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Stochastic processes in population studies Barrett, Marguerite Elaine 1962

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STOCHASTIC PROCESSES I B POPULATION STUDIES "by Marguerite Elaine Barrett,, B»Sc. University College of the West Indies, 1958 A Thesis Submitted i n P a r t i a l Fulfilment of the Requirements-f o r the Degree of MASTER OF ARTS i n the Department of Mathematics We accept t h i s thesis, as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA. September, 1962. In presenting t h i s thesis i n p a r t i a l f u lfilment of" the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of (Y\ti ITAI i,w> \Tc 5  The University of B r i t i s h Columbia, Vancouver 8, Canada. Date £g/>tfi4/nLftA IO, lc(^2~ i i . ABSTRACT This; paper develops a stochastic model f o r the growth of two int e r a c t i n g populations: when one species preys upon the other* The s p a t i a l d i s t r i b u t i o n of the populations i s considered, that of the prey "being assumed to be clustered and quasi-uniform* This l a t t e r d i s t r i b u t i o n i s discussed i n some d e t a i l , and i t i s found that, although i t has been suggested that clustering of the prey may "be a protective device against predators, any differences i n the stochastic models f o r clustered and unclustered populations l i e only i n the constant c o e f f i c i e n t s involved i n the formulation of the model* The approach used i n developing the proposed model i s that of Chiang* The size of the prey and predator popula-tions; are assumed to be random v a r i a b l e s X(t) and Y(t) respectively, and ce r t a i n assumptions are made concerning the b i r t h - r a t e and death-rate In ei t h e r population* These assumptions are based on the deterministic equations dx _ AjX - B-jX(x-l) - C 1xy , , , dt ~ - - • dv A^ -v - 3 2 v ( y - l ) dt " * * x+1 These equations; are modifications of equations published by L e s l i e i n 1958* D i f f e r e n t i a l equations are developed f o r the rate of change of the p r o b a b i l i t y that X(t)*x and Y(t)=y, by giving; t r a n s i t i o n p r o b a b i l i t i e s i n some small i n t e r v a l of i i i time and l e t t i n g the i n t e r v a l shrink to a point. Hence d i f f e r e n t i a l equations, f o r the rate of change of the j o i n t p r o b a b i l i t y generating function of X and Y, and f o r the rate of change of the j o i n t f a c t o r i a l moments of X and Y are obtained* Because of the complicated nature of these equations, however, no attempt i s made to solve them* Y i . . A C K I T O W L E I XTE M E N T S : A great debt of gratitude i s acknowledged to Dr* Schwartz and to Dr. S* Nash, both of the Department of Mathematics, f o r t h e i r help and encouragement during, the preparation of t h i s thesis* I also wish to thank the National Research Council f o r the grants* which made i t possible f o r me to do the necessary research* i v . TABLE, Off CONTENDS INTRODUCTION. CHAPTER I.. A DISCUSSION OF MATHEMATICAL MODELS FOR THE PREDATOR-PREY PROBLEM« 1. A "brief h i s t o r y . 2. Some e x i s t i n g theories of population c o n t r o l . 3. Chiang's stochastic model as applied to the Lotka-Volterra equations. i+. Monte Carlo methods and L e s l i e ' s modelo 5» Watt's deterministic model. CHAPTER I I . A STOCHASTIC MODEL FOR A CLUSTERED PREY POPULATION. A. POSTULATES. .1. The habitat of the two in t e r a c t i n g species. 2* The s p a t i a l d i s t r i b u t i o n of the predators. 3. The s p a t i a l d i s t r i b u t i o n of the clus t e r s of prey. il» Some comments on the number of individuals within a given c l u s t e r . 5. Possible d i s t r i b u t i o n s f o r the prey population. 6-7. Miscellaneous assumptions concerned with the dependence of the number of prey on the number of predators. 8~10» Miscellaneous assumptions concerned with movement and behaviour of individuals of eith e r species. .11. Death amongst the prey due to predation. 12. Death amongst the prey due to natural causes. 13« Death amongst the predators. .Il l , B i rths amongst the prey, 15» B i r t h s amongst the predators, B. FURTHER DEVELOPMENT OF THE MODEL. SOME RELEVANT DIFFERENTIAL EQUATIONS. 1. D i f f e r e n t i a l equations f o r d_£fx(t)} and d _ i fY(t)} dt dt 2* AV d i f f e r e n t i a l equation f o r d G,, v ( r , s ; t ) . dt 3. A d i f f e r e n t i a l equation f o r ^m^^-jCt) • 4. Some special cases of equation ( 2 , 2 2 ) « C, CONCLUDING REMARKS, BIBLIOGRAPHY. INTRODUCTION Every mathematical approach to population studies consists! i n b u i l d i n g , e x p l i c i t l y or otherwise, a\ mathematical model which may be eith e r deterministic or stochaatie. In ea r l y studies of population problems, however* ai deterministic outlook prevailed and an extensive deterministic theory of population dynamics now e x i s t s . Deterministic models have provided stimulus f o r both abstract and experimental work and. have encouraged investigation of problems i n terms of the concepts involved!. However, i n t u i t i v e l y at l e a s t , these models are unsatisfactory, f o r they make no allowances f o r the chance effects; which c l e a r l y influence the develop-ment of any natural population, and which should therefore be considered and accounted f o r i n a refinedl analysis; of the growth of a population. Stochastic models have both the p o t e n t i a l i t y and a c t u a l i t y of proceeding to more than one conclusion, a f a c t that seems to hold considerable; promise. The development of populations of two or more species which associate and interact with one another i n some way i s of great importance i n population ecology. I t has. been suggested that two species models are enjoyinga a less favourable reception amongst ecologists than formerly, owing i n part to the ov e r s i m p l i f i c a t i o n of t h e i r underlying assumptions. Nevertheless, a study of the effect of i n t e r -action between two species contributes much to the understand-ing, of the more general problem. The p a r t i c u l a r type of i n t e r a c t i o n which w i l l he investigated here i s that caused "by predation of one species upon the other. Stochastic models f o r predatory populations have been developed, hut one of the basic assumptions i n the formulation of these models has been that of random d i s t r i b u t i o n of each population* The in t e r a c t i o n between the species i s assumed to depend only on the numbers present In each population at the given time, and no mention is: made of the effect of the s p a t i a l d i s t r i b u t i o n s ; of the populations; on this, i n t e r a c t i o n * In recent years, i n t e r e s t has been shown i n the occurrence of clustering i n some species of animals* The present i n v e s t i g a t i o n was i n i t i a t e d by the suggestion that f i s h schooling may actu a l l y be 1 a protective device against predators, (Brock and R i f f enburgh, I960)* Three possible clustering situations may e x i s t : the prey population only may be clustered,, the predator population only may be clustered, or both prey and predator populations may be clustered* Of these a l t e r n a t i v e s , the f i r s t seems to be the one most usually encountered i n nature, and there also seems to be some in d i c a t i o n that c l u s t e r i n g of the prey does reduce predation* Brock and Riffenburgh, i n t h e i r investigar-t l o n of f i s h schooling, concluded that schooling of the prey reduced the chance of encounter with the predator, and hence the rate of consumption of the prey, whereas schooling of the predators; would act u a l l y reduce t h e i r scouting e f f i c i e n c y * In view of t h i s , i t was thought to be desirable to formulate 3. a stochastic model taking into account the s p a t i a l d i s t r i b u -t i o n s of both species: and the effect of cl u s t e r i n g of the prey, on the p r o b a b i l i t y of death by predatian* The; aim of t h i s paper, then, i s to develop a stochastic model which describes the growth of a population made up of two d i s t i n c t species S 1 and! S 2, as determined by a>. chance; mechanism of b i r t h , death from natural causes, and predation, when the prey species S^ e x i s t s i n c l u s t e r s and the predator species. Sg i s randomly d i s t r i b u t e d over the habitat* The f i r s t section of Chapter I provides, a short hi s t o r y of the general problem of population growth and the advances; made i n recent years i n the stochastic treatment of the problem* Subsequent sections of t h i s f i r s t chapter discuss e x i s t i n g theories of population c o n t r o l , and some relevant, deterministic and stochastic models* The proposed model i s formulated i n Chapter I I * The. approach used i n formulating t h i s model i s ; the same as that used by Chiang (195U) when he formulated a general stochastic model f o r two in t e r a c t i n g species* Chiang's model, which i s b r i e f l y discussed i n section 3 of Chapter I , assumes that the number© of individuals, i n each of two int e r a c t i n g species and Sg at time t , are random variables X(t) and Y(t) respectively, and r e s u l t s i n the derivation of a, d i f f e r e n t i a l equation f o r the function P„ ,.