UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1981_A6_7 C54.pdf [ 2.39MB ]
Metadata
JSON: 831-1.0080135.json
JSON-LD: 831-1.0080135-ld.json
RDF/XML (Pretty): 831-1.0080135-rdf.xml
RDF/JSON: 831-1.0080135-rdf.json
Turtle: 831-1.0080135-turtle.txt
N-Triples: 831-1.0080135-rdf-ntriples.txt
Original Record: 831-1.0080135-source.json
Full Text
831-1.0080135-fulltext.txt
Citation
831-1.0080135.ris

Full Text

A MULTIPLE AGE CLASS POPULATION MODEL WITH DELAYED RECRUITMENT by JOSEPH LOUIS CHUMA B.Sc, The University of 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 i n THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1981 (c) Joseph Louis Chuma, 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l 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 . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department 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 . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Joseph Louis Chuma Department o f Mathematics  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date August 14, 1981 ABSTRACT An exploited single-species population model with a density dependent reproductive function i s constructed, i n which recruitment to the adult breeding population may occur i n one of several possible age classes. The parent i s assumed capable of giving b i r t h only once. I t i s also assumed that 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 analysis of the multiply-delayed difference equation model i s carried out and a s u f f i c i e n t condition 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 indication of the complicated bifur c a t i o n structure of the model i s given by a series of computer simulation p l o t s . F i n a l l y , the method of Lagrange m u l t i p l i e r s i s used to f i n d the optimal equilibrium escapement l e v e l f o r the o r i g i n a l exploited population model. Colin W. Clark F. Y. M. Wan TABLE OF CONTENTS i i i Page. Abstract . ^ i Table of Contents i i i L i s t of Figures XY Acknowledgement * Y"!" Dedication • v i l Chapter I - INTRODUCTION 1 I I - THE MODEL 9 I I I - STABILITY .. 13 Special case: m = 0 17 Special case: m = 1 18 Example 1 . 24 Example 2 . 26 Computer simulations 27 IV - OPTIMALITY 46 V - DISCUSSION 49 BIBLIOGRAPHY 51 i v LIST OF FIGURES Figure Page 1 A schematic representation of the model f o r the speci a l case m = 1 10 2 The intersection of z n + m and A ( z m + X^z 1" -^ + ••• + X m) gives a r e a l root z^ > 1, when A > 1/(1 + X^ + • • • + A m) ... 15 3 The (X^,A) region of l o c a l s t a b i l i t y (shaded area) for n = 1 and m = 1 19 4 The (X^,A) region of l o c a l s t a b i l i t y (shaded area) fo r n = 2 and m = 1 20 5 A pl o t of A (X,) f o r m = 1 and n = 2, 3, 4, and 5. Note that A^(Xp + -1/(1 + X^ as n -+ +00 25 6a Simulation of the modeled population f o r n = 1 and m = 1. Equilibrium i s x* = 0.695 and i s stable 28 6b Simulation of the model f o r n = 1 and m = 1, showing a p l o t of yearlings versus age class two 29 7a Simulation of the model f o r n = 1 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.691 and i s stable 31 7b Simulation of the model f o r n = 1 and m = 4 showing a p l o t of yearlings versus age class two 32 8a Simulation of the model f o r n = 1 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.779 and i s not stable 33 8b Simulation of the model f o r n = 1 and m = 4 showing a p l o t of yearlings versus age class two 34 9a Sbmilation of the model f o r n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.410 and i s stable 36 V Figure Page 9b Simulation of the model f o r n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 100.0) 37 10a Simulation of the model f o r n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.689 and i s stable 38 10b Simulation of the model f o r n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 190.0) 39 11a Simulation of the model f o r n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.705 and i s not stable 40 l i b Simulation of the model with n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 200.0) 41 12a Simulation of the model with n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.738 and i s not stable 42 12b Simulation of the model with n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 225.0) 43 13a Simulation of the model with n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.754 and i s not stable 44 13b Simulation of the model with n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 240.0) ...... 45 ACKN0WLEDGEME3S1T I wish to thank Michael Bailey f o r suggesting the idea which became the basic model. I would also l i k e to express my appreciation to Dr. Colin Clark f o r h i s comments and h i s help i n preparing t h i s thesis. DEDICATION This thesis i s dedicated to Dr. Pauline van den Driessche, who provided me with i n s p i r a t i o n , encouragement, and the s k i l l s to pursue mathematical research. 1 CHAPTER I INTRODUCTION For many species of organisms, population growth i s discontinuous. The l i f e h i s t o r y of such organisms may be subject to strong seasonal or periodic influences. Also, f o r many species, recruitment to the breeding stock may only occur several months or 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 die before t h e i r descendents' l i v e s begin, f o r example, salmon or cicadas. To represent the population growth of such species, the only suitable model i s a difference, or difference-delay equation (Maynard Smith 1968, May 1973, Clark 1976a, Pielou 1977). Consider a b i o l o g i c a l population (P^ i n generation k) which has discrete and nonoverlapping generations. The population dynamics may be described by the f i r s t order difference equation P k + 1 = F ( P k ) , (1) which relates the population l e v e l P at time t = t k + ^ to the population l e v e l at a previous time t = t j , by means of some