  Solutions which,start on i t , on  equilibrium stay there.  o r  a t  a n  Solutions which start in the interior of the  domain bounded by the limit cycle converge to  E, . A l l the other  37  solutions converge to  E  . .  The last portrait was obtained for  b > b , i.e. when  Then solutions which start at an equilibrium or on Solutions which start under solutions converge to  E  T^  converge to  1^ > T^  T^ , remain there.  E^, and a l l the other  .  A fourth portrait that might have occurred would have  E^  as  an asymptotically unstable equilibrium and an asymptotocally stable limit cycle surrounding  E^.  It does not occur because  b > x^ .  The structural changes were derived from global considerations. However, at  b = x^  a Hopf bifurcation - a local phenomenon exists.  Its significance is consistent with the portrait derived from a global considerations.  r  38  VI.  INTERPRETATION  As we could see, there is a danger of driving the harvested population to  E^ .  Clearly, this is an undesirable equilibrium, an  equilibrium of a graveyard:  low animal density and no harvesting  activity.  Another feature is the possibility of fluctuations both in the effort and in the population.  With  b e (xQ,b)  these fluctuations  be undesirable but not fatal i f they are inside the limit cycle. they are outside the limit cycle they end up at  may  But i f  E., - a disaster.  If the situation is not too bad a collapse can be avoided by regulating the fishery, i.e. by controlling either  a  higher  E^  b  is the smaller the attraction domain of  (x(0),y(0))  we can determine  b  such that  or  b .  The  is . Given  (x(0),y(0))  are in the  interior of the limit cycle (in the second portrait) or under the third portrait) , or we need  b > x^  to achieve one of those.  Then  a collapse is inevitable. The main conclusion is that even from such a simple model i t can be seen that nature is not always forgiving; there is a danger of depleting this type of resource.  39  Appendix A:  Possible Values of  g(x)  g(x)  and  Q  can have either one or three roots.  is a function of where  R  R  and  The number of roots  Q . To find the domain in the  R-Q  plain  has 3 roots we solve simultaneously: g(x) = 0 ,  Then we obtain  R = R(x); Q = Q(x).  of the curves on which  g(x)  This is a parametric representation  has one double roots and one simple roots.  d where —|- (x) = 0 we have one triple root. These curves are the dx boundary which confines the domain where g(x) has 3 roots. 2  At  x  The equation  g(x) = 0  implies  The equation  ~r~ g = 0  implies *  X  X  R(l - —)  2 1 + x  =  0  2 X  Q  2,2  Hence 2,  2x  3  R(x) = (l + x YI y  i  By the nature of the problem  Q00  R  3  = x— - l  and  Q  are positive.  following properties can be derived from these forms:  The  40  x  1  +  x -»• To find  x  Q -* °° and R -»- %  +  implies  Q ->• °° and  +  R + 0  , .  we consider:  dR 2x (3 - x ) .; dx = . 2. 3 ' (1 + x ) 2  (  implies  2  Figure 9.  dQ _ 2 x ( - 3) _ dx .2..2 (x - 1) 2  2  X  dR _ dQ _ dx dx  Q  afc  ;  =  /J  The number of roots of g(x) as a function of various values for R and Q .  41  Appendix B: The Numerical Scheme A numerical scheme was used in order to verify the results which utilized asymptotic techniques.  The main part of the scheme was  an integration routine which used a predictor-corrector formula of fourth order.  The routine also measured a quantity proportional to the relative  error (namely, predicted value - corrected value ), and changes the time corrected value _3 step so that this quantity w i l l be smaller than 10  . When the time step  was decreased (or to start the solution) a fourth order Runge-Kutta method was used to generate the next three terms. The system was integrated with i n i t i a l (x,y) near E^' or E^, as the case required. Near E. the slopes of T. are x. - b m. = g'(x.) - a — x i ( i = 2, 3) . This observation was used to make 1  the integration slightly faster. The i n i t i a l initial  y was 10  x was taken to be on the line which goes through  the slope  and the E^ and has  TIK ( i = 2, 3) . To obtain  going backwards.  