( t ) , which1. •"•,«y i s defined as. the p r o b a b i l i t y that X(t)=« and Y(t)=y at time t * S i m i l a r assumptions w i l l be made here, and a d i f f e r e n t i a l equation, which i s a c t u a l l y a spe c i a l case of Chiang's d i f f e r e n t i a l equation, obtained* Hence;, by the introduction of generating functions, a series of d i f f e r e n t i a l equations f o r the f a c t o r i a l moments of X(t) and Y(t) w i l l be obtained. The earlier' part of Chapter I I i s devoted to the precise formulation of the basic assumptions made, as postulates* Assumptions^ concerning the p r o b a b i l i t y of a b i r t h i n a given species, and the p r o b a b i l i t y of a deaths by natural causes, i n some small i n t e r v a l of time (t,t+c£t), are prompted by L e s l i e ' s work on the predator-?prey problem, ( L e s l i e , 1958)* This i s discussed b r i e f l y i n sections. i+ of the f i r s t chapter'. The l a t t e r prat of Chapter I I deals with the development and discussion of the d i f f e r e n t i a l equations f o r d P (t>, v(r,s;t)andl d_m - , ( t ) ; and some dt x , y «^ t dt Ln»KJ s p e c i a l cases of these equations* Here G ^ ^ ( r , s ; t ) denotes the; -joint p r o b a b i l i t y generating function f o r X and Y, and "rh k " J ^ i s t n e (Jk»k)'tl1 f a c t o r i a l moment of (X,Y). CHAPTER I A DISCUSSION OF MATHEMATICAL MODELS FOR THE PREDATOR-PREY PROBLEM, 1. A BRIEF HISTORY. The need f o r a p r o b a b i l i s t i c approach to population studies was recognized as early as 1873 when the problem of ext i n c t i o n of surnames was studied by Francis Galtan and the Reverend H. W. Watson; and the idea of tr e a t i n g population sizes as random variables was conceived by Yule i n 1925. The pioneer papers of McKendrick (1926) and Kermack and McKendrick (1927 and l a t e r ) emphasized the importance of nonlinear processes i n the quantitative treatment of epidemics and population growth. B a r t l e t t (1949* 1955 a and b) and -Kendall (1949) have taken the matter up again and have drawn attention to the intransigence of these same processes. E a r l y stochastic models were formulated i n terms of discrete time, and i t was not u n t i l 1939 that F e l l e r i n i t i a t e d a systematic treatment of stochastic models f o r population growth processes, and discussed the p o s s i b i l i t y of t r e a t i n g population growth as a temporally continuous; stochastic process. The cardinal assumption now i s that the growth of a population can be represented by a random variable X(t) with the Markov property that p{x(t)=x | x ( t 0 ) = x Q l = P{x(t)=x | x ( t 0 ) = x 0 , X(t)=S(T) for every i * . t Q J • •' ( l . l ) In the past 25 years, the theory of stochastic processes has developed r a p i d l y and i n p a r t i c u l a r Markov processes have been extensively investigated. F e l l e r ' s 6. theory has heen extended' and a substantial amount of work has heen done on stochastic processes i n r e l a t i o n to natural phenomena* There i s now no r e a l d i f f i c u l t y i n the formula-t i o n of stochastic: models although they are more d i f f i c u l t to understand and apply than, t h e i r deterministic counterparts, e s p e c i a l l y i n animal, ecology where, even i n the deterministic formulation of Lotka and V o l t e r r a , many s i m p l i f i c a t i o n s had to he made* The l i t e r a t u r e dealing with the deterministic outlook on the general predator-prey problem i s extensive and w i l l not be l i s t e d i n i t s e n t i r e t y here* During the course of t h i s chapter, however, three models w i l l be discussed b r i e f l y * The Lotka-Volterra deterministic model, developed as early as 1923, i s the oldest, the simplest and the best known model f o r t h i s problem* I t i s discussed b r i e f l y , i n section 3* as i s i t s stochastic counterpart* L e s l i e found i t more useful to approach the problem by using Monte Carlo methods* His model was developed as recently as 1958. and forms the basis of the new/ stochastic model to be developed i n Chapter 2* L e s l i e ' s model i s discussed i n section ho F i n a l l y , i t seems desirable to mention, i n section 5, the work done by Watt: on the predator^prey problem^ since i t i s probably the most recent attempt to describe a l g e b r a i c a l l y the effe c t of population d e n s i t i e s on predation. The reader i s referred to Watt (1959) f o r f u r t h e r discussion of the deterministic approach to the problem. 2* SOME EXISTING THEORIES OF POPULATION CONTROL. There are many e x i s t i n g theories which attempt to describe or> explain the phenomenon of natural control of populations* In 1929, Thompson proposed a theory based on the assumption that control i s effected by the environ-ment, which varies continually i n space and time, t h i s v a r i a t i o n being caused p r i m a r i l y by the ceaseless natural f l u c t u a t i o n of physical f a c t o r s * He suggested that natural control i s the inevitable r e s u l t of the v a r i a t i o n between f a v o u r a b i l i t y and unfavourability with respect to the s p e c i f i c l i m i t a t i o n s of the organism* Although Thompson has r e -affirmed h i s theory i n more recent papers (1939. 1956), i t has- not had much of a reception amongst mathematicians*. Nicholson's: theory, f i r s t published i n 1933. i s simpler and more e a s i l y grasped i n i t s concept of control by density-dependent factors., and most mathematicians working; on t h i s problem have so f a r acquiesced i n i t * Nicholson* a theory holds that the natur a l control of a population of a p a r t i c u l a r species S i s the r e s u l t of density dependent actions a r i s i n g from some sort of competition* There are; three forms of competition:-( i ) the competition of other species f o r resources or space which are or could be used by S, that i s , i n t e r s p e c i f i c competition* ( i i ) the i n t r a - s p e c i f i c competition of a parasite or predator f o r S* ( i i i ) the i n t r a - s p e c i f i c competition of S i t s e l f * 8o More recently, Milne (1957) proposed a theory which he summarised as f o l l o w s ! -A p e r f e c t l y density dependent f a c t o r or process w i l l control increase of numbers endlessly* There i s only one such i n Nature f o r any species and that i s competitions between, i t s own individuals* This i s the ultimate c o n t r o l l i n g factor f o r increase* But i n Nature, most species, i n most places, f o r most of the time, are held f l u c t u a t i n g at population l e v e l s where t h i s kind of competition i s r e l a t i v e l y i n s i g n i f i -cant* That is,, the ultimate c o n t r o l l i n g f a c t o r i s seldom evoked* The suggestion, therefore, must be that control of increase i s , f o r most of the time i f not almost endlessly, a matter of combined action of f a c t o r s which are density independent and factors, which are imperfectly density dependent, each supplying the lack of the other* The ultimate c o n t r o l of decrease of numbers i s brought about by density independent factors* Milne's theory seems more r e a l i s t i c i n i t s descrip-t i o n of the natur a l phenomenon, but Nicholson's proposal lends i t s e l f more r e a d i l y to mathematical investigation* I t i s Nicholson).'s theory which has formed the basis of most of the existing; mathematical formulations and upon which the present investigation i s based. 