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A multiple age class population model with delayed recruitment Chuma, Joseph Louis 1981

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A MULTIPLE AGE CLASS POPULATION MODEL WITH DELAYED RECRUITMENT by JOSEPH LOUIS CHUMA B . S c , The U n i v e r s i t y o f V i c t o r i a , 1979  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department o f Mathematics)  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1981 (c) Joseph Louis Chuma, 1981  In p r e s e n t i n g  this  thesis i n partial  f u l f i l m e n t of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e of B r i t i s h Columbia, I agree that it  freely  the L i b r a r y  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree that p e r m i s s i o n for  University  f o r extensive  s c h o l a r l y p u r p o s e s may  for  financial  shall  Joseph L o u i s  Mathematics  The U n i v e r s i t y o f B r i t i s h 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 Date  August 14,  1981  thesis  Columbia  Chuma  my  It is thesis  n o t be a l l o w e d w i t h o u t my  permission.  Department of  further  be g r a n t e d by t h e h e a d o f  copying or p u b l i c a t i o n of t h i s  gain  I  make  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood that  shall  written  ABSTRACT  An e x p l o i t e d single-species population model w i t h a density dependent reproductive f u n c t i o n i s constructed, i n which recruitment to the a d u l t breeding population may occur i n one of s e v e r a l p o s s i b l e age c l a s s e s .  The parent i s assumed capable of g i v i n g b i r t h only once.  I t i s a l s o assumed t h a t a l l density dependence i s concentrated i n the f i r s t year of l i f e .  A l i n e a r i z e d s t a b i l i t y a n a l y s i s of the m u l t i p l y -  delayed d i f f e r e n c e equation model i s c a r r i e d out and a s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y i s derived f o r the general case, while necessary and s u f f i c i e n t conditions are found i n s p e c i f i c examples. Some i n d i c a t i o n o f the complicated b i f u r c a t i o n s t r u c t u r e o f the model i s given by a s e r i e s of computer s i m u l a t i o n p l o t s .  F i n a l l y , the  method o f Lagrange m u l t i p l i e r s i s used t o f i n d the optimal e q u i l i b r i u m escapement l e v e l f o r the o r i g i n a l e x p l o i t e d population model.  C o l i n W. F. Y. M.  Clark Wan  iii  TABLE OF CONTENTS  Page.  Abstract Table o f Contents L i s t o f Figures Acknowledgement Dedication  . ^ i i i i Y Y"!" X  *  •  v  i  l  Chapter I  -  INTRODUCTION  II  - THE MODEL  III  -  1 9  STABILITY S p e c i a l case: m = 0 S p e c i a l case: m = 1 Example 1 . Example 2 Computer simulations  ..  .  13 17 18 24 26 27  IV  - OPTIMALITY  46  V  - DISCUSSION  49  BIBLIOGRAPHY  51  iv  LIST OF FIGURES  Figure  1 2  Page  A schematic representation o f the model f o r the s p e c i a l case m = 1 The i n t e r s e c t i o n of z  n  +  m  10  and A ( z + X^z " ^ + ••• + X ) m  1  -  m  gives a r e a l root z^ > 1, when A > 1/(1 + X^ + • • • + A ) ...  15  The (X^,A) region o f l o c a l s t a b i l i t y f o r n = 1 and m = 1  (shaded area) 19  The (X^,A) region o f l o c a l s t a b i l i t y f o r n = 2 and m = 1  (shaded area)  m  3 4 5  20  A p l o t o f A (X,) f o r m = 1 and n = 2, 3, 4, and 5. Note t h a t A ^ ( X p + -1/(1 + X ^ as n -+ +  25  Simulation o f the modeled population f o r n = 1 and m = 1. E q u i l i b r i u m i s x* = 0.695 and i s s t a b l e  28  Simulation o f the model f o r n = 1 and m = 1, showing a p l o t o f y e a r l i n g s versus age c l a s s two  29  Simulation o f the model f o r n = 1 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.691 and i s s t a b l e  31  Simulation o f the model f o r n = 1 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two  32  Simulation o f the model f o r n = 1 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.779 and i s not s t a b l e  33  Simulation o f the model f o r n = 1 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two  34  Sbmilation o f the model f o r n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.410 and i s s t a b l e  36  00  6a 6b 7a  7b 8a  8b 9a  V  Figure 9b 10a  10b 11a  lib 12a  12b 13a  13b  Page Simulation o f the model f o r n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 100.0)  37  Simulation o f the model f o r n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.689 and i s s t a b l e  38  Simulation o f the model f o r n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 190.0)  39  Simulation o f the model f o r n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.705 and i s not s t a b l e  40  Simulation o f the model w i t h n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 200.0)  41  Simulation o f the model w i t h n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.738 and i s not s t a b l e  42  Simulation o f the model w i t h n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 225.0)  43  Simulation o f the model w i t h n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.754 and i s not s t a b l e  44  Simulation o f the model w i t h n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 240.0) ......  45  ACKN0WLEDGEME3S1T  I wish t o thank Michael B a i l e y f o r suggesting the idea which became the b a s i c model.  I would a l s o l i k e t o express my a p p r e c i a t i o n t o  Dr. C o l i n C l a r k f o r h i s comments and h i s help i n preparing t h i s t h e s i s .  DEDICATION  This t h e s i s i s dedicated t o Dr. Pauline van den Driessche, who provided me w i t h i n s p i r a t i o n , encouragement, and the s k i l l s t o pursue mathematical research.  1  CHAPTER I INTRODUCTION  For many species o f organisms, population growth i s discontinuous. The l i f e h i s t o r y o f such organisms may be subject t o strong seasonal o r p e r i o d i c i n f l u e n c e s . A l s o , f o r many species, recruitment t o the breeding stock may only occur s e v e r a l months o r years a f t e r b i r t h .  There  are species whose members reproduce only once i n t h e i r l i f e t i m e s and d i e before t h e i r descendents' l i v e s begin, f o r example, salmon o r cicadas. To represent the population growth of such species, the only s u i t a b l e model i s a d i f f e r e n c e , o r d i f f e r e n c e - d e l a y equation (Maynard Smith 1968, May 1973, C l a r k 1976a, P i e l o u 1977). Consider a b i o l o g i c a l population (P^ i n generation k) which has d i s c r e t e and nonoverlapping generations.  The population dynamics may  be described by the f i r s t order d i f f e r e n c e equation P  k + 1  = F(P ),  (1)  k  which r e l a t e s the population l e v e l P a t time t = t  k  +  ^ t o the population  l e v e l a t a previous time t = t j , by means o f some given stock-recruitment function, F(P).  