given stock-recruitment function, F(P). In most cases of i n t e r e s t , the function F(P) i s nonlinear and i s usually constructed to allow the population to grow rapid l y at low densities and l e v e l o f f or possibly decline at high densities. Many e x p l i c i t forms for the density dependent function.F(P) have been proposed i n the l i t e r a t u r e , and tables of s p e c i f i c forms used, with 2 references, can be found i n May, 1979 or May and Oster, 1976. One . iitportant example i s the Ricker model (Ricker 1954) i n which F(P) = P exp { r ( l - P/K)}. This Ricker model i s used extensively i n the management of the P a c i f i c salmon (Oncorhynchus species) populations (Clark 1976a). The dynamic behaviour of solutions of equation (1) fo r t h i s case i s s u r p r i s i n g l y complicated, but now seems to be w e l l understood, see Levin and Goodyear, 1980. C l e a r l y , equation (1), with F(P) as given above, always possesses a n o n - t r i v i a l equilibrium at P* = K. I t has been shown that t h i s equilibrium i s l o c a l l y stable provided that 0 < r 4 2; but i f r i s increased beyond the value 2, the equilibrium becomes unstable and a new stable l i m i t - c y c l e of period 2 bifurcates from the equilibrium. In f a c t , there i s an increasing sequence 2 = r-^ < 2.526 = < ••• such that when r increases past r n a new and stable cycle of period 2 n bifurcates from the equilibrium. 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 cycles of a r b i t r a r y period appear along with solutions that never s e t t l e i n t o any f i n i t e cycle. The region beyond r * has been c a l l e d dynamic chaos since solutions are e f f e c t i v e l y indistinguishable from random fluctuations. The term "chaos" was introduced i n the mathematical paper by L i and Yorke, 1975. They show that i f equation (1) has a solution of period 3, then f o r those same parameter values, solutions of any integer period can be found, as w e l l as solutions that never s e t t l e i n t o 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 structure of equations l i k e equation (1) see May, 1976 or May and Oster, 1976. Many r e a l populations have several d i s t i n c t but overlapping age classes, or the density dependent mechanisms operate with an e x p l i c i t time delay, say n generations. In t h i s case, a difference-delay equation of the form p i 1 1 = P, F ( P , ) (2) k+1 k k-n' v ' may be appropriate. Examples of such models, where the population at k+1 depends l i n e a r l y on the previous population and nonlinearly on a single population at some time i n the past, can be found i n Maynard Smith, 1968; May, Conway, Hassell, and Southwood, 1974; Clark, 1976b; and Beddington, 1978. Clark, 1976b, studied the delay equation = + F ( P ] c _ n ) ' which has applications as a model of baleen whale population dynamics. Equilibrium s t a b i l i t y and optimal ex 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 to k-n the breeding population at time k which was produced by the breeding population at time k-n. The l o c a l s t a b i l i t y of an equilibrium, P*, fo r t h i s difference-delay equation was shown to 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 of the recruitment function at the equilibrium, and the time delay i n the recruitment. Clark has shown that increased delay implies reduced s t a b i l i t y , i n the sense that increasing the delay reduces the region i n the parameter plane which possesses a stable equilibrium point. 4 Goh and Agnew, 1978, considered Clark's, 1976b, difference-delay model equation f o r a population i n which, recruitment to the breeding class takes place several generations a f t e r b i r t h . They employ a s p e c i f i c form for the recruitment function, F(P), namely, F(P) = 2 AP exp (-BP ), where A and B are pos i t i v e constants. I t i s shown that increasing the delay causes reduced s t a b i l i t y , while increasing the su 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 small, tends to 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 effects of 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 effects of the time delay. Harvesting of the modeled population i s studied and they conclude that, for populations that e x h i b i t a sharp peak i n t h e i r recruitment function, intermediate levels of constant e f f o r t harvesting can lead to 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 that give r i s e to 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 single species population in t e r a c t i n g with a maintained resource were studied. Considered f i r s t were discrete generation difference equation models with density dependent mortality and fecundity. I f the rate at which a population takes to return towards an equilibrium l e v e l i s c a l l e d the ch a r a c t e r i s t i c return time, t h i s paper contends "that i t i s the relationship of t h i s time to the time delays i n the system (e.g. the length of a generation) that determines whether the population approaches the equilibrium monotonically or 'overshoots' and o s c i l l a t e s about the equilibrium" 5 (May et a l . 