t  n  system (II.1.1) was integrated with time  e  This was done until the solution crossed the line  x = b . The value of T^ at x = b was not necessarily given by the routine because the integration was done with respect to time (rather than integrating 4^ dx  =  a  ^ / s — ) x g(x) - xy X  . Therefore, the routine took the '  value at the closest point to x = b, say at x = x,, . Incidentally, the value ^at of  x„ was either the last or the one next to the last value D  to be computed. The next step was integration of T^ . Now time went forwards  42  and the routine stopped integrating when the solution crossed the line x = x. . This time we were interested in the value of B  T  at  j 0  x = x , B  and used a third order Lagrangian interpolation formula to obtain i t . At this point comparison could be made between If  y. ± J  i s the y-value of J  T. x  at  x„, B  then  an  F = Y„ - Y„ 2 3  F = F(b) . To find  b  method of false position was used until iteration involves an integration of  and  . Obviously, each  T^ •  values ranging from -5  50 to  5000 and  the numerical solutions for b asymptotic "large  P"  a  from  10  P  P"  0.5,  to  10  . For  P > 10,  was not more than 5% away from the  approximation.  For  between 1 and 10 the results varied.  to the "large  0.014 to  3  P < 1  the numerical solution  was not more than 7% away from the asymptotic "small P" For  F .  |F| < 10 ^ . Then the  |F| < 10  1^  The program was run for R values from  (R, Q, a)  we had to find the zero of  This was done by the method of bisection until  Q  .  was the  interesting quantity. As i t was shown in the work, for any fixed we had  &  approximation.  Sometimes they were close  approximation, sometimes to the "small P"  approximation,  sometimes to both and sometimes to none. Yet even in the latter case, the computed  b  was larger than  X Q.  The formulas which were used in the scheme were based on Ralston [8].  3/2 P = aQ '  P" b  "large  P" b  "large P" relative error (%)  Q  "small P" i s good for P = 1.77  0.5 0.5 0.5  50 50 50  17.67 1.77 0.177  12.57 11.06 10.04  27.37 11.6 10.02  117 4.8 0.1  12.8 12.8 12.8  1.8 15 27  none i s good for P = 1.77  0.3 0.3 0.3  50 50 50  17.67 1.77 0.117  15.87 14.03 13.045  37.08 15.22 13.035  133 8.5 0.08  16.25 16.25 16.25  2.3 15 15  both are good for P = 1.77  0.108 0.108 0.108  50 50 50  17.67 1.77 0.117  22.57 22.35 21.74  41.58 23.48 21.67  84 5 0.3  22.58 22.58 22.58  "large P" i s good for P = 5.59  0.0389 0.0389 0.0389  500 500 500  55.90 5.59 0.56  155.80 150.78 131.21  2308.6 332.91 135.35  1381 120.8 3.2  Table:  A sample of the computer results.  computed b  "small  "small P" relative error (%)  R  155.99 155.99 155.99 .  0.04 1 3.9 0.1 3.5 18.9  LO  44  BIBLIOGRAPHY  Clark, C.W., "The Economics of Overexploitation", Science 181, 630-634 (1973). Clark, CW.,and G.R. Munro, "The Economics of Fishing and Modern Capital. Theory: A Simplified Approach", J. Environmental Economics and Management 2., 92-106 (1975) . McNaughton, S.S. and L.L. Wolf, General Ecology, Holt, Rinehart and Winston, Inc. (1973). Holling, C.S., "The Functional Response of Predators to Prey Density and i t s Role in Mimicry and Population Regulation", Memoirs of the Entomological Society of Canada 45_ (1965) . Schaeffer, M.B., "Some Considerations of Population Dynamics and Economics in Relation to the Management of Marine Fisheries", J. Fish. Res. Bd. Canada 14 (5), 669-681 (1957). Coddington, E.A. and N. Levinsbn, "Theory of Ordinary Differential Equations", McGraw-Hill Book Company, N.Y. (1955). Kop'ell, N. and L.N. Howard, "Plane Wave Solutions to Reaction-Diffusion Equations", Studies in Applied Mathematics _5_2 (4), (1973). Ralston, A., "A First Course in Numerical Analysis", McGraw-Hill Book Company, N.Y. (1965). Smith, V.L., "On Models of Commercial Fisheries", J. Polit. Econ. 7_7, 181-198 (1969) .  

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