I n most cases o f i n t e r e s t , the f u n c t i o n F(P) i s  nonlinear and i s u s u a l l y constructed t o allow the population t o grow r a p i d l y a t low d e n s i t i e s and l e v e l o f f o r p o s s i b l y d e c l i n e a t high d e n s i t i e s . Many e x p l i c i t forms f o r the density dependent function.F(P) have been proposed i n the l i t e r a t u r e , and t a b l e s o f s p e c i f i c forms used, w i t h  2  references, can be found i n May,  1979 o r May and Oster, 1976.  iitportant example i s the Ricker model (Ricker 1954) F(P) = P exp { r ( l - P/K)}.  One .  i n which  This Ricker model i s used e x t e n s i v e l y i n the  management of the P a c i f i c salmon (Oncorhynchus species) populations (Clark 1976a).  The dynamic behaviour of s o l u t i o n s of equation (1) f o r  t h i s case i s s u r p r i s i n g l y complicated, but now seems t o be w e l l understood, see Levin and Goodyear,  1980.  C l e a r l y , equation (1), w i t h F(P) as given above, always possesses a n o n - t r i v i a l e q u i l i b r i u m a t P* = K.  I t has been shown t h a t t h i s  e q u i l i b r i u m i s l o c a l l y s t a b l e provided t h a t 0 < r 4 2; but i f r i s increased beyond the value 2, the e q u i l i b r i u m becomes unstable and a new s t a b l e l i m i t - c y c l e of period 2 b i f u r c a t e s from the e q u i l i b r i u m . In f a c t , there i s an i n c r e a s i n g sequence 2 = r-^ < 2.526 = such that when r increases past r b i f u r c a t e s from the e q u i l i b r i u m .  <  •••  a new and s t a b l e c y c l e of period  n  2  n  However, the.sequence of b i f u r c a t i o n  values f o r r are bounded above by a c r i t i c a l value r * = 2.692, beyond which c y c l e s of a r b i t r a r y p e r i o d appear along w i t h s o l u t i o n s t h a t never s e t t l e i n t o any f i n i t e c y c l e .  The region beyond r * has been  c a l l e d dynamic chaos since s o l u t i o n s are e f f e c t i v e l y i n d i s t i n g u i s h a b l e from random f l u c t u a t i o n s . The term "chaos" was L i and Yorke, 1975.  introduced i n the mathematical paper by  They show t h a t i f equation (1) has a s o l u t i o n of  p e r i o d 3, then f o r those same parameter values, s o l u t i o n s of  any  integer period can be found, as w e l l as s o l u t i o n s t h a t never s e t t l e into a periodic cycle.  For a review of the irathematical d e t a i l s of  3  the b i f u r c a t i o n s t r u c t u r e o f equations l i k e equation (1) see May, 1976 o r May and Oster, 1976. Many r e a l populations have s e v e r a l d i s t i n c t but overlapping age c l a s s e s , o r the d e n s i t y dependent mechanisms operate w i t h an e x p l i c i t time delay, say n generations.  I n t h i s case, a d i f f e r e n c e - d e l a y equation  of the form p  may be appropriate.  i  11 k+1  = P, F ( P ,  k  k-n')  v  (2) '  Examples o f such models, where the population a t  k+1 depends l i n e a r l y on the previous population and n o n l i n e a r l y on a s i n g l e population a t some time i n the past, can be found i n Maynard Smith, 1968; May, Conway, H a s s e l l , and Southwood, 1974; C l a r k , 1976b; and Beddington,  1978.  C l a r k , 1976b, studied the delay equation  =  + ( ] _ ) ' F  P  c  n  which has a p p l i c a t i o n s as a model of baleen whale population dynamics. E q u i l i b r i u m s t a b i l i t y and optimal e x p l o i t a t i o n p o l i c i e s were discussed. Here, a i s the s u r v i v a l c o e f f i c i e n t and F(P. ) i s the recruitment t o k-n the breeding population a t time k which was produced by the breeding population a t time k-n.  The l o c a l s t a b i l i t y o f an e q u i l i b r i u m , P*,  f o r t h i s d i f f e r e n c e - d e l a y equation was shown t o depend seperately on the s u r v i v a l c o e f f i c i e n t , the slope o f the recruitment f u n c t i o n a t the e q u i l i b r i u m , and the time delay i n the recruitment.  C l a r k has shown  t h a t increased delay i m p l i e s reduced s t a b i l i t y , i n the sense t h a t i n c r e a s i n g the delay reduces the region i n the parameter plane which possesses a s t a b l e e q u i l i b r i u m p o i n t .  4  Goh and Agnew, 1978, considered C l a r k ' s , 1976b, d i f f e r e n c e - d e l a y model equation f o r a population i n which, recruitment t o the breeding c l a s s takes place s e v e r a l generations a f t e r b i r t h .  They employ a  s p e c i f i c form f o r the recruitment f u n c t i o n , F ( P ) , namely, F(P) = 2 AP exp (-BP ), where A and B are p o s i t i v e constants.  I t i s shown t h a t  i n c r e a s i n g the delay causes reduced s t a b i l i t y , while i n c r e a s i n g the s u r v i v a l c o e f f i c i e n t , when the delay between b i r t h and recruitment i s s m a l l , tends t o s t a b i l i z e the population.  However, when t h i s delay i s  longer and the s u r v i v a l c o e f f i c i e n t i s not near one, then the s t a b i l i z i n g e f f e c t s o f the s u r v i v a l c o e f f i c i e n t are overshadowed by the d e s t a b i l i z i n g e f f e c t s o f the time delay.  Harvesting o f the modeled population i s  studied and they conclude t h a t , for populations t h a t e x h i b i t a sharp peak i n t h e i r recruitment f u n c t i o n , intermediate l e v e l s o f constant e f f o r t harvesting can l e a d t o d e s t a b i l i z a t i o n . In the paper by May e t a l . , 1974, conditions t h a t give r i s e t o s t a b i l i t y and o s c i l l a t i o n s i n a s i n g l e species population i n t e r a c t i n g w i t h a maintained resource were s t u d i e d .  Considered f i r s t were  d i s c r e t e generation d i f f e r e n c e equation models w i t h d e n s i t y dependent m o r t a l i t y and fecundity.  I f the r a t e a t which a population takes t o  r e t u r n towards an e q u i l i b r i u m l e v e l i s c a l l e d the c h a r a c t e r i s t i c r e t u r n time, t h i s paper contends "that i t i s the r e l a t i o n s h i p o f t h i s time t o the time delays i n the system (e.g. the length o f a generation) t h a t determines whether the population approaches the e q u i l i b r i u m monotonically o r 'overshoots' and o s c i l l a t e s about the e q u i l i b r i u m "  5  (May e t a l . 1974:  747).  They found t h a t i n s t a b i l i t y follows from t h i s  r e t u r n time being too small compared with:.the\:time delays. Consideration i s then given t o m u l t i p l e age c l a s s models,  and  a model w i t h two overlapping age classes i s studied i n d e t a i l . course, the s t a b i l i t y properties of these m u l t i p l e age c l a s s  Of  populations  i s more complicated, but they are shown t o be s i m i l a r t o those o f m u l t i species systems.  The l o c a l s t a b i l i t y of an e q u i l i b r i u m i s shown t o be  determined by the dominant eigenvalue o f a matrix of parameters c h a r a c t e r i z i n g the slopes of the density dependent r e l a t i o n s h i p s between age c l a s s e s . Diamond, 1976, presented techniques f o r estimating the s i z e and shape o f regions of l o c a l s t a b i l i t y f o r d i f f e r e n c e equations.  