1974: 747). They found that i n s t a b i l i t y follows from t h i s return time being too small compared with:.the\:time delays. Consideration i s then given to multiple age class models, and a model with two overlapping age classes i s studied i n d e t a i l . Of course, the s t a b i l i t y properties of these multiple age class populations i s more complicated, but they are shown to be s i m i l a r to those of m u l t i -species systems. The l o c a l s t a b i l i t y of an equilibrium i s shown to be determined by the dominant eigenvalue of a matrix of parameters characterizing the slopes of the density dependent relationships between age classes. Diamond, 1976, presented techniques for estimating the s i z e and shape of regions of l o c a l s t a b i l i t y f o r difference equations. These techniques are based on a class of discrete Liapunov functions, and a " r e s t r i c t e d recipe" f o r finding these Liapunov functions along with 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 applied to a single species model with two age classes. Levin and May, 1976, gave s t a b i l i t y c r i t e r i a f o r the difference-delay equation (2). They presented general a n a l y t i c formulas describing the boundary between monotonic damping and o s c i l l a t o r y damping toward a stable equilibrium point, P*, and f o r the boundary seperating the regions of s t a b i l i t y and i n s t a b i l i t y of P*. As was found also by Clark, 1976b, May et a l . , 1974, 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 to be more prone to 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 presented by Levin and May, 1976, i s used i n t h i s t h e s i s . Extensions of the difference-delay model equation (2) to allow f o r m ultiple age spawning populations where density dependence i s expressed i n terms o f the population l e v e l s of both present and several preceding generations, i . e . , P, ,-, = P, F (P. , P. - , ... , P, ) , (3) k+1 k v k k-1' k-n ' have been considered by A l l e n and Basasibwaki, 1974, Ross, 1978, and Levin and Goodyear, 1980. A l l e n and Basasibwaki, 1974, studied a c l a s s of models incorporating mu l t i p l e age structure where recruitment to the population i s the product of fecundity and a s u r v i v a l r a t e . This f i r s t year s u r v i v a l r a t e was assumed to vary with the s i z e and structure of the population. The population a f t e r recruitment i s described by a l i f e t a ble with constant s u r v i v a l r a t e s . Necessary conditions f o r the s t a b i l i t y of an e q u i l i b r i u m and properties of o s c i l l a t i o n s about an unstable e q u i l i b r i u m were considered using a combination o f a n a l y t i c a l and simulation techniques. Ross, 1978, considered a s p e c i a l case of equation (3) of the form P, ,., = aP?" c . i n t h i s case, the model can be w r i t t e n as a k+1 k k-1 l i n e a r second order recurrence r e l a t i o n i n the logarithm o f the population and e x p l i c i t s o l u t i o n s to t h i s recurrence r e l a t i o n were derived and c l a s s i f i e d according to various parameter values. In the paper by Levin and Goodyear, 1980, a m u l t i p l e age spawning 7 population model with Ricker type stock-recruitment relationship was examined. Their model assumed that a l l density dependent effects occur w i t h i n the f i r s t year of l i f e . A density 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 applied to various s i m p l i f i e d models. Very complicated s t a b i l i t y properties were shown to be dependent on two opposing delays i n the system, the reproductive delay associated with deferring reproduction and the truncation delay associated with an eventual l e v e l i n g o f f of fecundity i n l a t e r age classes. The balance between these delays was shown to be at the root of the o v e r a l l dynamics of the system. Computer simulations found i n the paper of Levin and Goodyear, 1980, are very s i m i l a r to figures which may be found i n t h i s t h e s i s , and indicate some of the spectacular dynamics which can occur when an equilibrium i s i n a region of i n s t a b i l i t y . This thesis examines an exploited single-species population model with a density dependent reproductive function, i n which recruitment to the breeding population occurs i n one of m+1 (m = 0, 1, 2, ...) possible ages. In t h i s model however, the adult breeder can give b i r t h only once, and then dies. Harvesting i s assumed to occur only among the breeder population. An example of a species whose chara 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 to C a l i f o r n i a . The young chums quickly go to 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 mainly a f t e r three, four, o r f i v e growing seasons i n the ocean, although a very small percentage w i l l reproduce i n t h e i r second o r s i x t h year of l i f e . A l so, 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 rate seems to 'remain f a i r l y constant, that i s , a f t e r the f i r s t year of l i f e . Chum salmon return to c o a s t a l regions only as they are approaching maturity, so that there i s no harvesting o f chums i n l o c a l waters one o r more years before maturity (Ricker 1980). A model equation f o r the adult breeding population o f the form R = F CP. , p. ., ... , P. ) (i k v k-n' k-n-1 k-n-nv i s constructed f i r s t , where there are m+1 p o s s i b l e ages f o r breeding, n being the f i r s t p o s s i b l e age. Following t h i s i s an analysis of 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 s o l u t i o n , P*. A s u f f i c i e n t c ondition f o r l o c a l s t a b i l i t y of P* i s derived f o r general n and m, while necessary and s u f f i c i e n t conditions are examined i n d e t a i l f o r the s p e c i a l case of m = 1. Note that the case m = 0 reduces to Clark's, 1976b, model with s u r v i v a l c o e f f i c i e n t equal to zero. Considered next are a few s p e c i f i c examples f o r the density dependent reproductive function. F i n a l l y , the problem o f economically optimal e x p l o i t a t i o n of a population modeled by an equation s i m i l a r t o (4) i s considered and a formula determining the optimal 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 single-species population model with a density dependent reproductive function w i l l be constructed. Characteristics of the species to be modeled include recruitment to the mature adult (female) breeding population i n one, and only one, year of l i f e . Assume there are m+1 