3 . CHIANG'S STOCHASTIC MODEL AS: APPLIED TO THE LOTKAV-VOLTERRA EQUATIONS. In 1954, Chiang formulated a general stochastic model f o r two in t e r a c t i n g species. He, denoted by X the population, size of the f i r s t species S^, and "by Y the population s i s e of the second: species Sg, and treated X and Y as random variables assuming; non-negative i n t e g r a l values x and y respectively. The following assumptions were then made*-* I f X(t) and Y(t) denote the values of X and Y respectively at time t , then i n the time i n t e r v a l (t,t+£t), the p r o b a b i l i t y of a unit increase i n X, givem X(t)=x and Y(t)=y, i s X St + o(<5t); the p r o b a b i l i t y of a unit decrease i n X,, given X(t)=x and Y(t)=y, i s f i x ^ * + o($"fc); the p r o b a b i l i t y of a-, unit increase i n Y, given. X(t)=>x and Y(t)=y, i s A St + o ( J t ) ; the p r o b a b i l i t y of a unit decrease i n Y, given X(t)=x: and Y(t)=y, i s u.„ St + o(<£ t ) ; the p r o b a b i l i t y of the simultaneous occurrence of more than one event: ("birth? 1 or "death!1) i s o(S t); the p r o b a b i l i t y of no change i n X, given X(t)=x and Y(t)=y, i s 1. » (A + \K ) £t + o ( c ^ t ) ; the p r o b a b i l i t y of no change i n Y, given. X(t)=x and Y.(t)=y, i s 1 - ( A y + u.y) St + o(S t ) ; the p r o b a b i l i t y of no change i n either X or Y, given X(t)=x and Y(t)=y i s 1 - ( Xx+ p y ) St + o ( i t ) . 10. For' fu r t h e r discussion of these assumptions, the reader i s r e f e r r e d to Chiang (1954)« Denoting the j o i n t p r o b a b i l i t y that X(t)=sx andl Y(t)=y by P _ ( t ) , which i s a function of the time t , Chiang x, y derives, d i f f e r e n t i a l equations f o r P_ (t) as f o l l o w s s -x,y The event that X(t+S t)=x andi Y(t+St)=y can happen when, and only when, one of the fol l o w i n g composite events happens. ( l ) X(t)=sx, Y(t)=y, and i n the time i n t e r v a l ( t , t + i * t ) , no change occurred to S^ or S^. ( i i ) X(t)=*x-1, Y(t)=y, and i n the time i n t e r v a l (t,t+£t), there was a unit increase i n X and no change i n Y. ( i i i ) X(t)=x+1, Y(t)=y, and i n the time i n t e r v a l (t-,t+£t), there was a unit decrease in. X and no change i n Y. ( i v ) X(t)=cc, Y(t)=y - 1 , and i n the time i n t e r v a l (t,t+£t), there was no change i n X and a unit increase im Y. (v) X(t)=oc, Y(t)»y+1, and i n the time i n t e r v a l ( t,t+£t), there was no change i n X and a unit decrease i n Y. (vi) two or more changes, occur" to and Sg. Since the above events; are mutually exclusive, the p r o b a b i l i t y that any one event happens i s the sum of the corresponding p r o b a b i l i t i e s * Making the added assumption that the p r o b a b i l i t i e s of change of X and YT are independent, that i s that 11* p{x(t+ S t ) ^ and Y(t+ ^*t)sy1| X(t)«=xQ and Y(t)=y 0 J = P { x(t+^t) 3=x 1 | x(t)=x 0, Y(t)=y Q} . P f Y C t + ^ t J ^ / x C t ) ^ , Y(t)=y Q] = p{x(t+ i t)= X : L |x(t)=oc 0\. p(Y(t+ 5t)=y 1|Y(t)=y 0}, i t can be seen that P [Event ( i i ) } By s i m i l a r reasoning, P {Event ( i i i ) } = P x + 1 > y ( t ) • hx+lC*) + 0(«^ *)» P {Event (iv)^[ = P ^ y ^ t ) .' A : ? w l ( t ) + o(£t), and P {Event (v)} = P x # y + 1 ( t ) • My+i^) + o ( ^ ' t ) * The p r o b a b i l i t y of event (vi) can e a s i l y be found t o be o( S t ) , while the p r o b a b i l i t y of event ( i ) i s px.y ( t ) I1 - ( V » V V V * o ( * *>• Conse quent l y , P z > y ( t + i t ) = {l - (A x + ^ V ry) } px,y(*> + 5 t pjrt,y<*> + i % px+l,y ( t ) + V l *•* px.*-i<*> + /Vrt S t p * , y + i ( t ) + o( S t ) , and hence, <U>X y ( t ) = p„ r<*» ^ - »».,<*> dt x » y st->o 5t - -'< W V ^ p x , y ( t ) * ^ W * 5 •« + t»x+l *Wl,y ( t ) + V l p x , y - l ( t ) + 1*741 pz,y+l ( t )- ( 1 , 2 > Chiang (1954) showed how t h i s model can be used to obtain a stochastic analogue of the w e l l known Lotka-Volterra deterministic model, which i s based on the 12. equations ax dt (1.3a) p 2 ) y (1.3b) These equations, imply that, i f each species were l i v i n g i n the absence of the other, the prey population would maintain •a. constant i n t r i n s i c rate of growth \ whereas: the predator c a l l e d the c o e f f i c i e n t of defence of the prey, and ^ 2 the c o e f f i c i e n t of attack of the predator. This representa-tion! of the in t e r a c t i o n i s not sa t i s f a c t o r y , since i t implies, f o r example, the i n d e f i n i t e increase of the p r o b a b i l i t y of being attacked as x increases f o r constant y, and that the; c o e f f i c i e n t of attack i s independent of the number of predators. AtLso, the assumption of a constant b i r t h rate i n either* species is , open to c r i t i c i s m ; and no provision i s made i n these equa-tions; f o r i n t r a - s p e c i f i c competition i n ei t h e r species. Even, more serious is; the f a c t that no upper l i m i t i s placed on the r e l a t i v e rate of increase of the predator i n the second equation. these equations;, which were developed independently by Lotka i n 1923 and Volterra\ in. 1926, have made and continue to make a great contribution to population studies. Applying the assumptions; of these two mathematicians to h i s general stochastic model, Chiang assumed that population S 2 decrease at a constant rate _• Volterra. Fever.theless, i n spite of these and other c r i t i c i s m s , 13. \ X ' ^ y * Hk" H*** |*y= H2y' Equation.! (1.2) theni became f f c P x , y ( t ) - -( ^ x + J^xy + A 2xy * ^ y ) P x > y ( t ) + \(x-D P x . 1 > y C t ) + ^ ( » 1 ) 7 P x + 1 , y ( t ) ' • V ^ J P ^ I t ) + ^ 2 ( y + D P x > y + 1 ( ^ U ^ ) This equation describes a system of d i f f e r e n t i a l equations f o r P v „(t), x=s0,l ,2, . . . , y=0,l ,2,...j and hence there w i l l x»y be an i n f i n i t e number of such d i f f e r e n t i a l equations. Since P (t) i s defined as zero f o r x-cO or y^ 0 or both, x,y P - l Q ( t ) =s P Q ^ ( t ) as 0 are i n i t i a l conditions. To solve t h i s system of d i f f e r e n t i a l equations f o r P , r ( t ) , i t i s x,y necessary to know P x _ 1 > y ( t ) , P^^Ct), P x + 1 , y ( t ) and P X f y M L ( t ) . However, the introduction of the j o i n t p r o b a b i l i t y CO «... generating f u n c t i o n Q Y „(r,s:t) = r x s y P_ ,_(t) reduces equation (1.5) t o the form (1.6) i n which there i s only one function to consider. Denoting £{pc(tf \Y(tt} - I I A k P x , y ( t ) •by nk k ( t ) , the derivatives of the ;Joint moments are given hy • • > 2 £ ( A ) W * > + ^ ^ " ' ' ^ ( H K ^ 1 (1.7) Ik. When h=l and k=0, t h i s equation reduces t o $fl,oM = \ m 1 > Q ( t ) - > x m ^ C t ) , (1.8a) while i f h=0 and k=l, the equation becomes f-Pb,!^) = ^ ^ ^ ( t . ) - h 2Bb,i(t)t (Mb) where 0(t)- = &{x(t)\ , nfc jCt) = t{Y(t)J and «l,l<t) = t $ X ( t ) . Y ( t ) } . When dealing with random variables that a t t a i n only integer values, the use of f a c t o r i a l moments usually leads to simpler shorter formulas with fewer moments. Defining the f a c t o r i a l moment m^ k i ( t ) as i n Chapter I I , part B, Am (t) section 1, the equation; f or ^ Jh,kJ v analoguous to equation (1.7) i s fe^lcJCt> « f h ^  - k h 2 + hk( A 2 - ^ ) } m [ h # k ] ( t ) + h(h-l) A x * [ h _ l t k ] ( t ) + hk(k-l) A 2 " p ^ M ] (*) + k ( k - l ) ^ 2 m £ h + 1 > k - i ; | ( t ) + k V D I + W * ) This equation involves at most s i x f a c t o r i a l moments, whereas equation (1.7) always involves 2h+2k~l ordinary ' moments; i f neither h nor k i s zero, and 2h+2k such moments i f either h or k i s zero. Setting h-1, k=0, and then'h=0, k=l, r e s u l t s i n two equations which are i d e n t i c a l t o equations (l.8a) and (1.8b) since m 1 > 0 ( t ) s m ^ ( t ) , ^ i ^ * ) = " [ p . l ] ^ ' . . and m 1^ 1(t) = ^ l , ! ] ^ ) * 1 5 One frequently finds the following type of correspondence between the d i f f e r e n t i a l equations of a deterministic model and those of the analogous stochas-t i c model. The c o e f f i c i e n t s expressed i n terms of x and y i n the d i f f e r e n t i a l equations of the deterministic model are replaced by the expectations of the corresponding expressions i n X(t) and Y(t) as the c o e f f i c i e n t s i n the d i f f e r e n t i a l equations of the stochastic model. Comparing the deterministic equations (1.3a) and (1.3b) with the stochastic analogues (1.8a) and (1.8b) respectively, we see that t h i s type of correspondence holds i n the present case. Since X(t') and Y(t) are not independent, m l , l ( t ) =^{x(t)Y(t)\ ?*£{x(tfy£{Y(t)] i n general, and i t does not s u f f i c e to replace x by £{x(t)} and y by The main d i f f i c u l t y here i s the i n t r a c t a b i l i t y of the non-linear d i f f e r e n t i a l equations for the joint p r o b a b i l i t y d i s t r i b u t i o n and the j o i n t p r o b a b i l i t y generat-ing function. In view of t h i s d i f f i c u l t y , i t has so f a r been found more rewarding to approach the problem by Monte Carlo methods. Chiang's model, as formulated above, i s useful only i n cases where the population size at any instant depends only on the immediate history of the population. 