These  techniques are based on a c l a s s of d i s c r e t e Liapunov f u n c t i o n s , and a " r e s t r i c t e d r e c i p e " f o r f i n d i n g these Liapunov functions along w i t h an algorithm f o r c a l c u l a t i n g Liapunov contours i s given.  The  estimation  methods are a p p l i e d t o a s i n g l e species model w i t h two age c l a s s e s . Levin and May, delay equation (2).  1976,  gave s t a b i l i t y c r i t e r i a f o r the d i f f e r e n c e -  They presented general a n a l y t i c formulas d e s c r i b i n g  the boundary between monotonic damping and o s c i l l a t o r y damping toward a s t a b l e e q u i l i b r i u m p o i n t , P*, and f o r the boundary seperating regions of s t a b i l i t y and i n s t a b i l i t y o f P*. C l a r k , 1976b, May e t a l . , 1974,  the  As was found a l s o by  and Goh and Agnew, 1978,  an..increase  i n the e x p l i c i t time delay, n i n equation (2), leads the system t o be more prone t o o s c i l l a t i o n s and i n s t a b i l i t y .  An adaptation of the  6  l i n e a r i z e d s t a b i l i t y a n a l y s i s p r e s e n t e d by L e v i n and May, used i n t h i s  is  thesis.  E x t e n s i o n s o f t h e d i f f e r e n c e - d e l a y model e q u a t i o n for  1976,  (2) t o a l l o w  m u l t i p l e age spawning p o p u l a t i o n s where d e n s i t y dependence i s  e x p r e s s e d i n terms o f t h e p o p u l a t i o n l e v e l s o f b o t h p r e s e n t and s e v e r a l preceding generations, i . e . , P, ,-, = P, F (P. , P. - , ... k+1 k k k-1' v  have been c o n s i d e r e d by A l l e n and B a s a s i b w a k i , L e v i n and Goodyear,  (3)  , P, ), k-n ' 1974,  Ross, 1978,  and  1980.  A l l e n and B a s a s i b w a k i ,  1974,  s t u d i e d a c l a s s o f models i n c o r p o r a t i n g  m u l t i p l e age s t r u c t u r e where r e c r u i t m e n t t o t h e p o p u l a t i o n i s t h e p r o d u c t o f f e c u n d i t y and a s u r v i v a l r a t e . was  This f i r s t year s u r v i v a l r a t e  assumed t o v a r y w i t h t h e s i z e and s t r u c t u r e o f t h e p o p u l a t i o n .  The p o p u l a t i o n a f t e r r e c r u i t m e n t i s d e s c r i b e d by a l i f e t a b l e w i t h constant s u r v i v a l r a t e s .  Necessary  c o n d i t i o n s f o r the s t a b i l i t y o f  an e q u i l i b r i u m and p r o p e r t i e s o f o s c i l l a t i o n s about an u n s t a b l e e q u i l i b r i u m were c o n s i d e r e d u s i n g a combination  o f a n a l y t i c a l and s i m u l a t i o n  techniques. Ross, 1978, form P, ,., = aP?" k+1 k  c o n s i d e r e d a s p e c i a l case o f e q u a t i o n c  .  k-1  i  n  (3) o f t h e  t h i s c a s e , t h e model can be w r i t t e n as a  l i n e a r second o r d e r r e c u r r e n c e r e l a t i o n i n t h e l o g a r i t h m o f t h e p o p u l a t i o n and e x p l i c i t s o l u t i o n s t o t h i s r e c u r r e n c e r e l a t i o n were d e r i v e d and c l a s s i f i e d a c c o r d i n g t o v a r i o u s parameter v a l u e s . I n t h e paper by L e v i n and Goodyear, 1980,  a m u l t i p l e age  spawning  7  population model w i t h Ricker type stock-recruitment r e l a t i o n s h i p was examined.  Their model assumed t h a t a l l d e n s i t y dependent e f f e c t s  occur w i t h i n the f i r s t year of l i f e .  A d e n s i t y dependent L e s l i e matrix  was developed and l i n e a r i z a t i o n techniques a p p l i e d t o various s i m p l i f i e d models.  Very complicated s t a b i l i t y p r o p e r t i e s were  shown t o be dependent on two opposing delays i n the system, the reproductive delay associated w i t h d e f e r r i n g reproduction and the t r u n c a t i o n delay associated w i t h an eventual l e v e l i n g o f f o f fecundity i n l a t e r age c l a s s e s .  The balance between these delays was shown t o  be a t the r o o t o f the o v e r a l l dynamics of the system.  Computer  simulations found i n the paper o f L e v i n and Goodyear, 1980,  are  very s i m i l a r t o f i g u r e s which may be found i n t h i s t h e s i s , and i n d i c a t e some of the spectacular dynamics which can occur when an e q u i l i b r i u m i s i n a region of i n s t a b i l i t y . This t h e s i s examines an e x p l o i t e d s i n g l e - s p e c i e s population model w i t h a d e n s i t y dependent reproductive f u n c t i o n , i n which recruitment t o the breeding population occurs i n one o f m+1 1, 2, ...) p o s s i b l e ages.  (m = 0,  I n t h i s model however, the a d u l t breeder  can give b i r t h only once, and then d i e s .  Harvesting i s assumed t o  occur only among the breeder population. An example of a species whose c h a r a c t e r i s t i c s c l o s e l y approximate these i s chum salmon (Qncorhynchus k e t a ) , whose spawning grounds are from Alaska t o C a l i f o r n i a .  The young chums q u i c k l y go t o sea i n t h e i r  f i r s t year of l i f e , ranging f a r i n t o the P a c i f i c Ocean.  They mature  8  m a i n l y a f t e r t h r e e , f o u r , o r f i v e growing seasons a v e r y s m a l l p e r c e n t a g e w i l l reproduce of  life.  i n t h e ocean, a l t h o u g h  i n t h e i r second o r s i x t h y e a r  A l s o , t h e i r p e l a g i c annual n a t u r a l m o r t a l i t y r a t e seems  t o 'remain f a i r l y c o n s t a n t , t h a t i s , a f t e r t h e f i r s t y e a r o f l i f e . Chum salmon r e t u r n t o c o a s t a l r e g i o n s o n l y as t h e y a r e  approaching  m a t u r i t y , so t h a t t h e r e i s no h a r v e s t i n g o f chums i n l o c a l waters one o r more y e a r s b e f o r e m a t u r i t y  ( R i c k e r 1980).  A model e q u a t i o n f o r t h e a d u l t b r e e d i n g p o p u l a t i o n o f t h e R  k  = F CP. , p. ., ... k-n' k-n-1 v  i s c o n s t r u c t e d f i r s t , where t h e r e a r e m+1 n b e i n g t h e f i r s t p o s s i b l e age.  form  (i  , P. ) k-n-nv  p o s s i b l e ages f o r b r e e d i n g ,  F o l l o w i n g t h i s i s an a n a l y s i s o f t h e  l o c a l s t a b i l i t y o f an e q u i l i b r i u m s o l u t i o n , P*. c o n d i t i o n f o r l o c a l s t a b i l i t y o f P*  A  sufficient  i s d e r i v e d f o r g e n e r a l n and  m,  w h i l e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s a r e examined i n d e t a i l f o r t h e s p e c i a l case o f m = 1.  Note t h a t t h e case m = 0 reduces  to  C l a r k ' s , 1976b, model w i t h s u r v i v a l c o e f f i c i e n t e q u a l t o z e r o . C o n s i d e r e d n e x t a r e a few s p e c i f i c examples f o r t h e d e n s i t y dependent reproductive function.  F i n a l l y , the problem o f economically o p t i m a l  e x p l o i t a t i o n o f a p o p u l a t i o n modeled by an e q u a t i o n s i m i l a r t o is  (4)  c o n s i d e r e d and a f o r m u l a d e t e r m i n i n g t h e o p t i m a l e q u i l i b r i u m  escapement l e v e l , S*,  i s derived.  9  CHAPTER I I  THE MODEL  A s i n g l e - s p e c i e s population model w i t h a d e n s i t y dependent reproductive f u n c t i o n w i l l be constructed.  