possible breeding ages, with n being the f i r s t possible age of reproduction. A member of t h i s species dies immediately a f t e r giving b i r t h . Assume also that the period of years from b i r t h to maturity i s spent w e l l away from the spawning area and only mature members are. subject to harvesting as they return. F i x n and n+m as the f i r s t and l a s t possible ages f o r reproduction. Let P^ represent the parental, or adult breeding, population i n year k, while Q. , represents the population of j-year-olds i n year k. 1 / K I f i s the number of 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. Let a.. (0 < a.. ^  1) be the proportion of (n+j-1) -year-olds that reproduce, j = 1, 2, ... , m+1. Since n+m i s the l a s t possible breeding age, i t follows that = 1. Thus, i t i s possible to write: 10 •H k+1 'k+1 ^k+l Figure 1. A schematic representation of the model f o r the special case m = 1. k l^n,k 2 vn+l,k mrn+m-l,k vn+m,k Now, i f cjj i s the density independent natural s u r v i v a l rate from age j-1 to age j (0 < < 1, j = 1, 2, '... , n+m), then the following equations describe the age classes of the population: Ql,k+1 = o±f(Sk) (6) Qi,k+1 (7) Qn+j,k+1 = V j ( l - aj ) Qn+j-l,k' ^ = 1, 2, , m (8) where the term f ( S k ) i s the density dependent reproductive function. I t follows now from equations (5) and (8) that 11 Pk = alQn,k + a2 an +l ( 1- al ) Qn /k-l + a3an+2 ( 1 " a 2 } V l (1"al)Qn,k-2 +  + Vn+nhl(1-Vl )W2 (1-V2)"-Vl(1-al)Qn,k-rt + an+m(1-°m)an+m-l(1-Vl)'' * Vl ( Hl ) Qn,k- m' or in more compact notation m J P k = a l Q n , k + . Z 1 { a j + l Q n , k - j V I I 1 V i ( 1 " a J K (9) j = l J J i = l I t can also be e a s i l y seen that the population of n-year-olds s a t i s f i e s the following: Qn,k = ( a n V l a l ) f ( S k - n } - ( 1 0 ) For some s i m p l i c i t y of notation, define F(x) = ( a ^ ••• a n ) f ( x ) , (11) so that Qn,k +n = F< Sk>' ( 1 2 ) Thus i s obtained the difference-delay equation providing the population dynamics f o r the adult breeding population: m P.. = Z a.. 1F(S 1 • ) , (13) k+n j = 0 j+1 k-3" where a l = a l and 12 j - l n 1=1 a. = ct. U a n + i ( l - cu), j = 2, 3, ... , m+1. (14) 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 for an equilibrium solution of equation (13) with no harvesting, i.e., k k 13 CHAPTER I I I STABILITY S t a b i l i t y properties of an e q u i l i b r i u m s o l u t i o n , P*, of the model equation with no harvesting m P k + n = \ W ^ V j J ( 1 5 ) 3=0 are analyzed i n t h i s section. Considered f i r s t w i l l be a s u f f i c i e n t c o n d i t i o n f o r the l o c a l s t a b i l i t y o f P* and a numerical scheme f o r d e f i n i n g the boundaries of the l o c a l s t a b i l i t y regions w i l l be sketched. A d e t a i l e d examination of l o c a l s t a b i l i t y w i l l be presented f o r the t r i v i a l case m = 0 and the not so t r i v i a l case of m = 1. Let 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 the delay equation (15). Then m P* = F(P*) 2 a. ,. (16) j=0. 3 + ± Now l i n e a r i z e about P* by w r i t i n g P^ = P* + x^ ,, so that m P k + n " P * = \ ^ - K L ^ k - j ) " F < P * » 3=0 (17) and so m \ + n = j f o ( a j + 1 F ' ( P * ) x k _ j + 0 ( x ^ _ j ) ) , (18) where F'(P*) i s the d e r i v a t i v e of F evaluated a t P*. Define 14 A = a^F'(P*) (19) and Xj = *j+1Mr (20) Thus, the l i n e a r i z a t i o n i s obtained: m 1=1 J (21) k Express as z x^ to obtain the c h a r a c t e r i s t i c equation m zn+m_ A ( z m + z x _ ^ n - 3 ) = Q > ( 2 2 ) j = l ^ I t i s e a s i l y seen that P* i s l o c a l l y stable i f and only i f a l l roots o f the c h a r a c t e r i s t i c equation (22) have modulus |z| < 1. F i r s t i t s h a l l be proven that the c o n d i t i o n |A| < V d + A 1 + X2 + ••• +\ m) (23) 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 of P*. , m Let g(z) = z n m and l e t h(z) = - A ( z m + E X.z^"3). Suppose that j = l 3 | A | < 1/(1 + X 1 + X2 + '" + A ); then the following i s true: |h(z) | = | A | | z m + X^1 + + A j . So that |h(z) | < | A | (|z| m + A 1 | z | m - 1 + ••• + .\) , and so f o r |z| = 1: |h(z)| < | A | ( 1 + X± + ••• + A m) < 1 = |g(z) |. By Rouche's theorem, g(z) and g(z) + h(z) have the same number of zeros (namely n+m) i n the i n t e r i o r of the u n i t c i r c l e . Thus, the c o n d i t i o n (23) 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 of P*. 15 1 z o Figure 2. The intersection of z n + m and A(z r a+X 1z m~ 1+—+A m) gives a r e a l root z n > 1, when A > 1/(1+A,+-•-+A). Now suppose that A > 1/(1 + A^ + *•• + A ), and consider the n+m m m-' graphs of z n m and A ( z m + Z A.zm_-') , where z e R (see Figure 2) . j= l J Clearly, the interse c t i o n of these graphs always gives a r e a l root ( > 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 to introduce the notion of 'region of s t a b i l i t y 1 , but f i r s t more notation i s needed to f a c i l i t a t e ease m of presentation. Let A ^ = I A.., then i t i s possible to speak of the parameter plane (A^A). A (A^A) region of l o c a l s t a b i l i t y w i l l be the set of parameter values f o r which P* i s l o c a l l y stable. From the preceeding discussion i t i s possible to conclude that, for any n, the upper bound of the (Am,A) region of l o c a l s t a b i l i t y i s always A = 1/(1 + A^). By equation (23), the lower bound, A = A n ( A 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 A2/ , A m ) , may be numerically determined from the c h a r a c t e r i s t i c equation (22). Let G(z) = c z q + c , z q - 1 + ••• + c,z + c_ (25) q q-1 1 0 — I cr —1 where c^ > 0. Define the inverse polynomial G (z) = z^G(z ), then G - 1(z) = c Q z q + c ^ - 1 + ••• + c - j Z + c . (26) The roots of G~"'"(z) are the inverses of the roots of G(z) with respect to the c i r c l e |z| = 1. In addition, ( G ^ C z ) ) " 1 = G(z). Let G~ 