16. 4. MONTE CARLO METHODS AND LESLIE'S MODEL. The Monte Carlo method i s concerned with the generation of an a r t i f i c i a l r e a l i z a t i o n of a stocha&tie process by a sampling procedure which i s determined by the underlying p r o b a b i l i t y structure of the stochastic process. In 1957$ B a r t l e t t used t h i s approach i n h i s i n v e s t i g a t i o n of the stochastic process analagous to the Lotka-Volt err a deterministic model. His investigation revealed stochastic f l u c t u a t i o n which led to eventual e x t i n c t i o n of the predator' species, either before the prey or by starvation a f t e r the prey. This result i s not supported by actual observation of the natural s i t u a t i o n . I t i s also noted i n B a r t l e t t * a paper that although the o r i g i n a l deterministic equations (1,3) indicate ai stationary state when x = P2 / /'^2  sn®' y = ^"l/ ^1 ' t h i s stationary state i s unstable. In. 1958, L e s l i e published an account of h i s work on the predator-prey problem using Monte Carlo methods. He suggested that the equations, dt = a l x " * l y ) x -(1.10a) g = ( r 2 - b 2y/x)y (1.10b) are a more r e a l i s t i c representation of the problem than the Lotka-Volterra equations, the chief assets of these new 2 2 equations being that they contain terms i n x and y thus allowing f o r i n t r a - s p e c i f i c ; competition; and that they provide an upper l i m i t on the relative; rate of increase of 17. the predator; that i s , as x->oo , J . "^r2 ' t h e i n t r i n s i c -rate of increase of the predator. Also, when x-*0, 1 dy — • ^ j r , corresponding to the disappearance of the predator i n the absence of the prey. The stochastic representation of this, system by Monte Carlo methods i s more s a t i s f a c t o r y i n that the; stationary state i s i n this; case stable. Furthermore, the chances of random e x t i n c t i o n appear to be n e g l i g i b l e once the population sizes are i n the region of the equilibrium l e v e l s . This stochastic model has been discussed by L e s l i e and Gower i n a j o i n t a r t i c l e , ( L e s l i e and Gower, i960), and they claim quite good agreement between experimental and t h e o r e t i c a l r e s u l t s . 18 c 5. WATT'S: DETERMINISTIC MODEL In 1959, Watt gave a comprehensive h i s t o r y of the predator-prey problem, discussing and c r i t i c i z i n g many of the deterministic models e x i s t i n g at the time. He then developed a new model which yielded the r e s u l t . N A = PK(1 - e"^ 1* 1" 1 3) (1.11) where, using Watt's: notation, N^ = the number of prey attacked, HS'o = the i n i t i a l number of prey vulnerable to attack, P = the number of predators a c t u a l l y searching, K = the maximum number of attacks that can be made per P during the period the N 0 are vulnerable. This model suggests that the r e l a t i v e l y simple expression axy which i s used i n both the Lotka--Volterra equations and i n L e s l i e ' s equations to represent the death of prey due to predation could with advantage be replaced by an expression l i k e k y p ( l - e ^ " * ) where p i s the p r o b a b i l i t y that a given predator i s hungry at time t , and k, a, and b are constants. The values of the constants a and h compared with the order of magnitude of x and y w i l l determine how closely/Watt's formula approximates to the o r i g i n a l Lotka-rVolterra formulation. In an even more recent paper, Watt (i960) c r i t i c i z e s the assumption made by Lotka and Volter r a and by Lesl i e and others of a constant b i r t h - r a t e amongst in d i v i d u a l s 19. of a given species. I n t u i t i v e l y , i t would seem l i k e l y that population density as wel l as environmental factors would, affect i n some way the rate of reproduction of a>. species* Watt;' s paper1 encompasses a comprehensive and c r i t i c a l study of the subject. He favors: an analysis of the e f f e c t of density on fecundity made by F u j i t a i n 1954 and improves on Fujita's: r e s u l t . 20. CHAPTER I I A STOCHASTIC MODEL FOR A CLUSTERED PREY POPULATION. The "basic assumptions which are the foundation of a proposed stochastic model f o r a clustered prey popula-t i o n w i l l now he formulated, these formulations being made i n precise mathematical d e t a i l . The main difference between the proposed model and e x i s t i n g models i s i n the assumption of cl u s t e r i n g amongst the prey, and t h i s aspect of the problem w i l l be considered at some length. A. POSTULATES. 1 . The habitat of the two i n t e r a c t i n g species. The habitat of the two inte r a c t i n g species i s treated as a three-dimensional coordinate system which can be divided i n t o an i n f i n i t y of bounded regions of convenient shape. Each region w i l l contain a certain number of predators and a ce r t a i n number of clusters of prey. 2. The s p a t i a l d i s t r i b u t i o n of the predators. The predator population S 2 i s uniformly d i s t r i b u t e d over the habitat. In the region H under consideration, the number- of predators e x i s t i n g at time t i s represented by a random variable Y ( t ) . The d i s t r i b u t i o n of predators over the region H i s assumed to be Poisson, the number of in d i v i d u a l s i n any subregion depending; only on the volume of the subregion and on the expected number of predators per unit volume, and not on the shape or loc a t i o n of the subregion. I f G Y(s|V) denotes the p r o b a b i l i t y generating 21. function of Y f o r a region of volume V, where G Y(s|V) = T \ e Y|v} = 8 Y P{Y(t)=yJvJ , then, since Y(t) has a Poisson d i s t r i b u t i o n , the generating function: Q Y(s|v) = e V T K ^ D (2.2.1) where represents the expected number of predators per unit volume, and i s a function of the time t , say "<\ = " ^ ( t ) . Also, P $Y(t)=y"l = e~ V T l (VTp y , y=0,l,2,... • J y i 0 , otherwise. (2.2.2) 3» The s p a t i a l d i s t r i b u t i o n of the c l u s t e r s of prey. The prey population i s clustered and i t i s assumed that the d i s t r i b u t i o n of prey over the h a b i t a t - i s quasi-uniform. A quasi-uniform d i s t r i b u t i o n , as described by Neyman and Scott (1952) presents, i n general, the following p i c t u r e : • a d i s t r i b u t i o n containing a Poisson d i s t r i b u t i o n of single c l u s t e r s with the expected number of c l u s t e r centers per unit volume equal to P ^ say; and a Poisson d i s t r i b u t i o n , independent of the f i r s t , of double c l u s t e r s , with the expected number of c l u s t e r centers per unit volume equal to ? 2 say; and so on f o r t r i p l e , quadruple etc. c l u s t e r s , a l l of these Poisson d i s t r i b u t i o n s being completely independent. So the prey are assumed to be grouped into c l u s t e r s , and the c l u s t e r s again grouped into higher aggregates or 22. multiple c l u s t e r s , c a l l e d s i n g l e , double, t r i p l e etc. c l u s t e r s . In general, a multiple c l u s t e r containing v clusters w i l l be c a l l e d a v - f o l d cluster. In accordance with the assumptions of qiuasi-unif ormity, the number of prey i n any given c l u s t e r i s completely independent of the number of prey i n a l l other cl u s t e r s . I t i s also to be assumed here that the composition of each multiple cluster i s independent of the composition of a l l other multiple c l u s t e r s . The. number of single, double, t r i p l e , etc. cl u s t e r s e x i s t i n g at time t are denoted by the random variables N ^ t ) , NgOOf N^(t),... respectively, and i t i s noted that the expected values »?^ , a r e also functions of the time t ; that i s i?k= ^ k ( t ) say, k=l,2,3»... The random variable U(t) i s defined to be: kN k(t) and to have expected value {* = /A.(t) per unit volume. In f a c t , U(t) represents the t o t a l number of clusters of prey e x i s t i n g at time t . To s a t i s f y the necessary conditions f o r quasi-uniformity, the numbers lr^, Pg, <?Y» are subject to the r e s t r i c t i o n s that ^ 0 f o r k=l,2,3>».« and that the s e r i e s ^ k converges to some non-negative number ir> say. Cle a r l y , the exact form of the d i s t r i b u t i o n of U(t) w i l l depend on the s p e c i f i c assumptions made about the l ? k , k=l,2,3,... Two simple p o s s i b i l i t i e s w i l l now be considered. 