C h a r a c t e r i s t i c s o f the  species t o be modeled i n c l u d e recruitment t o the mature a d u l t (female) breeding population i n one, and only one, year o f l i f e .  Assume there  are m+1 p o s s i b l e breeding ages, w i t h n being the f i r s t p o s s i b l e age o f reproduction. giving birth.  A member o f t h i s species d i e s immediately a f t e r  Assume a l s o that the p e r i o d o f years from b i r t h t o  maturity i s spent w e l l away from the spawning area and only mature members are. subject t o harvesting as they r e t u r n . F i x n and n+m as the f i r s t and l a s t p o s s i b l e ages f o r reproduction. Let P^ represent the p a r e n t a l , o r a d u l t breeding, population i n year k, w h i l e Q. , represents the population o f j-year-olds i n year k. 1 /  If  K  i s the number o f mature stock harvested i n year k, then the  escapement w i l l be  = P^ - H^.  The model can be represented  schematically as i n Figure 1. L e t a.. (0 < a.. ^ 1) be the p r o p o r t i o n o f (n+j-1) -year-olds t h a t reproduce, j = 1, 2, ... , m+1. breeding age, i t f o l l o w s t h a t write:  Since n+m i s the l a s t p o s s i b l e = 1.  Thus, i t i s p o s s i b l e t o  10  •H  k+1  'k+1  ^k+l  Figure 1. A schematic representation o f the model f o r the s p e c i a l case m = 1.  k  l^n,k  2 n+l,k  mrn+m-l,k  v  n+m,k  v  Now, i f cjj i s the density independent n a t u r a l s u r v i v a l r a t e from age j-1 t o age j (0 <  < 1, j = 1, 2, '... , n+m), then the f o l l o w i n g  equations describe the age c l a s s e s o f the population:  Q  Q  l,k+1  =  o f(S ) ±  (6)  k  (7)  i,k+1  n+j,k+1 = V  l - j n + j - l , k ' ^ = 1, 2, a  Q  j  (  )Q  , m  where the term f ( S ) i s the density dependent reproductive f u n c t i o n . k  I t f o l l o w s now from equations (5) and (8) t h a t  (8)  11  k l n,k 2 n l - l n k-l  P =a Q  a  +  a  (1  a  )Q  +  +  /  3 n+2  a  a  ( 1  " 2 a  }  V l " l n,k-2 (1  a  )Q  +  Vn+nhl -Vl W2 -V2 "-Vl - l n,k-rt  +  )  (1  (1  )  (1  a  )Q  +  n+m-° n+m-l-Vl'' * Vl l n,k- '  a  (1  )a  (1  )  (H  )Q  m  m  or in more compact notation J  m  P  k = l n,k a  Q  .j = l j l n , k - ij =Vl V i Z  +  { a  Q  1  J  +  I I  "J a  ( 1  1  J  K  (9)  I t can a l s o be e a s i l y seen t h a t the population o f n-year-olds s a t i s f i e s the f o l l o w i n g :  Q  n,k  =  (  a  nVl  a  l  ) f ( S  k-n }  ( 1 0 )  For some s i m p l i c i t y o f n o t a t i o n , define F(x) = ( a ^ ••• a ) f ( x ) , n  (11)  so t h a t  Q  n , k n = < k>' F  S  ( 1 2 )  +  Thus i s obtained the difference-delay equation p r o v i d i n g the population dynamics f o r t h e a d u l t breeding population:  P.. k+n = where a  and  l  = a  l  m Z a.. F(S j+1 1  j = 0  k-3"• ) , 1  (13)  12  a. = ct.  j-l U n a  1=1  n + i  ( l - cu), j = 2, 3, ... , m+1.  What follows next i s a consideration of s t a b i l i t y c r i t e r i a f o r an equilibrium s o l u t i o n of equation k  k  (13) with no harvesting, i . e . ,  (14)  13  CHAPTER I I I  STABILITY  S t a b i l i t y p r o p e r t i e s o f an e q u i l i b r i u m s o l u t i o n , P*, o f t h e model e q u a t i o n w i t h no h a r v e s t i n g m P  k  +  n  \ W ^ V j J 3=0  =  are analyzed i n t h i s s e c t i o n .  (  Considered f i r s t w i l l be a  1  5  )  sufficient  c o n d i t i o n f o r t h e l o c a l s t a b i l i t y o f P* and a n u m e r i c a l scheme f o r d e f i n i n g t h e b o u n d a r i e s o f t h e l o c a l s t a b i l i t y r e g i o n s w i l l be s k e t c h e d . A d e t a i l e d e x a m i n a t i o n o f l o c a l s t a b i l i t y w i l l be p r e s e n t e d f o r t h e t r i v i a l c a s e m = 0 and t h e n o t so t r i v i a l c a s e o f m = 1. L e t P* be a n o n - t r i v i a l  e q u i l i b r i u m f o r t h e d e l a y e q u a t i o n (15).  Then m P* = F(P*) 2 a . ,. j=0. 3  Now  linearize  +  (16)  ±  about P* by w r i t i n g P^ = P* + x^,, so t h a t m P  k n " * P  =  +  \ ^-KL^k-j) 3=0  "  F  (17)  < *» P  and s o m \  +  n  = j  f (a o  j + 1  F'(P*)x _ k  j +  (18)  0(x^_ )), j  where F'(P*) i s t h e d e r i v a t i v e o f F e v a l u a t e d a t P*.  Define  14  A  = a^F'(P*)  (19)  and  Xj = *j M +1  Thus, t h e l i n e a r i z a t i o n  is  (20)  r  obtained: m (21)  1=1  J  k as z x^ t o o b t a i n t h e c h a r a c t e r i s t i c  Express  equation  m z  n+m_  A  (  z  m  +  z  x  _^n-  j=l It i s easily  seen t h a t P*  o f the c h a r a c t e r i s t i c  Q >  ( 2 2 )  ^  < V d  only i f a l l roots  (22) have modulus |z| <  F i r s t i t s h a l l be proven t h a t t h e  + A  1  + X  1.  condition  +  2  ••• +\ )  m  m  (23)  m  i s s u f f i c i e n t f o r l o c a l s t a b i l i t y o f P*. , L e t g(z) = z and l e t h(z) = - A ( z + n  =  i s l o c a l l y s t a b l e i f and  equation  |A|  3 )  m E X.z^" ). j=l 3  Suppose t h a t  3  |A|  < 1/(1  + X  + X  1  |h(z) | = | A | | z A |z| 1  < 1 =  m - 1  +  m  +  2  + X^  '"  1  ••• + .\) , and  |g(z) |.  +  + Aj.  So t h a t  so f o r |z| = 1:  By Rouche's theorem, g(z)  number o f z e r o s the c o n d i t i o n  + A ); t h e n t h e f o l l o w i n g i s t r u e :  (namely n+m)  (23)  |h(z) | < | A | (|z|  |h(z)| < | A | ( 1 + X and g(z)  + h(z)  ±  m  +  +  ••• +  have t h e same  i n the i n t e r i o r o f the u n i t c i r c l e .  i s s u f f i c i e n t t o ensure the l o c a l s t a b i l i t y o f  Thus, P*.  A) m  15  1  Figure 2. The i n t e r s e c t i o n o f z gives a r e a l r o o t z  z  n + m  o  and A ( z + X z ~ + — + A ) ra  m  1  1  m  > 1, when A > 1/(1+A,+-•-+A).  n  Now suppose t h a t A > 1/(1 + A^ + *•• + A ), and consider the n+m graphs o f z and A ( z nm  m-' Z A.z -') , where z e j=l m  m  +  m_  R (see Figure 2) .  J  C l e a r l y , the i n t e r s e c t i o n o f these graphs always gives a r e a l r o o t ( > 1) of the c h a r a c t e r i s t i c equation (22). I t would seem appropriate here t o introduce the n o t i o n o f 'region of s t a b i l i t y , but f i r s t more n o t a t i o n i s needed t o f a c i l i t a t e ease m of presentation. L e t A ^ = I A.., then i t i s p o s s i b l e t o speak o f the 1  parameter plane ( A ^ A ) .  A (A^A) r e g i o n o f l o c a l s t a b i l i t y w i l l be  the set o f parameter values f o r which P* i s l o c a l l y s t a b l e . From the preceeding d i s c u s s i o n i t i s p o s s i b l e t o conclude t h a t , f o r any n, the upper bound o f the (A ,A) r e g i o n o f l o c a l s t a b i l i t y m  i s always A = 1/(1 + A^).  By equation (23), the lower bound, A = A ( A ) , n  m  16  must always s a t i s f y  v v i - ^ + v-  (24)  A modified Schur-Cohn c r i t e r i o n (see Freeman, 1965)  i s now  presented,  by which the l o c a l s t a b i l i t y of P*, f o r s p e c i f i c values of the parameters (A, A-p A /  , A ) , may be numerically determined from the  2  m  characteristic  equation (22). Let G(z) = c z q  + c  q  ,z q-1  q - 1  + ••• + c,z + c_ 1 0 —I  where c^ > 0.  cr —1 (z) = z^G(z ), then  Define the inverse polynomial G G (z) = c z - 1  q  Q  + c ^  -  1  (25)  -jZ + c .  + ••• + c  (26)  The roots of G~"'"(z) are the inverses of the roots of G(z) w i t h respect to the c i r c l e |z| = 1.  In addition, ( G ^ C z ) ) "  G~ (z)/G(z) = B 1  = G(z).  