1(z)/G(z) = B Q + G^ 1(z)/G(z). (27) The remainder, G~ 1(z), w i l l be a polynomial of degree q-1 and the quotient term Bg = CQ/C^. Continue i n t h i s way: G^ 1(z)/G i(z) = 6 ± + GTj 1(z)/G i(z), (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 condition that a l l roots of the equation G(z) = 0 l i e i n the i n t e r i o r of the u n i t c i r c l e i n the z-plane 17 i s that a l l of the following are s a t i s f i e d : (a) G(l) > 0 (b x) G(-l) < 0 f o r q odd (h^) > 0 f o r q even (c) | B i | < 1, i = 0, 1, 2, ... , q-2. Application of t h i s method with G(z) replaced by the c h a r a c t e r i s t i c equation (22) i s straightforward when parameter values are known, however i t does not seem feasable to obtain closed form expressions for the s t a b i l i t y region with general n and m. One interesting r e s u l t can nevertheless be gleaned from condition (a). I f G(z) = zn+m _ A ( z m + ^ m-1 + ... + ^ ^ then condition (a) requires that A < 1/(1 + A^ + ••• + A ) = 1/(1 + A ), which i s merely the upper bound of the (A^A) region of s t a b i l i t y derived previously. This modified Schur-Cohn c r i t e r i o n w i l l also be made use of i n the detailed analysis of some s p e c i a l cases f o r m and n to follow. Special case: m = 0. In t h i s case the population being modeled i s one that has n age classes, but only the n age class reproduces. The adult breeding population is. described by P. = Q . = F (P, ) (29) k n,k k-n so that an equilibrium i s simply given by P* = F(P*). The c h a r a c t e r i s t i c 18 equation (22) becomes z n - F*(P*) = 0. Thus, i t i s c l e a r that P* i s l o c a l l y s t a b l e i f and only i f -1 < F'(P*) < 1. (30) Note that t h i s s p e c i a l case with m = 0 i s Clark's, 1976b, model with the s i m p l i f i c a t i o n of the s u r v i v a l c o e f f i c i e n t being zero. The next s p e c i a l case i s more i n t e r e s t i n g , r e q u i r i n g more rigorous a n a l y s i s . S p e c i a l case: m = 1. In t h i s case the population being modeled i s one that has n+1 age cl a s s e s , and only members of the f i n a l two can reproduce. The adult breeding population i s described by P k + 1 = a l F < P k - n + l > + a2 F( Pk-n> ( 3 1 ) so that an e q u i l i b r i u m i s given by P* = ( a x + a 2 ) F ( P * ) . (32) The c h a r a c t e r i s t i c equation (22) i n t h i s case becomes z n + 1 - Az - A A = 0. (33) From general r e s u l t s already derived i t i s known that the upper bound of the (A-^A) region of l o c a l s t a b i l i t y f o r P* i s A = 1/(1 + A 1 ) , f o r a l l n; while the lower bound depends on n and s a t i s f i e s A n ( A 1 ) < -1/(1 + X±). (34) Now, closed form expressions f o r two of these lower bounds, A^(A^) and A 2 ^ i ^ ' w i l l ke derived using the modified Schur-Cohn c r i t e r i o n (Freeman 1 9 6 5 ) as presented e a r l i e r . In the case under consideration, G(z) = z - Az - A-^ A. As already stated, condition (a) requires that A < 1 / ( 1 + A-^), which i s merely the upper bound derived previously. For n even, condition (b^) requires that A ( l - A-^) < 1 ; and f o r n odd, condition (b 2) requires that A ( A X - 1 ) < 1 . For n = 1 , a l l that condition (c) requires i s that [3Q| < 1 . Since G _ 1(z)/G(z) = + (-Az11 - A ^ z + ( 1 - A 2 A 2 ) ) / G ( z ) , i t follows that 3Q = -X-jA, and since they w i l l be needed shortly i t i s found that G^-(z) = -Az 1 1 - A x A 2 z + ( 1 - A 2A 2) so that G^z) = ( 1 - A 2 A 2 ) z n -AjA z - A. Therefore, condition (c) requires that |A^A| < 1 . Combining t h i s with conditions (a) and (b 2) gives the (A^,A) region of Figure 3 . The (A^,A) region of l o c a l s t a b i l i t y (shaded area) for 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, condition (c) requires that | g Q | < 1 and |B-J < 1. Since G~ 1(z)/G 1(z) = - A / ( l - * 2A 2) + ( z ) / G ^ z ) , where G"1^) = ( - ^ A 3 / ( l - x j A 2 ) ) z n _ 1 - \±A2z + ( 1 - A 2 A 2 - A 2 / ( l - A 2A 2)) , i t follows that 3^ = - A / ( l - A-jA ). Combining t h i s with |A^A| < 1 and conditions (a) and (b^) gives the (A^,A) region of l o c a l s t a b i l i t y f o r n = 2: (1 - A + 4A 2' )/2A 2 < A < 1/(1 + A x ) (35) (see Figure 4 ) . In attempting to 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 solve a cubic equation, f o r n = 4 a quartic, etc.; hence i t i s not feasable to make use of t h i s Schur-Cohn c r i t e r i o n f o r finding closed form expressions f o r A n ( ^ ^ o r n — However, i t i s useful to note here that the s t a b i l i t y region f o r any n must contain the region f o r A Figure 4. The (A^,A) region of l o c a l s t a b i l i t y (shaded area) fo r n = 2 and m = 1. 21 n+1. I f t h i s f a c t i s combined with the s u f f i c i e n c y condition (23), the progression of lower bounds, A n ( A ^ ) , follows: A 1 ( A 1 ) « ••• < A n(A x) < T A ^ U ^ < . . . < -1/(1 + A ^ . (36) Further r e s u l t s f o r n >_ 2, employing another approach, are now derived. Following Levin and May, 1976, and Clark, 1976b, consider again the c h a r a c t e r i s t i c equation (33), z - Az - A^A = 0, with n ^  2, ifl and express z as Re , R > 0. Note that f o r A = A (A^) > equation (33) ifl must have a root z - e , i . e . , with R = 1. Rewrite equation (33) i n the form 1 = Az ~ n + Z \ l Z - { n ¥ l ) . (37) i 0 Substitute z = e and equate r e a l and imaginary parts to obtain s i n ( n e ) + A-jSin{ (n+l)fl} = 0 (38) and Acos ( n e ) + AA-jCosf (n+1) 6} = 1. (39) Equation (38) has a unique root 9 = 9 n(X 1) such that ir/(n+l) < ^ ( X j ) < Tr/n, although of course, there are i n f i n i t e l y many other roots > 6 n ( ^ ) Given some A^ > 0, equation (38) can be solved f o r 9 n(^^) and thus A = A n(X^) can be found from equation (39). I t i s shown here that t h i s process determines the lower boundary of the s t a b i l i t y region. R e c a l l that f o r n >^  2 i t has been demonstrated that