23. a) One may.suppose, f o r s i m p l i c i t y , that only single clusters; e x i s t , that i s , v>2 = i?3 = . . . = 0, > 0 . Obviously, the r e s t r i c t i o n s placed on the p k and necessary fo r quasi-uniformity are t r i v i a l l y s a t i s f i e d . Since N k£t) = 0 f o r k= 2 , 3 , 4 , . . . , i t follows that N x ( t ) « U( t ) , and from the assumption of quasi-uniformity, U(t) i s a Poisson v a r i a t e . • - . Hence, P $U(t)=u] = e ~ V ^ ( V ^ Y - , u= 0 , l , 2 , . . . U* 0 , otherwise, (2*3.1) where -^ = The p r o b a b i l i t y generating function of U(t), denoted by Gu(ss|v), w i l l i n t h i s case be G U(S)V) = e V ^ s ^ . (2,3,2) b) Another p o s s i b i l i t y i s to assume that v>v = ffpk f o r k=l , 2 , 3 , « . • where 0 < p < l , and 0~ = o-fa) i s a function of the time t * In t h i s case, ^ 9 = ? ' = - c- log (l-p) ' ^ 0 , and the r e s t r i c t i o n s placed on the ? k are again s a t i s f i e d . Now, G N (sjv) = exp { V l> k(s-l)f = exp J Vo-pJ^s-l)), k = l , 2 , 3 . . . . , oo k and U(t) = | ^ kN k(t) . 21+'. oo This i s the p r o b a b i l i t y generating function of the Nega-tive Binomial d i s t r i b u t i o n , where p{u(t)=u] = ( V < T + u - 1 ) p u ( l . - p ) V < r , u=0,l,2, 0 , otherwise. (2.3.1+) It may be noted here that the Poisson d i s t r i b u -t i o n obtained f o r the d i s t r i b u t i o n of U(t) In (a) i s actually a l i m i t i n g case of the Negative Binomial d i s t r i -bution obtained i n (b), where p-*0 and o=*»<x> i n such a way that op remains equal to the fixed constant jm. . It i s clear from the d e f i n i t i o n of the random variables U(t) and X(t) above that P ^ U(t)=u|x(t)=x"J- =0 f o r u x. Hence x ^ P {u(t)=u|x(t)=x| = 1 . 25. 4. Some comments on the n&mber of individuals within a given c l u s t e r . The number of in d i v i d u a l s i n the i - t h cluster at time t i s denoted by W ±(t>, 1=1,2,3»....U(t). w\(t) i s assumed to be a sequence of mutually independent random variables. C l e a r l y , the W^Ct) must s a t i s f y the conditions 1 £ W i(t) < X(t) - U(t) + 1 f o r i = l , 2 , 3 , . . . U ( t ) , (2.4.1) 8 1 1 ( 1 U(t) ^ W ±(t) = X(t) . (2.4.2) I t i s also natural to suppose that the d i s t r i b u t i o n of WjCt) i s independent of i , so that -I ^ W^t) | X(t)=x, U(t)=u | = x/u ' ( 2 . 4 . 3 ) Postulates 1 to 4 , together with some assumption about the d i s t r i b u t i o n of Wi consistent with postulate 4, determine the marginal d i s t r i b u t i o n s of X and Y at a given instant of time t , but give no information concerning the jo i n t d i s t r i b u t i o n of X and Y. Some possible d i s t r i b u t i o n s f o r the w i l l now be investigated. 5, Possible d i s t r i b u t i o n s f o r the pre.v population. Consider the general assumption of quasi-uniformity of the prey population, made i n postulate 3* I t i s to be assumed that the composition of each multiple c l u s t e r i s 26. independent of the composition of a l l other multiple c l u s t e r s . In the discussion which follows, we deal, f o r convenience, with a unit volume of habitat. Prom postulate J>, N v ( t ) represents the number of v - f o l d c l u s t e r s e x i s t i n g at time t ; and U(t) represents the t o t a l number of in d i v i d u a l clusters e x i s t i n g at time t;. OO Hence U(t) = XL v 2 T(t). (2.5.1) v=l v and j*-(t) = £fa(t)\ Nov/ l e t N(t) represent the t o t a l number of multiple clusters e x i s t i n g at time t j and l e t P = \P(t) a i fN(t)} . Then N(t) = XL. N (t) ; (2.5-2) v=l v and J>(t) = XL >>v(t) . v=l v Let be a random variable denoting the number of i n d i v i d u a l clusters i n the i - t h multiple c l u s t e r , i = l , 2 , 3 , « « . Then, f o r any multiple c l u s t e r , F { v 1 ( t ) a v ] . = J i , and G v (s) = I\A I} i = XL i i s v . (2 .5 .3) v=l p Prom the assumption of quasi-uniformi-ty, -P_ n P { N v ( t ) * n v | = e y>v v , n v = 0,l , 2,...j n v ; 27. so that G N (s) = exp j \> v ( s - l ) ^  , (2 . 5.4) and therefore, Q K(s) = e x p j / ( s - l ) ^ . (2.5.5) Also, G n(s) = T T G H ( s v ) u v=l % - exp H frv(sv-l)j v=l ( J = exp \ a v— since V = ^ from ( 2 . 5 . 2 ) , That i s , O u ( B ) = exp j p [ G V i ( a ) - l ] | . (2 .5.6) Let: U„.(t), w=l , 2 , 3 , . « . he a random variable denoting the number of clusters of size w e x i s t i n g at time. t , and l e t ^ w = ^ w ( t ) he the expected value of U w ( t ) . I t i s reasonable to assume that the random variables U w ( t ) , w=l , 2 , 3 , . . . are completely independent• U(t) = FL U w ( t ) , (2.5.7) w=l and therefore Let W i j ( t ) , i=l ,2,3..»«t d=l>2,3#..,» D e a random variable denoting the number of prey i n the j - t h c l u s t e r of the i - t h multiple c l u s t e r . Then, f o r any multiple c l u s t e r , P W i ^ * ) ^ = i l w (2.5.8) 28* Let X v i ( t ) , v=l,-2.3> ••••»<- i=l ,2 ,3»«o. be a random variable denoting the number of prey i n the i - t h v - f o l d c l u s t e r . Then X ^Ct) = Z f w ± j ( t ) . d~i V Therefore G v (s) = T T Gw ( s) A v i 3=1 v v i d = ( f ; i > s w l V . (2.5*9) (w=l /x | Now, l e t X ^ t ) = ^ - X v i ( t ) = the t o t a l number of prey i n v-f o l d c l u s t e r s . Here 0 6 vN„(t) < X„r(t) ^ °o . v v G X <"8> 8 5 5* s X v p { X v ^ t ) = ? x v \ V V v L n^T lw=l ^ ) J = Y e-"v U 1 4 » ^ \ T J S = exp W i f e * ' H i -' ( 2 . 5 . oa But the t o t a l number of prey X(t) = ^ ( t ) » and the X y ( t ) are independent. 10) So <V(s) 29. TT <*x (s) Y=l V = exp = exp Aw s^  8)) ~ I-- 1 (2.5.11) Now, consider some spec i a l cases. a) F i r s t , consider postulate (3a). The assumption here i s that p„ = 0 f o r P=2,3.4, • and ;>0. Hence U(t) = N(t) = N,(t). r v i ? Also, V 1 ( t ) = 1 f o r a l l i , and therefore © v (s)= £js J = s« F i n a l l y , W i ; J(t) = 0 f o r d=2,3,U,... ) W ±(t) f o r 3=1. Using these r e s u l t s , equation (2.5.H) gives G x(s) = Q N ( Q v ( % ( & ) ) ) 1 «$» G u ( 0 w i ( B ) ) e x p {/X|?w1(s) - (2.5.12) Some additional assumptions w i l l now he made concerning the d i s t r i b u t i o n of W i(t). al) The d i s t r i b u t i o n which f i r s t suggests i t s e l f as a d i s t r i b u t i o n f o r the W± i s the truncated Poisson d i s t r i b u t i o n , so that 30.-P^W i(t)= !w^ * e " 1 ^ w f o r w=l , 2 , 3 , . . . C l - e ~ ? > w l ^ V i f . 1 = 1 , 2 , 3 , . otherwise J ..U(t) 1 f o r w=0 i f i > U(t) 0 otherwise where the parameter £ = ^ ( t ) i s a function of the time t< (2.5.13) The p r o b a b i l i t y generating function of Wi G w (s) = £ s w P j w ^ t ) ^ } v vi w=l , ^ 1 J 1-e Therefore, = - £ - L ( ® f B - 1) (2.5.1U) -5 Q x(s/V) = exp | V^j" e~%U%B-l) - l j , from (2.5-12) e x p | V ^ j e ^ 8 " 1 ) - l j j (2.5.15) This i s the p r o b a b i l i t y generating function of Neyman*s Type A contagious d i s t r i b u t i o n , (Neyman, 1939) • Suppose that ~ ? * w f o r w=l,2,3,... p[wi(t)=w |u(t)=ul = e j f j ( l - e ~ 5 ) w i i f i=l,2,3,. otherwise: »u 1 f o r w=0 ) I i f 1 > u 0 otherwise j (2.5*16) 31. This i s consistent with (2.5.13) above. Suppose also that W.(t) and W.(t), given U(t)=u, are independent i f i - ^ J . Then P {x(t)=x|u(t)=uj-u - • Z L J J P |w i(t)=w i [U(t)-u] ^x,u u where Qx > u= |(w1,w2,.. .w^ ) : MV± ^ 1 and ± 2 j w±=x ^  . Thus P |x(t)=x |u(t)=uj "v" u f w* Sc,u ^ ( l - e " 1 ) W i . •- e " U \ ft**>x 7] x l (2.5.17) ( l - e " f ) u x ! % u u x r f w^. i = l Now ZZ^ x . ,u u x ff w i • i = l x = the p r o b a b i l i t y of f i n d i n g zero c e l l s empty i n the random d i s t r i b u t i o n of x things amongst u c e l l s . = 7Z (-Dv 0(i-v) x (2.5.18) v=0 u (Feller, 1 9 5 7 ; PP 91-92) So that P jx(t)=x |u(t)=uj ° ~ U \ ( H f i i y ^ ( - D V ( ^ ) ( l - 5 ) X (2.5.19) ( l - e ~ V x . ^ 0 32. Further discussion along these l i n e s i s greatly-f a c i l i t a t e d by the introduction of differences of zero and S t i r l i n g numbers of the second kind. Define &y* = ( y + l ) x - y* , and A R + 1 y* = A r ( y + l ) x - A ¥ , r=l,2,3,... Then i t can e a s i l y be shown by mathematical induction that ^ ( - D v ( p ( y + r - v ) x = In p a r t i c u l a r , putting y=0, | ^ ( - 1 ) V ( $ ) ( r - v ) X = A r 0 x (2.5.20) The S t i r l i n g numbers of the second kind are related to the numbers &r0x by means of the equation' r ! S(x,r) = A r 0 x (2.