1  Let  + G^ (z)/G(z). 1  Q  (27)  The remainder, G ~ ( z ) , w i l l be a polynomial of degree q-1 and the 1  quotient term Bg = CQ/C^.  Continue i n t h i s  G^ (z)/G (z) = 6 1  i  ±  way:  + GTj (z)/G (z), 1  i  (28) i = 0, 1, 2, ... , q-2 where G^(z) = G(z). The necessary and s u f f i c i e n t c o n d i t i o n t h a t a l l roots of the equation G(z) = 0 l i e i n the i n t e r i o r o f the u n i t c i r c l e i n the z-plane  17  i s t h a t a l l of the f o l l o w i n g are s a t i s f i e d : (a)  G(l) > 0  (b )  G(-l) < 0  (h^)  > 0  x  f o r q even  < 1, i = 0, 1, 2, ...  |B |  (c)  f o r q odd  i  , q-2.  A p p l i c a t i o n of t h i s method w i t h G(z) replaced by the c h a r a c t e r i s t i c equation (22) i s s t r a i g h t f o r w a r d when parameter values are known, however i t does not seem feasable t o o b t a i n c l o s e d form expressions f o r the s t a b i l i t y region w i t h general n and m.  One  interesting  r e s u l t can nevertheless be gleaned from c o n d i t i o n (a). n+m _ ( m  z  A  z  +  ^  m-1  +  ...  +  ^  If  G(z) =  ^ then c o n d i t i o n (a) r e q u i r e s that  A < 1/(1 + A^ + ••• + A ) = 1/(1 + A ), which i s merely the upper bound o f the (A^A)  r e g i o n o f s t a b i l i t y derived p r e v i o u s l y .  This modified  Schur-Cohn c r i t e r i o n w i l l a l s o be made use of i n the d e t a i l e d a n a l y s i s of some s p e c i a l cases f o r m and n t o f o l l o w .  S p e c i a l case:  m = 0.  I n t h i s case the population being modeled i s one t h a t has n age c l a s s e s , but only the n  age c l a s s reproduces.  The a d u l t breeding  population i s . described by P. = Q . = F (P, ) k n,k k-n so t h a t an e q u i l i b r i u m i s simply given by P* = F ( P * ) .  (29) The c h a r a c t e r i s t i c  18  equation  (22) becomes z  - F*(P*) = 0.  n  Thus, i t i s c l e a r t h a t P* i s  l o c a l l y s t a b l e i f and o n l y i f  -1 < F'(P*) < 1.  (30)  Note t h a t t h i s s p e c i a l c a s e w i t h m = 0 i s C l a r k ' s , 1976b, model w i t h the s i m p l i f i c a t i o n o f the s u r v i v a l c o e f f i c i e n t being zero.  The n e x t  s p e c i a l c a s e i s more i n t e r e s t i n g , r e q u i r i n g more r i g o r o u s a n a l y s i s .  Special case:  m = 1.  I n t h i s c a s e t h e p o p u l a t i o n b e i n g modeled i s one t h a t has n+1 age c l a s s e s , and o n l y members o f t h e f i n a l two c a n reproduce.  The  a d u l t b r e e d i n g p o p u l a t i o n i s d e s c r i b e d by  P  k 1  =  a  +  l < k-n l> F  P  +  +  a  2 ( k-n> F  P  ( 3 1 )  so t h a t an e q u i l i b r i u m i s g i v e n by  P* =  The c h a r a c t e r i s t i c e q u a t i o n  z  (a + a )F(P*). x  (22) i n t h i s c a s e  n + 1  (32)  2  becomes  - Az - A A = 0.  (33)  From g e n e r a l r e s u l t s a l r e a d y d e r i v e d i t i s known t h a t t h e upper bound o f t h e (A-^A) r e g i o n o f l o c a l s t a b i l i t y f o r P* i s A = 1/(1 + A ) , f o r 1  a l l n; w h i l e t h e lower bound depends on n and s a t i s f i e s  A ( A ) < - 1 / ( 1 + X ). n  1  ±  (34)  Now, c l o s e d form expressions f o r two o f these lower bounds, A  2 ^ i ^ ' w i l l ke derived using the modified Schur-Cohn  A^(A^)  and  criterion  (Freeman 1 9 6 5 ) as presented e a r l i e r . - Az - A-^A.  In the case under c o n s i d e r a t i o n , G(z) = z  As  already s t a t e d , c o n d i t i o n (a) r e q u i r e s t h a t A < 1 / ( 1 + A-^), which i s merely the upper bound derived p r e v i o u s l y .  For n even, c o n d i t i o n (b^)  requires t h a t A ( l - A-^) < 1 ; and f o r n odd, c o n d i t i o n (b ) r e q u i r e s 2  that A ( A  X  - 1) < 1.  For n = 1 , a l l t h a t c o n d i t i o n (c) r e q u i r e s i s t h a t [3Q| < 1 . + (-Az - A ^ z + ( 1 - A A ) ) / G ( z ) , i t f o l l o w s  Since G ( z ) / G ( z ) = _1  11  2  2  t h a t 3Q = -X-jA, and since they w i l l be needed s h o r t l y i t i s found t h a t G^-(z) = -Az AjA z  11  - A.  - A A z + ( 1 - A A ) so t h a t G^z) 2  2  2  x  = (1 - A A ) z 2  2  n  -  Therefore, c o n d i t i o n (c) r e q u i r e s t h a t | A ^ A | < 1 .  Combining t h i s w i t h c o n d i t i o n s (a) and (b ) gives the (A^,A) r e g i o n o f 2  Figure 3 .  The (A^,A) r e g i o n o f l o c a l s t a b i l i t y (shaded area) f o r n = 1 and m = 1 .  20  l o c a l s t a b i l i t y f o r n = 1 (see Figure 3 ) . For n = 2, c o n d i t i o n (c) r e q u i r e s t h a t | g | < 1 and |B-J < 1. Q  Since G ~ ( z ) / G ( z ) = - A / ( l - * A ) + 1  2  ( z ) / G ^ z ) , where G" ^) =  2  1  1  (-^A /(l - xjA ))z 3  2  - \Az  n _ 1  + (1- A A  2  2  ±  2  - A / ( l - A A ) ) , i t follows 2  2  2  that 3^ = - A / ( l - A-jA ). Combining t h i s w i t h |A^A| < 1 and c o n d i t i o n s (a) and (b^) g i v e s the (A^,A) r e g i o n o f l o c a l s t a b i l i t y f o r n = 2: (1 -  A+  4A ' ) / 2 A < A < 1 / ( 1 + A ) 2  2  x  (35)  (see Figure 4 ) . I n attempting t o use t h i s c r i t e r i o n f o r n = 3 i t i s necessary to s o l v e a cubic equation, f o r n = 4 a q u a r t i c , e t c . ; hence i t i s n o t feasable t o make use o f t h i s Schur-Cohn c r i t e r i o n f o r f i n d i n g c l o s e d form expressions f o r  A n  ( ^ ^  o r n  —  However, i t i s u s e f u l t o note  here t h a t t h e s t a b i l i t y region f o r any n must contain the r e g i o n f o r A  Figure 4.  The (A^,A) r e g i o n o f l o c a l s t a b i l i t y (shaded area) f o r n = 2 and m = 1.  21  n+1.  I f t h i s f a c t i s combined w i t h t h e s u f f i c i e n c y c o n d i t i o n  t h e p r o g r e s s i o n o f lower bounds,  A n  (A^),  follows:  A ( A ) « ••• < A ( A ) < T A ^ U ^ 1  1  n  F o l l o w i n g L e v i n and May, the c h a r a c t e r i s t i c equation  1976,  the  a n o t h e r approach,  (36)  a r e now d e r i v e d .  and C l a r k , 1976b, c o n s i d e r a g a i n  (33), z  ifl and e x p r e s s z a s Re , R > 0. ifl must have a r o o t z - e  < . . . < -1/(1 + A ^ .  x  Further r e s u l t s f o r n > _ 2, employing  (23),  - Az - A^A = 0, w i t h n ^ 2,  Note t h a t f o r A = A (A^) > e q u a t i o n (33)  , i . e . , w i t h R = 1.  Rewrite equation  (33) i n  form  1 = Az~ + Z \ n  l Z  -  { n ¥ l )  .  (37)  i0 Substitute z = e and equate r e a l and i m a g i n a r y p a r t s t o o b t a i n sin(ne)  + A-jSin{ (n+l)fl} = 0  (38)  + AA-jCosf (n+1) 6} = 1.  (39)  and  Acos(ne)  Equation  (38) has a unique r o o t 9 = 9 ( X ) such t h a t ir/(n+l) < ^ ( X j ) n  1  < Tr/n, a l t h o u g h o f c o u r s e , t h e r e a r e i n f i n i t e l y many o t h e r r o o t s > G i v e n some A^ > 0, e q u a t i o n  (38) can b e s o l v e d f o r 9 ( ^ ^ )  thus A = A ( X ^ ) can b e found from e q u a t i o n n  t h i s process determines  n  (39).  6 (^) n  and  I t i s shown h e r e  that  t h e lower boundary o f t h e s t a b i l i t y r e g i o n .  R e c a l l t h a t f o r n >^ 2 i t has been demonstrated  t h a t t h i s lower boundary  i s i n t h e r e g i o n 0 > (A-]_) > -1. A  n  So, b y examining t h e d e r i v a t i v e o f  R w i t h r e s p e c t t o A-^ i t i s shown t h a t a t R = 1, dR/dA t h a t 0 > A > -1.  