t h i s lower boundary i s i n the region 0 > An(A-]_) > -1. So, by examining the d e r i v a t i v e o f R with respect to A-^  i t i s shown that a t R = 1, dR/dA1 > 0; provided that 0 > A > -1. Thus, as A-^  increases, R can only cross the boundary R = 1 from below, so that once a root leaves the s t a b i l i t y region i t cannot reenter. ifi Again, l e t z = Re , R > 0, and su b s t i t u t e i n t o equation (37). Seperate r e a l and imaginary parts to obtain R n + 1 = ARcos(ne) + AA1cos{(n+1)6} (40) 0 = ARsin(nG) + AA^sin{(n+1)6}. (41) From these equations i t follows that R = -A l Sin{(n+1)0}/sin(n0) (42) and A n = ( - D ^ A s i ^ ^ e j s i n O J / s i n " 7 * " 1 ! (n+1) 6}. (43) D i f f e r e n t i a t e each side of equation (43) with respect to 6 and equate to obtain (n/A x)(dAj/dO) = n 2cot(n6) + cot(6) - (n+1)2cot{(n+1)6}. (44) Now d i f f e r e n t i a t e equation (42) with respect to 8, so (1/R) (dR/de) = (1/A^ (dAj/de) + (n+1)cot{ (n+1) 6} - n - c o t ( n 6 ) , (45) but dA,/d8 can be found from equation (44), so that 23 (iVR) (dR/de) = cot(9) - (n+l)cot{ (n+1)9}. (46) ifi Now f o r R = 1, put z = e into equation (33), seperate r e a l and imaginary parts, and obtain the following: cos{(n+l)0} = Acos(9) + AA-L (47) sin{(n+1)0} = Asin(9). (48) Considering equations (38) and (39), i t i s e a s i l y seen that cot{(n+l)9} = -{1 - Acos(n9)}/Asin(n0), (49) while from equations (47) and (48) obtain cot (e) = -{1 - Acos(nG) - A 2A 2}/Asin(n0). (50) Use the i d e n t i t i e s given by equations (49) and (50) to determine the following from equation (46): (n/R) (dR/d9)| x = (n{l - Acos(n9)} + A 2A 2)/Asin (n9), (51) and from equation (44): (n/Xx) (dA-j/dG) 1 ^ = ( n 2 + 2n{l - Acos(n0)} + A 2A 2)/Asin(n0). (52) So, 2 2 / , /n-.x / j , - ,/-,, ,| n { l - Acos(n8)} +AnA / c o> (A,/R) (dR/dA,)l p_-| = -5— • -— 1 5-— 1 O3) • n " + 2n{l - Acos(n9)} + ApT 24 but -1 < A < 0, so 1 - Acos(ne) > 0. Thus, i t follows from equation (53) that (Xj/R) (dR/dX-^ \^=1 > 0, but X1 > 0, so i t i s proven that (dR/dX^ 1 > 0 when 0 > A > -1. As a consequence of t h i s r e s u l t , the lower bound of the s t a b i l i t y region i n the (X-^A) plane can be found using equations (38) and (39). Although closed form expressions f o r A n ( X ^ ) do not seem feasable f o r n ;> 3, graphs of the lower bound can e a s i l y be plotted f o r any n >: 2 (see Figure 5). Note that IT/(n+1) < 9 n(X 1) < nyn, so n0 n(X 1) TT as n +°°. Thus as expected, f o r any X^ > 0, A n ( ^ ) + - V ( 1 + ^j) as n -> +°°. Some s p e c i f i c examples of the density dependent reproductive function F(P) w i l l now be discussed, applying some of the resu l t s that have been derived. Example 1. As a f i r s t example, consider a quadratic, or l o g i s t i c type, reproductive function. Suppose that f (P) = r P ( l - P/K), (54) where r i s the average fecundity per adult and K i s the carrying capacity of the adult population. The equilibrium population, P*, of equation (16) i s given by P* = K ( l - B ^ n / r ) , (55) where 25 26 v V = Y r \ ( 1 + V - (56> Note that f o r P* > 0, i t i s required that r > B . R e c a l l that ^ m,n A = a-jF' (P*) , so i t i s : e a s i l y seen tha t A = (2 - r/B ) / ( l + A ). (57) ' i r ^ n " nr 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 of P*, equation (23), i n t h i s example becomes B m < r < 3B m , (58) m,n m,n and f o r the s p e c i a l case of m = 1, i t follows from equation (36) that there e x i s t s a p o s i t i v e number e (B, ) such that P* i s stable i f and c n l , n only i f B < r < 3B, + e . (59) l , n l , n n a r e s u l t very s i m i l a r t o one derived by Clark, 1976b. Example 2. Consider i n t h i s example a reproductive function that i s of exponential, or Ricker, type, i . e . , f(P) = rP exp (-P/K). (60) Now, the e q u i l i b r i u m population, P*, of equation (16) i s given by P* = K log ( r / B ^ n ) , ^ (61) 27 where B i s as i n Example 1. 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 m,n required that r > B m,n* In t h i s example A = {1 + l o g ( B i n ^ / r ) } / ( l + A J , (62) so that the s u f f i c i e n t condition f o r l o c a l s t a b i l i t y of P* becomes K l o 9 ( r ) < 2 + l oS ( Bm,n>- (63) As i n the f i r s t example, equation (36) implies that there i s a p o s i t i v e number e (B, ) such that P* i s l o c a l l y s t a b l e i f and only i f This concludes the 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 the model equation (15). Following are several computer simulations of the model, where the density dependent reproduction function f (P.) was chosen from example 1. Parameter values were chosen as l i k e l y estimates to give some i n d i c a t i o n of the very complicated dynamics a r i s i n g from the b i f u r c a t i o n structure. Very s i m i l a r f i g u r e s may be found i n the paper by Levin and Goodyear, 1980. Computer simulations. The simulations presented here employ the reproduction function f(Xj) = rXj(1 - Xj) of Example 1, where the ca r r y i n g capacity, K, has been scaled out, i . e . , P. = Kx.. The f i g u r e s a l l come i n p a i r s , the l p g ( B 1 ) < log(r) < 2 + I p g ^ ) + e n. (64) 3 28 Figure 6a. Simulation of the modeled population f o r n = 1 and m = 1. Equilibrium i s x* = 0.695 and i s stable. 29 Figure 6b. Simulation of the model for n = 1 and m = 1, showing a p l o t of yearlings versus age class two. 30 f i r s t of each p a i r shows the dynamics o f the adult breeder population and the second shows one-year-olds p l o t t e d against two-year-olds. In the f i r s t simulation, shown i n Figures 6a and 6b, a simple case with m = 1 and n = 1 i s considered. Parameters chosen were: a-^  = 0.7, a-^  = 0.2, O2 = 0.4, and r = 20.0. Thus the e q u i l i b r i u m i s x* = 0.695, A = -0.589, A-j^  = 1.171, and •B1 = 6.098. The r value i s outside the s u f f i c i e n c y range of equation (58) but w i t h i n the s t a b i l i t y bounds as given i n Figure 3. 