-5.21). (Riordan, 1958; pp 33, equation 38), u Let Q(x,u) = -JIT (2.5.22) l - i w^. Then x. Q(x,u) = 21 x: t ^x,u £ ^ wi* = Z ^ ( - 1 ) V C K U - V > X f r o m (2.5.18) v=0 v • b^O* from (2.5.20) = vtl S(x,u) from (2.5*21) 33 That is, Q(x,u) • /^V = ui S(x,u) , (2.5.23) xT x; and P { X(t) »x jU(t) =u^ * e " U \ * X # S(x,u) (2.5.24) ( l - e ^ * * x i Recalling that P ^U(t)=u^ = e" ^ M-u where /< =/^(t) from (2 .3-1) , P ^U(t)=u[x(t)=x] =» P(u(t)=n"^ . p(x(t)=xlu(t)=:tt} Z~PJU(t)=vV P{x(t)=x|u(t)=v\ v=0 J ^ - ^ b r V • s(x,v) « S(x,u) - (2.5.25) I t may be noted here that, since 8(0,0)=!, S(0,u)=Q f o r u ? 0 , and S(x,u)=0 f o r u>x, equation (2.5.25) implies that p{u(t)=0 |x(t)=o] = 1, that P$U(t)=u|x(t)=oJ • 0 f o r u>0, and that p[u(t)=Uj'x(t)=x} = 0 f o r u>x. These r e s u l t s are what would be expected t r i v i a l l y from the d e f i n i t i o n s of U(t) and X ( t ) , and from the foregoing discussion. Consider now the conditional d i s t r i b u t i o n of' the W j L(t), given X(t) and U ( t ) . 34. P ^W i(t)=w|x(t)=x, U(t)=u^ P ^ W ^ t ) ^ O X(t)=x |u(t)=ul P ^X(t)=x | U(t)=*u] The denominator of t h i s f r a c t i o n i s given by equations* (2.5 .I9) and (2.5*24)• The numerator PJW i ( t )=w 0 X(t)=x ju(t)=u^ P J W^(t)«w 0 X(t)»x J U(t)=u J since the random v a r i a b l e s W^Ct), 1=1 ,2 ,3 . . . .u , are independent and have a d i s t r i b u t i o n which i s independent of 1. Thus P ^ Wjl t) =w 0 X ( t ) =x J U(t) =uI 88 ^ w "y> e"^ when w=l,2,3# ( W ^ ) wi d - e " * ) W j : = e * " U \ * X Q(x-w,u-l) . from (2*5*22) ( W * ) * wi = e" u : § ^ x ( u - l H S(x-w,u-l) (2.5*26) ( l - e ~ 5 ) u wi (x-w)i Therefore P ^ ( t j s w Jx(t)=x, U(t)=uj = ( u - l ) ? wi (x-w)i x i S(x-*w. u - l ) when w=l,2,3». u i S(x,u) 35. P {W^(t)=w |x(t)=x, U(t)=u^ =0 otherwise. The S t i r l i n g numbers S(x,u)=0 f o r x-cu, and hence S(x«!^ r,u*-1) = 0 f o r w> x-u+1* So P |*Wi(t)=w [ X(t)=x, U(t)=.u| a 0 f o r w >x«u+l, which i s i n accordance with previous assumptions about W 1(t) i n postulate I).. I t may also he v e r i f i e d that i (w^t) | X(t)=x, U(t>=u } = | , as stated i n (2.1+.3). For, t h i s conditional expectation of W i(t), I [w^t) j X(t)=x, U(t)=uj. x-u+1 . 2 1 w ( w) S(x-*wtu-l) using (2.5.27) w=l u S(x,u) x-u+1 — * — x 2 1 (££). S(x-w,u-l) . u S(x,u) w=l w + Using the r e s u l t ( 0 s( v»*0 = S(/x+l,m+l) , v=m (Jordan, 1939; ppl87, equation x-1 S(x,u) = Zl S(v,u-1) v=u-l x-u+1 Z w=l (jjpj) S(x-w,u-l) putting v=x-w, That i s , 36. x-u+1 S(x,u) « 2 T ( w l i ) S(x-w,u-l) Hence £ ^ W t(t) | X(t)«x, U(t)=u| uS(x,u) u S(x,u) a2) Another more general p o s s i b i l i t y , and one which includes case (al) as a p a r t i c u l a r l i m i t i n g case, i s to assume that W^(t) has a truncated negative binomial d i s t r i b u t i o n . In t h i s case, P J W ^ t ) ^ * ; * - 1 ) P w a - P ? f o r w=l,2,3,... l - ( l - p f 1=1,2,3, U(t) f o r w=0 fo r w=l,2,3,..» f o r w=0 > otherwise where the parameter $ = 5 ( t ) . The p r o b a b i l i t y generating function of W i(t) i s (2.5.28) (1-ps) - 1 (1-p)" 1 - 1 (2,5.29) 37. and the p r o b a b i l i t y generating function of X(t) i a QfcCIV) = *™ {V$^J-l]/^- ( 1 _ E , / ] J ( 2 .5.30) The d i s t r i b u t i o n of X(t) i s then the Poisson Pascal d i s t r i b u t i o n of which Neyman's Type A contagious d i s t r i b u t i o n i s known to be a l i m i t i n g case, ( K a t t i and Gurland, 1961). Now, consider the postulate (3b) which allows multiple clusters instead of just single clusters as i n (3a). Here i t i s assumed that P v = °-PY , v= l , 2 , 3 , . . . v where 0 < p < 1 and c= ^ ( t ) i s a function of the time t $ , = 21 ei£ v = * °* iog (I-P) > o . v=l v Prom equation (2.5.3), G v (s) = £ | j B T - " v i v=l P = log (1-pa) . log (l-p) Recal l i n g equation (2.5«5), % ( s | v ) = exp {Vv>(s-1)} . Hence, using equation (2.5.11), 38. jlog (l-p)j I-PG W (s) Dl) Suppose (t) has a truncated Poisson d i s t r i b u t i o n i d e n t i c a l to that of W ±(t) i n ( a l ) . (See equation (2.5.13)) Then >-3 (e 5 s - 1) from (2.5.1k) 1-e •as e ^ — 1 e - 1 Hence G x(s]Y) = [log (l-p)J In Gurland* s notation, t h i s d i s t r i b u t i o n i s Poisson v logarithmic v truncated Poisson. But a Poisson v logarithmic d i s t r i b u t i o n i s equivalent to a negative binomial d i s t r i b u t i o n ; so we have a negative binomial v truncated Poisson d i s t r i b u t i o n . (Gurland, 1957). b2) Suppose W i^(t) has the more general truncated negative binomial d i s t r i b u t i o n discussed i n (a2). Then and G, 1L (s) = "id G x(e|V) = (l*.ps)~' 5-(1-P) l - p . fa-oar* *i\ ( I - P ) - ^ - i . * from (2.5.29) / l o g (l-p;/ 39. Using Garland* s: notation, t h i s i s a Poisson v logarithmic v negative binomial d i s t r i b u t i o n . Postulate 5, concerning possible d i s t r i b u t i o n s f o r the prey population, may be summarized, as follows,: The quasi-uniform prey population i s considered to be made up of multiple clusters as stated i n postulate 3. The composition of each multiple c l u s t e r i s independent of the composition of a l l other multiple c l u s t e r s * R e c a l l i n g that X(t) represents; the t o t a l number of i n d i v i d u a l s e x i s t i n g i n the prey population at time t j N(t) represents the t o t a l number of multiple clusters of prey e x i s t i n g at time t ; V^('t) represents the number of i n d i v i d u a l c l u s t e r s i n the i - t h multiple c l u s t e r at time t ; and W^Ct) represents: the number of prey i n the j - t h c l u s t e r of the i - t h multiple c l u s t e r at time t ; i t i s shown that the relationship e x i s t i n g between the p r o b a b i l i t y generating functions of these random variables, is; G x(s) = G N ;(G V (G w (»))). 2.5.11) i i«J Some special cases are then considered. F i r s t i t is. shown that i f the assumptions of postulate (3a) are made, so that only single c l u s t e r s e x i s t , then G x(s) = (*)) where W i(t) represents the number of individuals: i n the i - t h c l u s t e r at time t , and U(t) represents the number of single clusters of prey e x i s t i n g at time t . 40. The further assumptions of a Poisson d i s t r i b u t i o n f o r U(t) and a truncated Poisson d i s t r i b u t i o n f o r the W ^ t ) , i=l,2 ,3..«..U(t), made i n ( a l ) , lead to a Neyman*s Type A contagious d i s t r i b u t i o n f o r X ( t ) . Further consideration of these s p e c i f i c assumptions and the introduction of S t i r l i n g numbers of the second kind lead to formulae f o r P {x(t)=x|u(t)=u} , see (2 .5.17), (2 .5.19), and (2 . 5 . 2 4 ) ; P {tJ(t)=u|x(t)=x } , see (2 , 5 . 2 5 ) ; and P{w i(t)«w| X(t)=x, U(t)=uJ , see (2 .5.27). I t i s then v e r i f i e d that I {%(*') I X(t)=x, U(t)=u? = x . 1 u Alternative assumptions of a Poisson d i s t r i b u t i o n f o r U(t) and a truncated negative binomial d i s t r i b u t i o n f o r the W^t), i = l , 2 , 3 , . . . , U ( t ) , made i n (a2), lead to a Poisson Pascal d i s t r i b u t i o n f o r X ( t ) . The assumptions made i n postulate (3b) are then considered. These assumptions allow multiple clusters, of prey instead of just single c l u s t e r s . The s p e c i f i c assumptions made i n postulate (3b) together with the assumption of a truncated Poisson d i s t r i b u t i o n f o r the W ^ ( t ) , i = l , 2 , 3 , . . . , j=l,2 , 3 , . . . # lead to a Poisson v logarithmic v truncated Poisson d i s t r i b u t i o n f o r X ( t ) . The s p e c i f i c assumptions made i n postulate (3b) together with the assumption of a truncated negative binomial d i s t r i b u t i o n f o r the W i 3 ( t ) , i = l , 2 , 3 , . . . , d = l , 2 , 3 , . . . . r e s u l t i n a Poisson v logarithmic v negative binomial d i s t r i b u t i o n f o r X ( t ) . Miscellaneous assumptions concerned with the dependence of  the number of pre.v on the number of predators. 6. In order to prevent the model from becoming too unwieldy, i t has been found necessary to make the somewhat u n r e a l i s t i c assumption that the conditional d i s t r i b u t i o n of W ^ t ) , given X ( t ) , Y ( t ) , and U ( t ) , i s independent of Y ( t ) ; that i s P ^Wi(t)=w |x(t)=x, Y(t)=y, U(t)=u} « P{ Vi±(t)=w | X(t)=x, U(t)auj " (2,6.1) 7o For s i m i l a r reasons i t i s postulated that P{U(t)=u|X(t)=x, Y(t)=y] = P{ U(t)=u| X(t)=x} (2.7.1) Postulates 3,h, and 5 have dealt with the d i s t r i b u t i o n s of X ( t ) , U(t), and W i(t) i n space. So f a r , the dependence of the process on. time has; been completely unspecified. This aspect of the problem w i l l now be dealt with, and the proposed model t i e d i n with Chiang's stochastic model. Miscellaneous assumptions concerned with the movement and  behaviour of i n d i v i d u a l s of either species, 8, A cl u s t e r of prey i s assumed to move and behave-as a single u n i t . The behaviour of individuals within the cluster w i l l be of no p a r t i c u l a r interest" here. The density of prey within a clu s t e r i s assumed to be constant f o r a given species of animal, and the clu s t e r s 142. are assumed to be approximately spherical i n shape. Each clu s t e r i s assigned some point, approximately at the geometrical center of the cl u s t e r , known as the cl u s t e r center. A.cluster i s assumed to l i e e n t i r e l y w i t h i n any given region i f i t s c l u s t e r center l i e s within t h i s region, 9. Individual predators and clusters of prey are assumed to move randomly, and the searching of the predator population to be completely at random, as systematic searching of each i n d i v i d u a l of the population does not constitute organized searching of the population. This i s c e r t a i n l y true of f i s h populations, although among some higher animals:, organized searching, within say family groups, may e x i s t . 