1  > 0; p r o v i d e d  Thus, as A-^ i n c r e a s e s , R c a n o n l y c r o s s t h e boundary  R = 1 from below, s o t h a t once a r o o t l e a v e s t h e s t a b i l i t y r e g i o n i t cannot r e e n t e r . ifi Again,  l e t z = Re  , R > 0, and s u b s t i t u t e i n t o e q u a t i o n (37).  S e p e r a t e r e a l and imaginary  R  From t h e s e e q u a t i o n s  parts to obtain  = ARcos(ne) + AA cos{(n+1)6}  (40)  0 = ARsin(nG) + AA^sin{(n+1)6}.  (41)  n + 1  1  i t follows that  R = -A in{(n+1)0}/sin(n0)  (42)  lS  and A  n  = (-D^Asi^^ejsinOJ/sin" *" ! 7  D i f f e r e n t i a t e each s i d e o f e q u a t i o n  1  (n+1) 6}.  (43)  (43) w i t h r e s p e c t t o 6 and equate  to obtain  (n/A )(dAj/dO) = n c o t ( n 6 ) + cot(6) - (n+1) cot{(n+1)6}. 2  2  x  Now d i f f e r e n t i a t e e q u a t i o n  (44)  (42) w i t h r e s p e c t t o 8, s o  (1/R) (dR/de) = (1/A^ (dAj/de) + (n+1)cot{ (n+1) 6} - n - c o t ( n 6 ) ,  b u t dA,/d8 c a n be found from e q u a t i o n  (44), s o t h a t  (45)  23  (iVR) (dR/de) = cot(9) - (n+l)cot{ (n+1)9}.  (46)  ifi Now  f o r R = 1, put z = e  i n t o equation (33), seperate r e a l and  imaginary p a r t s , and o b t a i n the f o l l o w i n g : cos{(n+l)0} = Acos(9) + AA-L  (47)  sin{(n+1)0} = A s i n ( 9 ) .  (48)  Considering equations (38) and  (39), i t i s e a s i l y seen t h a t  cot{(n+l)9} = -{1 - Acos(n9)}/Asin(n0), w h i l e from equations (47) and  (49)  (48) o b t a i n  cot(e) = -{1 - Acos(nG) - A A } / A s i n ( n 0 ) . 2  (50)  2  Use the i d e n t i t i e s given by equations (49) and  (50) t o determine the  f o l l o w i n g from equation (46): (n/R) (dR/d9)|  = ( n { l - Acos(n9)} + A A ) / A s i n (n9), 2  x  (51)  2  and from equation (44): (n/X ) (dA-j/dG) 1 ^ x  = ( n + 2 n { l - Acos(n0)} + A A ) / A s i n ( n 0 ) . 2  2  2  (52)  So,  /,  •  -.x  (A,/R) /n  /j,-,/-,,  ,|  (dR/dA,)l p_-|  =  2 2 n { l - Acos(n8)} +A A  -5— • -— 1 55-— n " + 2n{l - Acos(n9)} + ApT n  O> ) 3  1  /co  24  but -1 < A < 0, so 1 - Acos(ne) > 0. t h a t (Xj/R) (dR/dX-^ \^  =1  (dR/dX^ 1 >  > 0, but X  1  Thus, i t f o l l o w s from equation (53)  > 0, so i t i s proven t h a t  0 when 0 > A > -1.  As a consequence o f t h i s r e s u l t , t h e lower bound o f t h e s t a b i l i t y r e g i o n i n the (X-^A) plane can be found using equations (38) and (39). Although c l o s e d form expressions f o r ( X ^ ) do not seem feasable f o r A  n  n ;> 3, graphs o f t h e lower bound can e a s i l y be p l o t t e d f o r any n >: 2 (see Figure 5). Note t h a t IT/(n+1) < 9 ( X ) < nyn, so n 0 ( X ) n  as n  1  +°°. Thus as expected, f o r any X^ > 0,  n -> +°°.  n  A  1  TT  ( ^ ) + V ( 1 + ^j) as -  n  Some s p e c i f i c examples o f the d e n s i t y dependent reproductive  f u n c t i o n F(P) w i l l now be discussed, applying some o f the r e s u l t s t h a t have been derived.  Example 1. As a f i r s t example, consider a q u a d r a t i c , o r l o g i s t i c reproductive f u n c t i o n .  type,  Suppose t h a t f (P) = r P ( l - P/K),  (54)  where r i s the average fecundity per a d u l t and K i s t h e c a r r y i n g c a p a c i t y o f t h e a d u l t population. The e q u i l i b r i u m population, P*, o f equation P* = K ( l - B ^ / r ) , n  where  (16) i s given by (55)  25  26  V  v  =  Y  r  \  (  1  +  V-  (56  Note t h a t f o r P* > 0, i t i s r e q u i r e d t h a t r > B . ^ m,n  >  Recall that  A = a-jF' (P*) , so i t i s : e a s i l y seen t h a t  A =  (2 - r / B ) / ( l + A ). ' ir^n" nr  (57)  The s u f f i c i e n t c o n d i t i o n f o r l o c a l s t a b i l i t y o f P*, e q u a t i o n  (23), i n  t h i s example becomes  B  and  m,n m  < r < 3B  m,n  m  ,  (58)  f o r t h e s p e c i a l c a s e o f m = 1, i t f o l l o w s from e q u a t i o n  (36) t h a t  t h e r e e x i s t s a p o s i t i v e number e (B, ) such t h a t P* i s s t a b l e i f and n l,n c  only i f  B  l,n  < r < 3B, + e . l,n n  a r e s u l t v e r y s i m i l a r t o one d e r i v e d by C l a r k ,  (59)  1976b.  Example 2. C o n s i d e r i n t h i s example a r e p r o d u c t i v e exponential,  o r Ricker,  function that i s of  type, i . e . , f ( P ) = r P exp (-P/K).  Now,  the equilibrium population,  P*, o f e q u a t i o n  P* = K l o g ( r / B ^ ) , n  (60)  (16) i s g i v e n by  ^  (61)  27  where B i s a s i n Example 1. m,n required that r > B m,n*  Again,, f o r a f e a s i b l e e q u i l i b r i u m i t i s  I n t h i s example  A = {1 + l o g ( B  i n  ^/r)}/(l + A J ,  (62)  so t h a t t h e s u f f i c i e n t c o n d i t i o n f o r l o c a l s t a b i l i t y o f P* becomes  K  As  l  o  9  (  r  )  i n t h e f i r s t example, e q u a t i o n  number e (B,  <  2  +  lo  (63)  S m,n>(B  (36) i m p l i e s t h a t t h e r e i s a p o s i t i v e  ) such t h a t P* i s l o c a l l y s t a b l e i f and o n l y i f  lpg(B  1  ) < log(r) < 2 + I p g ^  (64)  ) + e . n  T h i s c o n c l u d e s t h e a n a l y s i s o f l o c a l s t a b i l i t y f o r t h e model e q u a t i o n (15).  Following  a r e s e v e r a l computer s i m u l a t i o n s o f t h e model, where  t h e d e n s i t y dependent r e p r o d u c t i o n 1.  f u n c t i o n f (P.) was chosen from example  Parameter v a l u e s were chosen a s l i k e l y e s t i m a t e s t o g i v e some  i n d i c a t i o n o f the very complicated structure.  dynamics a r i s i n g from t h e b i f u r c a t i o n  V e r y s i m i l a r f i g u r e s may be found i n t h e paper b y L e v i n and  Goodyear, 1980.  Computer The  simulations. s i m u l a t i o n s p r e s e n t e d h e r e employ t h e r e p r o d u c t i o n  function  f(Xj) = r X j ( 1 - Xj) o f Example 1, where t h e c a r r y i n g c a p a c i t y , K, has been s c a l e d o u t ,  i . e . , P. = Kx.. 3  The f i g u r e s a l l come i n p a i r s , t h e  28  Figure  6a.  Simulation of the modeled population f o r n = 1 and m = 1. E q u i l i b r i u m i s x* = 0.695 and i s s t a b l e .  29  Figure 6b.  Simulation of the model f o r n = 1 and m = 1, showing a p l o t of y e a r l i n g s versus age c l a s s two.  30  f i r s t o f each p a i r shows t h e dynamics o f t h e a d u l t b r e e d e r and t h e second shows o n e - y e a r - o l d s  plotted against  two-year-olds.  I n t h e f i r s t s i m u l a t i o n , shown i n F i g u r e s 6a and w i t h m = 1 and n = 1 i s c o n s i d e r e d . a-^ = 0.2,  O2 = 0.4,  and r = 20.0.  A = -0.589, A-j^ = 1.171, and •B  1  s u f f i c i e n c y range o f e q u a t i o n as g i v e n i n F i g u r e 3.  population  6b, a s i m p l e  Parameters chosen were:  case  a-^ =  0.7,  Thus t h e e q u i l i b r i u m i s x* = 0.695, = 6.098.  The r v a l u e i s o u t s i d e  the  (58) b u t w i t h i n t h e s t a b i l i t y bounds  C l e a r l y , the e q u i l i b r i u m i s stable, but  the  approach t o i t i s o s c i l l a t o r y . F i g u r e s 7a and  7b show t h e dynamics f o r n = 1 and m = 4, where t h e  parameters c h o s e n were: = 0.2,  a  2  =  = 0.05,  ••• = CT5 = 0.5,  c a s e i s x* = 0.691  = 0.3,  = 0.6,  and r = 50.0.  