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 to i t i s o s c i l l a t o r y . Figures 7a and 7b show the dynamics f o r n = 1 and m = 4, where the parameters chosen were: = 0.05, = 0.3, = 0.6, = 0.9, = 0.2, a 2 = ••• = CT5 = 0.5, and r = 50.0. The e q u i l i b r i u m i n t h i s case i s x* = 0.691 while A = -0.192, A 4 = 5.493, and B 4 1 = 15.401. Again, the r value i s j u s t outside the s u f f i c i e n c y range of equation (58), but the e q u i l i b r i u m i s stable, though the approach to i t i s o s c i l l a t o r y . Figures 8a and 8b show the dynamics of a population exactly the same as f o r Figures 7a,b except f o r the r value, which i n t h i s case i s taken f u r t h e r outside the range o f equation (58), namely, r - 70.0-Here x* = 0.779, A = -0.392, while A^ and B 4 are unchanged. There seems to be no p e r i o d t o the o s c i l l a t i o n s of Figure 8a, while i n Figure 8b some age structure i s evident i n the precession of points around the curve. Figures 9 through 13 i n d i c a t e the dynamics f o r a population with n = 2 and m = 4. For each, the parameters •a1 = 0.02, a 0 = 0.3, = 0.5, 31 o a r r v j a o a Q _ O CM O . CO o a _ a co 9-0 1 1 ZL'O t?9-0 NonuindOd 9S"0 3NI033cig r-iinatj a a Q *'0 Figure 7a. Simulation of the model for n = 1 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.691 and i s stable. 32 Figure 7b. Simulation of the model f o r n = 1 and m = 4 showing a p l o t of yearlings versus age class two. 33 Figure 8a. Simulation of the model for n = 1 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.779 and i s not stable. 34 Figure 8b. Simulation of the model for n = 1 and m = 4 showing a p l o t of yearlings versus age class two. 35 = 0.8, a-^  = 0.2, and a 2 = * * * = o"g = 0.4 remain unchanged. The r value i s increasing though, through the values r = 100.0, 190.0, 200.0, 225.0, and 240.0 i n succeeding figures. For Figures 9 and 10, r i s with i n the s t a b i l i t y region, while i t appears that i n Figure 11 there has been b i f u r c a t i o n to quasi-periodic o s c i l l a t i o n s of about seven years duration. In Figure l i b the c y c l i c nature of the age structure f o r one-and two-year-olds i s evident. Figure' 12a indicates a period of approximately f i f t y years, while Figure 12b shows an increasing complexity i n age structure. 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 analysis of the complicated b i f u r c a t i o n structure of the model i s beyond the scope of t h i s thesis, but i t i s hoped that these figures give some idea of t h i s complex behaviour. 0' I T 0 9*0 N Q I l b Y l d Q d 3Nia33°eJg Figure 9a. Simulation of the model for n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.410 and i s stable. 37 UD CO in S010 cJU3A 0 /U Figure 9b. Simulation of the model f o r n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 100.0) 38 Figure 10a. Siinulation of the model for n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.689 and i s stable. 39 Figure 10b. Simulation of the model f o r n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 190-0) 40 o a r CM a a CM a a _ a CM Q _ O CO 0" I I 1— 9*0 9*0 NOIlbindOd 9NIQ33°cjg 2"0 iinau a O'O Figure 11a. Simulation of the model for n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.705 and i s not stable. 41 Figure l i b . Simulation of the model with n = 2 and m = a p l o t of yearlings versus age class two. 4 showing (r = 200.0) 42 Figure 12a. Simulation of the model with n = 2 and m = 4 showing parental population versus time. Equilibrium i s x* = 0.738 and i s not stable. 43 Figure 12b. Simulation of the model with n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 225.0) 44 Q • 00 CM CM o o _ o CM ) 1 1 1 r 0"! 8'0 9'0 v"0 Z'O O'O N O I l U T l d O d 9NIQ33dg linati Figure 13a. Simulation of the model with n = 2 and m = 4 showing parental population versus time. E q u i l i b r i u m i s x* "= 0.754 and i s not st a b l e . o o 45 o Figure 13b. Simulation of the model with n = 2 and m = 4 showing a p l o t of yearlings versus age class two. (r = 240.0) 46 CHAPTER IV OPTIMALITY In t h i s section, the problem of economically optimal ex p l o i t a t i o n of a population modeled by the difference-delay equation (13) i s discussed. The method of Lagrange m u l t i p l i e r s i s used to derive formally an optimal equilibrium condition. Suppose that the adult breeding population modeled by equation (13) i s subject to exploitation, where H^ . i s the number of animals harvested i n year k, so that m \ + 1 = a j + 1 F ( P k + 1 _ n _ j - H ^ ^ . ) , (65) 1c — 0/ 1 ^  2/ ••• * Clea r l y , must s a t i s f y the f e a s i b i l i t y conditions 0 i \ = P k ' k = °/ !/ 2/ • • • • (66) Given a harvest p o l i c y (H Q, H-^ , H 2, . . . ) , the future stock l e v e l s (P-^, P 2, •••) 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, k = 1, 2, ... , n+m-1 (67) and the i n i t i a l stock l e v e l , P Q, are known. Let 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 adult population 47 of s i z e P. Following Clark, 1976a, assume that net economic revenue i n year k i s r Pk n ( P k , H k ) = J {p - C(P)}dP, (68) V H k where p = p r i c e and C(P) = un i t harvest cost when the population l e v e l i s P. In the usual fishery-production model (Clark 1976b), C(P) = c/P, where c i s a constant. I t i s assumed that the maximization of the discounted present value of net economic y i e l d 00 J = £ 6 kn(P,,H v) (69) k=0 k K i s the objective of ex p l o i t a t i o n . Here, 6 = 1/(1 + i ) i s the annual discount factor, where i = in t e r e s t rate. An optimal equilibrium condition i s now derived using the method of Lagrange m u l t i p l i e r s (Clark 1976a, 1976b). Consider the Lagrangean expression: °° k m L = Z (6 K n(P k,H k) - ^{P - Z a l F ( P k + 1 _ n - H k ) } ) . (70) k=0 j=0 j J Ignoring constraints, the necessary conditions are: 3L/3P k =0 (k > 1) (71) and dL/dE^ =0 (k > 0). (72) /Assume there i s an equilibrium solution to these conditions with 48 m escapement l e v e l S, so that P = P = E a. F(S) and H, = P, - S. J _Q D+l K K I t follows from equations (71) and (72) then that k m 6 n p + F ' ( S ) _ E o a j + l W l + j = ? k _ 1 (73) k_ m 6 \ - F ' ( S ) _ E o a j + l W l + j = 0, (74) f o r k = 1, 2, ... . Thus, ? k = 6 k + 1 ( n p + n R ) . (75) Hence, i t follows from equations (75) and (73) that m . i = {(n p + n H)/n H}F'(s) ^ 5 a j + 1 . (76) The optimal equilibrium 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 will be examined in light of the results obtained herein and some conclussions wi l l be drawn. Extrapolations will be considered whereby this work might be extended or generalized. A model has been constructed which describes a species in which a member may reproduce in 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, in fact, a l l density dependence was assumed to be concentrated in 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 in the (A^,A) parameter plane, where Am reflects the effect of the l i f e parameters of the species while A is 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 spec i a l cases for 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 increasing the delay leads to 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, increasing always takes a point out of the stable region. For the case n = 1 and m = 1 however, there i s a range of values f o r A (-2 < A < -1) where increasing from zero takes a point from i n s t a b i l i t y to s t a b i l i t y and back to i n s t a b i l i t y . Two examples for the density dependent reproductive function employing commonly used functions were exhibited. A study of the global behaviour of the model and a detailed examination of the b i f u r c a t i o n structure i s beyond the scope of t h i s thesis, however, some inter e s t i n g computer simulation plots were given, showing the complicated dynamics possible with the model and indi c a t i n g that the bi f u r c a t i o n structure i s indeed complex. F i n a l l y , the exploited population model was considered again, and a condition derived, giving the optimal equilibrium escapement l e v e l f o r the general case. An examination of the optimal approach paths to t h i s escapement l e v e l would be int e r e s t i n g , but, unfortunately, i s also beyond the scope of t h i s thesis. BIBLIOGRAPHY A l l e n , R.L. and Basasibwaki. "Properties of age structure models for f i s h populations." Journal of the Fisheries Research Board of  Canada, 31; 1119-1125, 1974. an der Heiden, U. "Delays i n physiological systems." Journal of  Mathematical Biology, 8: 345-364, 1979. Beddington, J.R. "On the dynamics of Sei whales under exploitation." International Whaling Ggnroission, S c i e n t i f i c (Committee Report, 28: 169-172, 1978. Bellman, Richard. Introduction to the mathematical theory of control  processes, Volume I I . New York: Academic Press, 1971. Canon, M.D., CD. Galium, J r . , and E. Polak. Theory of optimal control  and mathematical programming. New York: McGraw-Hill Book Company, 1970. Clark, CW. Mathematical bioeconomics. New York: John Wiley and Sons, 1976a. Clark, CW. "A delayed-recruitment model of population dynamics, with an application to 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 difference equations." Journal of Theoretical  Biology. 61: 287-306, 1976. Freeman, H. Discrete-time systems. New York: John Wiley and Sons, 1965. Goh, B.S. and T.T. Agnew. " S t a b i l i t y i n a harvested population with delayed recruitment." Mathematical Biosciences, 42: 187-197, 1978. Levin, S.A. and CP. Goodyear. "Analysis of an age-structured fishery model." Journal of Mathematical Biology, 9: 245-274, 1980. Levin, S.A. and R.M. May. "A note on difference-delay equations." Theoretical Population Biology, 9: 178-187, 1976. L i , T.-Y. and J.A. Yorke. "Period three implies chaos." American Mathematical Monthly, 82: 985-992, 1975. 52 May, R.M. S t a b i l i t y and complexity i n model ecosystems. Princeton: Princeton University Press, 1973. May, R.M. " B i o l o g i c a l populations with nonoverlapping generations: stable points, stable cycles, and chaos." Science, 186: 645-647, 1974. May, R.M. "Simple mathematical models with very complicated dynamics." Nature, 261: 459-467, 1976. May, R.M. "Simple models for single populations: an annotated bibliography." 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. Hassell, and T.R.E. Southwood. "Time delays, density dependence, and single species o s c i l l a t i o n s . " Journal of Animal Ecology, 43: 747-770, 1974. May, R.M. and G.F.: Oster. "Bifurcations and dynamic complexity i n simple ecological models." American N a t u r a l i s t , 110: 573-599, 1976. Maynard Smith, J . Mathematical ideas i n biology. London: Cambridge University Press, 1968. Pielou, E.C. Mathematical ecology. 2nd ed. New York: John Wiley and Sons, 1977. Ricker, W.E. "Stock and recruitment." Journal of the Fisheries  Research Board of Canada, 11: 559-623, 1954. Ricker, W.E. "Ocean growth and mortality of pink and chum salmon." Journal of the Fisheries Research Board of Canada, 21(5): 905-931, 1964. Ricker, W.E. "Changes i n the age and si z e of chum salmon (Oncorhynchus keta)." Canadian Technical Report of Fisheries and Aquatic  Sciences, No. 930, 1980. Ross, G.G. "A note on dynamics of populations with history-dependent b i r t h rate." B u l l e t i n of Mathematical Biology, 40: 123-131, 1978. Solomon, M.E. Population dynamics. London: Edward Arnold (Publishers) Limited, 1969. Williamson, M. The analysis of b i o l o g i c a l populations. London: Edward Arnold (Publishers) Limited, 1972. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080135/manifest

Comment

Related Items