10. The p r o b a b i l i t y that any given predator i s hungry at time t i s assumed to be constant and independent of t . 11. Death amongst the prey due to predation. The problem of formulating an expression f o r the p r o b a b i l i t y of death by predation i n the prey population w i l l now be considered. This p r o b a b i l i t y , together with the p r o b a b i l i t y of death due to natural causes amongst the prey, sp e c i f i e s the f ^ x of Chiang's model. In e a r l i e r models f o r unclustered populations, i t has been assumed that the p r o b a b i l i t y of death by predation i s d i r e c t l y proportional to the number of prey and the number of predators e x i s t i n g at time t . Brock and Rifferiburgh suggested i n t h e i r paper (i960) that, considering schooling as a protective device amongst f i s h populations, there were 43. two factors operating at cross purposes: i f a single f i s h i s regarded as a group of size one, schooling reduces the number of groups and consequently the frequency of encounter with predators; hut schooling increases the size of a u n i t , thus increasing the chance of detection. These two f a c t o r s are considered i n the following approach to the problem* Consider a c l u s t e r of prey, center C, and l e t <S V be the volume l y i n g between two concentric spheres with common center- 0 and r a d i i z and z+ Sz respectively* Prom postulate 2, the p r o b a b i l i t y that a predator w i l l be located i n this; region of volume at time t , i s d_G Y(s | S V) j ds x |s=0 - n ( t ) . e " ^ ( t ) where "f\(t) i s the expected number of predators per unit„ volume at time t i * Thus £jaY(s|$V) da s=0 = 4TT0 t^(t) z 2 <5z +o(<5z) since $V = k ^ {(z+Sz) 3 - z 3 \ . 3 By s i m i l a r reasoning, the p r o b a b i l i t y that the region of volume contains more than one predator at instant of time t i s o(S z ) * The p r o b a b i l i t y that a predator at ai distance z from the center of at cl u s t e r of size w w i l l sight i t i n the i n t e r v a l of time (t,t+ St) w i l l be a function of w, z and £ ft, but w i l l be. assumed to be independent of t« For f i x e d w and z, i t i s natural to suppose that t h i s p r o b a b i l i t y i n f i n i t y provided the predator i s able to sight such a cluster at a l l . The assumption which i s therefore made here i s that the pr o b a b i l i t y that a predator at a d i s -tance z from the center of a cluster of size w w i l l sight where c i s some po s i t i v e constant. Thus, the p r o b a b i l i t y that a predator at a distance z from the center of a cl u s t e r of size w w i l l sight i t i n the i n t e r v a l of time (t,t+ %t) i s cjzf(z,w) £t + o( %t). (2 . 1 1 . 1 ) Two suggestions as to the precise form of the function jz£(z,w) w i l l be made l a t e r on. The p r o b a b i l i t y of a death due to predation i n a cluster of size w i n the i n t e r v a l of time ( t , t + $ t ) i s then where Is some pos i t i v e constant. Hence, the pr o b a b i l i t y of a death due to predation i n the i - t h cluster i n the i n t e r v a l of time ( t , t + & t ) , given that X(t)=x, Y(t)=y, and U(tj=u, i s x-u+1 i s zero i f St i s zero, and approaches one as§t tends to i t i n the i n t e r v a l of time (t,t+£t) i s ^ ( z ,w)•(l-e~ c S t j 0 = yP(x,u) + o( §t) say, and t h i s expression i s independent of i . 45. The p r o b a b i l i t y of a death by predation i n the whole population i n the i n t e r v a l of time (t,t+<£t), given that-X(t)=x, Y(t)=y, and U(t)=u i s therefore ] T U y + o(£t) i = l =s yu P(x,u) + o(<$*t). (2.11*4) And the p r o b a b i l i t y of a death by predation i n the whole population i n the i n t e r v a l of time ( t , t + 5 t ) , given that X(t)=x, and Y(t)=y i s x yu P(x,u) . P^U(t)=u]x(t)=x^ + o.(£t), r e c a l l i n g postulate 7» x x-u+1 • c l y , S t S ^U(t)=uJx(t)=oc}^[p{w i(t)= Sw|x(t)==x,U(t)=u^ • J z2jrf(z,w)dz / + o(<ft), 0, J r e c a l l i n g postulate 6© (2.11.5) Before proceeding fu r t h e r , i t i s desirable to specify more p r e c i s e l y the form of jzf(z,w). Two p o s s i b i l i t i e s are considered here, ( i ) A. function which suggests i t s e l f almost immediately i s of the form jzf(z,w) = 1 i f 0J « z: 4, * V (w) i f OL y (w) * z ^  f> f (w) 0 i f (3 y(w) < z ^ °o (2.11.6) 46. where i t i s assumed, f o r s i m p l i c i t y , that y>(w) = (j^w)^ , and f "*1 i s the density of prey i n the c l u s t e r , assumed, constant. This choice of function seems f a i r l y reasonable, but i s quite a r b i t r a r y . I t does have the advantage that the d e f i n i t e i n t e g r a l 1 z^(z,w)dz which appears i n the J0 expression (2.11.5) above i s given by J z jrf(z,w)dz s K* & w, 0 J where K' = 2_ (3^- * 5 15 ( 2 > 2 - * 2 This fact s i m p l i f i e s expression (2.11*5) considerably, ( i i ) Another possible function has been suggested by Dr. S. Nash, who considered the image of any object seen by a predator to be an area on the surface of a sphere 8 whose center i s the predator's eyes, and which has a r b i t r a r y radius R. The p r o b a b i l i t y that an object i s sighted by the predator may then be assumed to be proportional to the area of i t s image on 8 • I f the further assumption i s made that the radius of a clus t e r of prey of size w i s proportional to ?/w, say equal to 3|pw , and i f the radius of SQ i s chosen, f o r convenience, t o be J z - ( f w) , where z i s the distance of the center of the cl u s t e r being sighted from the predator, then the area of the image on S Q of a spherical c l u s t e r of size w at a distance z from the predator i s calculated to be: The t o t a l surface area of S Q i s klT ^  zZ - (y w)* J . Hence the following f u n c t i o n was t e n t a t i v e l y proposed. ft{z,m) = 1 i f 0 ^  z < 3j pw , 1 - J l - ^ 3Jpw) 2 i f z > 3 | p ^ (2.11.8) This function has to he modified, however, f o r , i f jrffz.w.) r ° 9 2 i s defined as ahove, \ z /zf(z,w)dz i s i n f i n i t e . Also, J 0 there i s probably some minimum s o l i d angle that an object must intercept i n a viewer , !s v i s i o n , i f the object i s to make any impression on the viewer. I f the s o l i d angle i s below that psychological threshold, the object i s not seen. I t i s therefore suggested that >rf(z,w) = 1 i f 0 6 Z - L yfw", Jl-ft~2 - A-fVe¥/z ) 2 i f < z * (i j j f i r , 0 i f j?3fpw' 4. . (2.11.9) When jrf(z,w) i s thus redefined, to f z2^(z,w)dz = /Bj> w (2.11.10) 0 3 I n either case then, i t happens that the d e f i n i t e i n t e g r a l | z j(^(z,w)dz i s d i r e c t l y proportional to w, say 0 equals c2w/. The expression (2.11.5) f o r the p r o b a b i l i t y of a death by predation i n the whole population i n the i n t e r v a l of time (t,t+ £t), given X(t)=x and Y(t)=y, now becomes 14S. c^y St u PJu(t)=u|x(t)=xJ °2 W p{ w i( t)= !w| x( t)=x,U(t)=i u=0 w=l + o ( 5 t ) u=0 C ] Ly i t C2 ^ 0 U ?