w h i l e A = -0.192, A  4  =  0.9,  The e q u i l i b r i u m i n t h i s  = 5.493, and B  4  1  = 15.401.  A g a i n , t h e r v a l u e i s j u s t o u t s i d e t h e s u f f i c i e n c y range o f e q u a t i o n  (58),  b u t t h e e q u i l i b r i u m i s s t a b l e , though t h e approach t o i t i s o s c i l l a t o r y . F i g u r e s 8a and  8b show t h e dynamics o f a p o p u l a t i o n e x a c t l y t h e  same as f o r F i g u r e s 7a,b  except f o r the r value, which i n t h i s case i s  t a k e n f u r t h e r o u t s i d e t h e range o f e q u a t i o n Here x* = 0.779, A = -0.392, w h i l e A^ and B  (58), namely, r a r e unchanged.  4  70.0There  seems t o b e no p e r i o d t o t h e o s c i l l a t i o n s o f F i g u r e 8a, w h i l e i n F i g u r e 8b some age around t h e  s t r u c t u r e i s evident i n the precession o f p o i n t s  curve.  F i g u r e s 9 t h r o u g h 13 i n d i c a t e t h e dynamics f o r a p o p u l a t i o n w i t h n = 2 and m = 4.  F o r each, t h e parameters •a = 0.02, 1  a  0  = 0.3,  =  0.5,  31 o a rrvj  a o  a _  Q  O  CM  O  . CO  o  _  a a co  a  9-0  1  1  ZL'O  t?9-0  9S"0  NonuindOd 3NI033cig  r-  iinatj  Figure 7a. Simulation o f the model f o r n = 1 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.691 and i s s t a b l e .  a Q  *'0  32  Figure 7b.  Simulation of the model f o r n = 1 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two.  33  Figure  8a.  Simulation of the model f o r n = 1 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.779 and i s not s t a b l e .  34  Figure 8b.  Simulation of the model f o r n = 1 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two.  35  = 0.8, a-^ = 0.2, and a  2  = * * * = o"g = 0.4 remain unchanged.  The r  value i s i n c r e a s i n g though, through the values r = 100.0, 190.0, 200.0, 225.0, and 240.0 i n succeeding f i g u r e s .  For Figures 9 and 10, r i s  w i t h i n the s t a b i l i t y region, w h i l e i t appears t h a t i n Figure 11 there has been b i f u r c a t i o n t o q u a s i - p e r i o d i c o s c i l l a t i o n s o f about seven years duration.  I n Figure l i b the c y c l i c nature o f the age s t r u c t u r e f o r one-  and two-year-olds i s evident.  Figure' 12a i n d i c a t e s a p e r i o d o f  approximately f i f t y years, w h i l e Figure 12b shows an i n c r e a s i n g complexity i n age s t r u c t u r e . F i n a l l y , Figures 13a and 13b r e f l e c t an r value deep i n t o the region of i n s t a b i l i t y .  An a n a l y s i s of the complicated  b i f u r c a t i o n s t r u c t u r e of the model i s beyond the scope of t h i s t h e s i s , but i t i s hoped t h a t these f i g u r e s give some idea o f t h i s complex behaviour.  T  0' I  0  9*0  NQIlbYldQd  3Nia33°eJg  Figure 9a. Simulation o f the model f o r n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.410 and i s s t a b l e .  37  UD  CO  in  S010 Figure 9b.  cJU3A  0/U  Simulation o f the model f o r n = 2 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two. (r = 100.0)  38  Figure  10a.  Siinulation of the model f o r n = 2 and m = 4 showing parental population versus time. E q u i l i b r i u m i s x* = 0.689 and i s s t a b l e .  39  Figure 10b.  Simulation of the model f o r n = 2 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two. (r = 190-0)  40 o a  r  CM  a a CM  a a _ a CM  Q  _ O CO  a  0" I  Figure 11a.  I  9*0  1— 9*0  NOIlbindOd 9NIQ33°cjg  2"0  iinau  O'O  Simulation o f the model f o r n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.705 and i s not s t a b l e .  41  Figure l i b .  Simulation of the model w i t h n = 2 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two. (r = 200.0)  42  Figure 12a.  Simulation of the model w i t h n = 2 and m = 4 showing p a r e n t a l population versus time. E q u i l i b r i u m i s x* = 0.738 and i s not s t a b l e .  43  Figure 12b.  Simulation o f the model w i t h n = 2 and m = 4 showing a p l o t o f y e a r l i n g s versus age c l a s s two. (r = 225.0)  44  Q  • 00 CM  CM  o o _ o CM  )  0"!  1  1  1  8'0  9'0  v"0  NOIlUTldOd Figure  13a.  9NIQ33dg  o o  r  Z'O  linati  O'O  S i m u l a t i o n o f t h e model w i t h n = 2 and m = 4 showing p a r e n t a l p o p u l a t i o n v e r s u s time. E q u i l i b r i u m i s x* "= 0.754 and i s n o t s t a b l e .  45 o  Figure 13b.  Simulation of the model w i t h n = 2 and m = 4 showing a p l o t of y e a r l i n g s versus age c l a s s two. (r = 240.0)  46  CHAPTER IV  OPTIMALITY  In t h i s s e c t i o n , the problem of economically optimal e x p l o i t a t i o n of a population modeled by the d i f f e r e n c e - d e l a y equation (13) i s discussed.  The method o f Lagrange m u l t i p l i e r s i s used t o d e r i v e  formally an optimal e q u i l i b r i u m c o n d i t i o n . Suppose t h a t the a d u l t breeding population modeled by equation i s subject t o e x p l o i t a t i o n , where H^. i s the number of animals  (13)  harvested  i n year k, so t h a t m \  =  + 1  a  j + 1  F(P  k + 1  _ _ n  j  - H^^.)  ,  (65)  1c — 0/ 1 ^ 2/ ••• * Clearly,  must s a t i s f y the f e a s i b i l i t y conditions  0  i  \  = k' P  k  Given a harvest p o l i c y (H , H-^, H , Q  (P-^, P , 2  2  = °/ !/ / 2  ••• •  (66)  . . . ) , the future stock l e v e l s  •••) are determined by-equation  (65) i f the h i s t o r i c a l escapement  levels S_  k  = P_  k  - H_ ,  k = 1, 2, ... , n+m-1  k  and the i n i t i a l stock l e v e l , P , are known. Q  (67)  L e t TI = II(P,H) be the net  economic revenue r e s u l t i n g from a harvest H taken from an a d u l t population  47  of s i z e P.  Following C l a r k , 1976a, assume t h a t net economic revenue  i n year k i s rk P  n(P ,H ) = J k  {p - C(P)}dP,  k  (68)  V k H  where p = p r i c e and C(P) = u n i t harvest c o s t when the population l e v e l i s P.  I n the usual f i s h e r y - p r o d u c t i o n model (Clark 1976b), C(P) = c/P,  where c i s a constant.  I t i s assumed t h a t the maximization o f the  discounted present value o f n e t economic y i e l d 00  J =  £ 6 n(P,,H )  (69)  k  v  k=0  k  K  Here, 6 = 1/(1 + i ) i s the annual  i s the o b j e c t i v e o f e x p l o i t a t i o n .  discount f a c t o r , where i = i n t e r e s t r a t e . An optimal e q u i l i b r i u m c o n d i t i o n i s now derived using the method o f Lagrange m u l t i p l i e r s (Clark 1976a, 1976b).  Consider the Lagrangean  expression: L =  °° k Z (6 n(P ,H ) - ^{P k=0 K  k  k  m  -  Z  j=0  a  l F  (P  k + 1  _  n  j  - H  k  )}). J  (70)  Ignoring c o n s t r a i n t s , the necessary conditions a r e : 3L/3P = 0  (k > 1)  (71)  dL/dE^ = 0  (k > 0).  (72)  k  and  /Assume there i s an e q u i l i b r i u m s o l u t i o n t o these conditions w i t h  48  escapement l e v e l S, so t h a t P  m E a. F(S) and H, = P, - S.  = P =  D+l  J_Q  K  K  I t f o l l o w s from equations (71) and (72) then t h a t k  6n  m  p  +  F'(S)_E a o  k_  j  +  l  W  l  +  j  =  ? k  _  (73)  1  m  6\-F'(S)_E  a  o  j  +  l  W  l  +  p  + n ).  = 0,  j  (74)  f o r k = 1, 2, ... . Thus, ?  k  = 6  k + 1  (n  (75)  R  Hence, i t f o l l o w s from equations (75) and (73) i = {(n  p  m + n )/n }F'(s) ^ H  H  that  . 5  a  j + 1  .  (76)  The optimal e q u i l i b r i u m escapement l e v e l may now be found from equation (76) when the functions H(P,H) and F(S) are chosen.  49  CHAPTER V  • DISCUSSION  In this section, the research w i l l be examined i n light of the results obtained herein and some conclussions w i l l be drawn. Extrapolations w i l l be considered whereby this work might be extended or generalized. A model has been constructed which describes a species i n which a member may reproduce i n only one age class, the ages ranging from n to n+m.  The natural survival rates from age j to j+1 were assumed to be  density independent, i n fact, a l l density dependence was assumed to be concentrated i n the f i r s t year of l i f e .  Thus, a difference-delay  equation, with m delays, describing the adult breeding population was produced. A possible extension of this study would be a generalization to allow the survival rates to also be density dependent. Stability of the unexploited model was studied next and, for general n and m, a simple condition sufficient for stability of an equilibrium was derived.  This condition was shown to provide the  upper bound for the region of local stability i n the (A^,A) parameter plane, where A  m  reflects the effect of the l i f e parameters of the  species while A i s a measure of the slope of the reproductive function at equilibrium. From this result, i t can be concluded that for fixed A positive, increasing A  m  reduces stability.  The lower bound of this  stability region, for arbitrary n and m, was seen to be more intractable. However, a numerical scheme was presented for determining stability,  50  when s p e c i f i c parameters are given. Two s p e c i a l cases f o r m were analyzed i n d e t a i l and some closed form expressions were derived f o r the lower bound.  I t was c l e a r l y  seen here that i n c r e a s i n g the delay leads t o reduced s t a b i l i t y and, except f o r the case n = 1, f o r f i x e d A, i n c r e a s i n g p o i n t out o f t h e s t a b l e region.  always takes a  For t h e case n = 1 and m = 1 however,  there i s a range o f values f o r A (-2 < A < -1) where i n c r e a s i n g  from  zero takes a p o i n t from i n s t a b i l i t y t o s t a b i l i t y and back t o i n s t a b i l i t y . Two examples f o r t h e density dependent reproductive f u n c t i o n employing commonly used functions were e x h i b i t e d .  A study o f the g l o b a l behaviour  of the model and a d e t a i l e d examination of the b i f u r c a t i o n s t r u c t u r e i s beyond the scope o f t h i s t h e s i s , however, some i n t e r e s t i n g computer s i m u l a t i o n p l o t s were given, showing the complicated dynamics p o s s i b l e w i t h the model and i n d i c a t i n g t h a t the b i f u r c a t i o n s t r u c t u r e i s indeed complex. F i n a l l y , the e x p l o i t e d population model was considered again, and a c o n d i t i o n derived, g i v i n g the optimal e q u i l i b r i u m escapement l e v e l f o r the general case.  An examination o f the optimal approach paths  t o t h i s escapement l e v e l would be i n t e r e s t i n g , but, unfortunately, i s a l s o beyond the scope o f t h i s t h e s i s .  BIBLIOGRAPHY  A l l e n , R.L. and Basasibwaki. "Properties o f age s t r u c t u r e models f o r f i s h populations." J o u r n a l of the F i s h e r i e s Research Board o f Canada, 31; 1119-1125, 1974. an der Heiden, U. "Delays i n p h y s i o l o g i c a l systems." Mathematical Biology, 8: 345-364, 1979.  Journal of  Beddington, J.R. "On the dynamics of S e i whales under e x p l o i t a t i o n . " I n t e r n a t i o n a l Whaling Ggnroission, S c i e n t i f i c (Committee Report, 28: 169-172, 1978. Bellman, Richard. Introduction t o the mathematical theory of c o n t r o l processes, Volume I I . New York: Academic Press, 1971. Canon, M.D., CD. Galium, J r . , and E. Polak. Theory of optimal c o n t r o l and mathematical programming. New York: McGraw-Hill Book Company, 1970. C l a r k , CW. Mathematical bioeconomics. Sons, 1976a.  New York:  John Wiley and  C l a r k , CW. "A delayed-recruitment model o f population dynamics, w i t h an a p p l i c a t i o n t o baleen whale populations." Journal of Mathematical Biology, 3: 381-391, 1976b. Diamond, P. "Domains of s t a b i l i t y and r e s i l i e n c e f o r b i o l o g i c a l populations obeying d i f f e r e n c e equations." Journal of T h e o r e t i c a l Biology. 61: 287-306, 1976. Freeman, H. Discrete-time systems. 1965.  New York:  John Wiley and Sons,  Goh, B.S. and T.T. Agnew. " S t a b i l i t y i n a harvested population w i t h delayed recruitment." Mathematical Biosciences, 42: 187-197, 1978. L e v i n , S.A. and C P . Goodyear. "Analysis of an age-structured f i s h e r y model." J o u r n a l of Mathematical Biology, 9: 245-274, 1980. L e v i n , S.A. and R.M. May. "A note on d i f f e r e n c e - d e l a y equations." T h e o r e t i c a l Population Biology, 9: 178-187, 1976. L i , T.-Y. and J.A. Yorke. "Period three i m p l i e s chaos." Mathematical Monthly, 82: 985-992, 1975.  American  52  May, R.M. S t a b i l i t y and complexity i n model ecosystems. Princeton U n i v e r s i t y Press, 1973.  Princeton:  May, R.M. " B i o l o g i c a l populations w i t h nonoverlapping generations: s t a b l e p o i n t s , s t a b l e c y c l e s , and chaos." Science, 186: 645647, 1974. May, R.M. "Simple mathematical models w i t h very complicated dynamics." Nature, 261: 459-467, 1976. May, R.M. "Simple models f o r s i n g l e populations: an annotated b i b l i o g r a p h y . " F o r t s c h r i t t e der Zoologie, 25: 95-107, 1979. May, R.M., G.R. Conway, M.P. H a s s e l l , and T.R.E. Southwood. "Time delays, d e n s i t y dependence, and s i n g l e species o s c i l l a t i o n s . " J o u r n a l o f Animal Ecology, 43: 747-770, 1974. May, R.M. and G.F.: Oster. " B i f u r c a t i o n s and dynamic complexity i n simple e c o l o g i c a l models." American N a t u r a l i s t , 110: 573599, 1976. Maynard Smith, J . Mathematical ideas i n b i o l o g y . U n i v e r s i t y Press, 1968. P i e l o u , E.C. Mathematical ecology. Sons, 1977.  2nd ed.  London:  New York:  Cambridge  John Wiley and  R i c k e r , W.E. "Stock and recruitment." J o u r n a l o f the F i s h e r i e s Research Board o f Canada, 11: 559-623, 1954. R i c k e r , W.E. "Ocean growth and m o r t a l i t y o f pink and chum salmon." J o u r n a l o f the F i s h e r i e s Research Board o f Canada, 21(5): 905-931, 1964. Ricker, W.E. "Changes i n the age and s i z e o f chum salmon (Oncorhynchus k e t a ) . " Canadian Technical Report o f F i s h e r i e s and Aquatic Sciences, No. 930, 1980. Ross, G.G. "A note on dynamics o f populations w i t h history-dependent b i r t h r a t e . " B u l l e t i n o f Mathematical Biology, 40: 123-131, 1978. Solomon, M.E. Population dynamics. L i m i t e d , 1969.  London:  Edward Arnold (Publishers)  Williamson, M. The a n a l y s i s o f b i o l o g i c a l populations. Edward Arnold (Publishers) L i m i t e d , 1972.  London:  

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