{u(t)=ujx(t)=ac}. & | ' w i(t)jx(t)=x, U(t)=u| + o(£t) = c^ycSt C g X + o(£t) = J ^ x y S t + o(St), where = c-jCg , (2*11.11) rec a i l i n g that { W±Ct) ) X(t)=x, U(t)=u| = 2 £ro m postulate h, and that x 2~] P|u(t)=u]x(t)=x} - 1 • u=0 B r i e f l y then, postulate 11 states that the p r o b a b i l i t y of a death by predation i n the prey population i n the i n t e r v a l of time (t,t+<£t), given that X(t)=x and Y(t)=y, i s /* 1 3xy St + o(<St), I t appears that, from the mathematical point of view, and under the conditions which have been assumed to e x i s t i n the foregoing postulates, cl u s t e r i n g of the prey population does not affect the type of model needed to represent the s i t u a t i o n . The only difference between (2*11*11) and the corresponding r e s u l t f o r randomly d i s t r i b u t e d populations i s possibly i n the constant c o e f f i c i e n t P l 3 * U9P 12. Death amongst the pre.v due to natural causes. L e s l i e ' s model, mentioned e a r l i e r , suggests that the p r o b a b i l i t y of a natural death amongBt the prey i n the i n t e r v a l of time (t,t+£t), given X(t)=x and Y(t)=y, may be supposed to be of the form ( K'-QX + ^ ^ x 2 ) + o( <£t), the term i n x being introduced to allow f o r " i n t r a -s p e c i f i c " competition. In the proposed model, i t i s found to be more convenient to write t h i s p r o b a b i l i t y i n the form ] Pllx + * * - i 2 x ( x ~ 1 ) ] * t + o( £ t). I t i s postulated then that the p r o b a b i l i t y of a natural death amongst the prey i n the i n t e r v a l of time ( t , t + ^ t ) , given that X(t)=x and Y(t)=y, i s f/*1:Lx + ^ 1 2 x ( x - l ) } S t + o(<ft). Postulates 11 and 12 imply that the p- x of Chiang's, model i s given by P-x " /^nX + ^ 1 2 x ( x - l ) + /^^xy. 13« Death amongst the predators. Returning again to Le s l i e ' s model, one may suppose that the p r o b a b i l i t y of a death amongst the predators i n the i n t e r v a l of time ( t , t + £ t ) , given that X(t)=x and Y(t)=y, i s O 2 1 y + * 2 2 y 2 A ) S t + o( St). But, i n t h i s case, the p r o b a b i l i t y would not be defined f o r x=0. In the proposed model, a s l i g h t modification i s made, and the p r o b a b i l i t y of a death amongst the predators in-the i n t e r v a l of time (t,t+ S t ) , given that X(t)=x and Y(t)=y, i s assumed to be " f / * - ^ + jU --.vf.v-l'W St + o(£t). x+1 50* This modification serves to eliminate any d i f f i c u l t y when x=0 and to simp l i f y the d i f f e r e n t i a l equations obtained l a t e r on. B r i e f l y then, y.^ = j*217 + ^ 2 2 Y l z = l ) • _ x+1 l h . B i r t h s amongst the -prey* In formulating a postulate f o r the p r o b a b i l i t y of a b i r t h i n the prey population i n the i n t e r v a l of time (t,t+ <St), given that X(t)=x and Y(t)=y, a peri o d i c function g-^ Ct) i s introduced to account f o r the fact that the reproductive process i s often seasonal. I t i s assumed that t h i s p r o b a b i l i t y i s A^g^t) x St + o( St) ; so that / l x = ^ g ^ t ) x • 15, Births amongst the predators. S i m i l a r l y , the p r o b a b i l i t y of a b i r t h i n the predator population i n the i n t e r v a l of time (t,t+£t), given that X(t)=x and Y(t)=y, i s assumed to be ^2&2^ y + °(St) , where g g ( t ) likewise i s a periodic function. Then *y " ' W * ) Y • B. FURTHER DEVELOPMENT OF THE MODEL. SOME RELEVANT DIFFERENTIAL EQUATIONS. B r i e f l y , the implication of the above postulates i s that *x s \ s i ( t ) x 5 f x = t1 l l x + ^ 1 2 x ( x - l ) +y" 1 3xyj \ = A 2 g 2 ( t ) y ; fi7..a ^ 2 1 y + f 2 2 z l g * ) (2.16) 51. 1. D i f f e r e n t i a l Equations f o r § ^ X ^ and ^ -^ /^ C*)} , Substituting the above expressions f o r A , M , A and u i n equation (1.2), t/s.y^ = > A ( t ) { (x-l)i»x_lfy(t) - *P x, y(t)} • ^ 2 s 2 ( t ) { ( j - D P ^ C t ) -yP x > y(t>} + f»u{(x+i)P x + 1,y(t) - ^ x , y ( t ) ] • /. 1 2 j(«DzP x + 1 > y(t) - x( x-l)P x > y(t)} • f-22 | I j a l k P x , y + 1 ( t ) - £to)P x, y(t)J (2.i7) This equation, l i k e equation (1.5), represents an i n f i n i t e system of d i f f e r e n t i a l equations which cannot be solved unless the functions p x«i y(*)» p x + l j P v ,r (t) and P„ „.n(t) are known. The derivatives of the x,y—JL x,y+j. expected value of X ( t ) , and of the expected value of Y ( t ) , with respect to time, may be obtained from equation (2.17), i n terms of the j o i n t momenta of X(t) and Y ( t ) . Before attempting t h i s , however, i t i s convenient to define f a c t o r i a l powers of x and y, and t o use f a c t o r i a l moments instead of the ordinary moments used by Chiang i n h i s model based on the Lotka-Volterra equations. Define x^ n3 = x( x - l ) (x-2)... (x-n+l), where n i s a p o s i t i v e integer. Then, ( x + n ) W = (x+n) [ n l x ,. where m and n are posi t i v e integers, and m> n. 52. Setting n=0, i t i s natural to define x 1 ^ as x?Ol = 1 # Setting ra=0, i t i s natural to define x^~ n^ as x M = i (x+n) [ n J = 1 (x+1)(x+2)(x+3)••. (x+n) F a c t o r i a l moments may now be defined as follows*! A , k l ( t ) = t { f K ( t ^ { Y ( t ) j t k 3 } f o r ^^.....-2,-1, L^'^J "* ' 0,1,2, m [ h , k ] ( t ) = 2fx(X-l)(X-2)......(X-li+l)Y(Y-l)(Y-2-)« ..(Y-k+l) \ when h,k=l, 2 , 3 • * * • If we accept the visual convention that J empty products,, equal 1, we can use the l a t t e r equation f or h-"8rlk.V'0.'. Multiplying equation (2.17) by x, and summing over x and y results i n the equation - ^ 1 2 m i ; 2 , 0 J ( t ) f hi^n,!]^* (2.18a) A s i m i l a r procedure, t h i s time multiplying by y, gives - f ^ c - i ^ j t * ) ' ( 2 ' 1 8 b ) These d i f f e r e n t i a l equations f o r d m „ n - i ( t ) dt '••L,UJ and d m m -.-i(t) bear the expected s i m i l a r i t y to the dt LU'-LJ deterministic equations on which the model i s based; namely dx = | A l g l ( t ) ~ x - ^ 1 2 x ( x - l ) - ^ 1 3 x y (2.19a) and dv. = { ^ 2 g 2 ( t ) - / l 2 i \ y - MaoZfoci) dt ^ ^ x+1 (2.19b) 53. 2. A d i f f e r e n t i a l equation f o r ^ X,Y^r'a;t^ A more general equation giving the rate of change of m^ k ^C*) with respect to time, may be obtained by introducing the p r o b a b i l i t y generating function G, ^ < x, Y(r,s;t) = ^ r V ^ C t ) I f h and k are non-negative integers, Jti k s h+k = r s ^ s k *'* G •(r , s ; t ) ; m [ h , k ] ( t ) r = l s=l (2.20) Also, ^ ^ x 1 " 1 1 ^ 2 1 ^ , r(t) x,y' dr where the constant of integration must be zero. Thus ' J] ^ ^ y ^ V V ^ y O s ) r ^ L . G x y ( y ,s;t) d f . J d s 2 = s_ r f =0 r=l Therefore, m (t) r=0 j ^ x , Y ( r , s ; t ) dr 3=1 (2,21) 1 54, Mow, writ i n g equation (2.17) i n terms of the generating function G X ^(T,a;t), which i s abbreviated to G f o r convenience, = ^TgnCt) ( r 2 SG - r £G 1 ^ 2 g 2 ( t ) { s 2 i S - s ^ G \ £t a ^ 1 I d r d r J + * * I as os J l«^ r <?r J ( ^ r 2 d r 2 J p=r par 1 5>V S . PJ=6^ that i s , - p-^rCr-1) - / ^ 1 3 ( r - l ) s ^ ""G ^ r 2 ' ^  d r ^ s ? = r ^ 2 2 s l s - l ) f ^ 5 - % > y ( j > ,s;t) dp • (2.22) ^ J ^ s 2 A d i f f e r e n t i a l equation f o r ^ " ( h . k ^ * ^ Recalling that mr. ^-.(t) = a n * k G Y v ( r , s ; t ) r=l 3=1 where h , k ^ 0, dm,-. . (t) may be calculated from equation dt (2.22) by f i n d i n g a H + K + 1 G # 55. The l a s t term of equation (2.22) may he wri t t e n i n the form - ju s ( s - l ) 2 1 Y.x(wrl) T^B*'2? „(t) ^ x y x+1 x , y - -^22 L S z L Z z i ) ^ e * - (t) ^ x y x+1 , y (2.23) This f a c i l i t a t e s the subsequent d i f f e r e n t i a t i o n of equation (2.22). Using Leibnitz*s rule f o r obtaining higher derivatives of a product; namely that a zn j=0 \ d z 3 / \ d z n 3 / where d°u i s defined as u, and s i m i l a r l y f o r v; equation (2.22) gives + 1 1 ( ^ 1 ) ^ ^ ^ ) ^ ^ % X 2 2 21J . ^ & k + 1 + k { A 2 g a ( t ) ( 2 s * i ) - r c x p ^ <^ r c^ s + k ( k - l ) A 2 g 2 ( t ) ^ >h+k~1Q - ^ ^ ( r - l ) ^ h+k+2, G (to be continued on the next page) 56. - h /V>(2r-l) a h + k + 1 0 - h(fc-l) JU 1 0 ^ n + k G *r* + 1<)s k ^ s k - / ^ ( r - l ) «> n + k+ 2G - h u__ B ^ h + k + 1 G 3 ^ + 1 ^ 8 k + 1 3 < ^ s k + 1 3 ^r h + 1<*s k 3 ^ r h J s k -/*22 E l x M r x - h | y M 8 ^ k - ( y - l ^ - ^ (2.2k) The l a s t terra of equation (2.24) may be written i n the form - rzz T £.ifc=l> WkV^a*-*Px _(t> x y x+l x » y + f 2 2 Z Z i£3k) W ^ r ^ y - 1 - ^ ft) . ' x y x+1 , y Now/v x+1 ~ i=0 Also y ( y - l ) ~ (y-k)(y-k~l) + 2k(y-k) + k ( k - l ) , ancL y-1 =: (y-k-l) + k . So the expression f o r the l a s t term of equation (2.2k) may be written - i s * C - D ^ Z Z x ^ - ^ V - 1 1 ; y l k + 2 ^ s 7 - k ^ i=0 x y 1 + 2 k ^ k + 1 V - k + k ( k - l ) / k V - k - yCk+2]8y-l-k - k y ^ V - ^ P ^ / t ) . 57. Hence, setting s=r=l i n equation (2.2k), Ifl*,^ ( h * l g l ( t ) " ^ " 12 * ^ 1 3 + a 2 g 2 ( t ) - k p 2 1 ] m [ M | t ) + h ( h - l ) ^ 1 g x ( t ) n i | - h ^ 1 > k - j ( t ) + ^ ( k - l J ^ g g g C t ^ ^ t ) - * K i 2 f f iLh +i,kj <*> - h h ^ ^ k + r i ( t ) h " k ^ 2 2 ^ 0 ( " * 1 ) i l i r i : { m t h - i . l , k + l ^ t > + ^ 1 > m i ; h - i ^ i , k ^ t > } • (2.25) When k=0, equation (2.25) becomes • ^ - ^ o ^ - ^ ^ ( t ) - i * n * ( h ^ ) hiz] m r h , o ] ( t ) + h(h-l)A 1 g 1 ( t ) m t h ^ l j 0 - j ( t ) * n P l 2 > + l , o / ^ - h fixft*,!} (t)* (2.26a) ' And when h=0, equation (2.25) "becomes I ^ O . k j W " k { A 2 g 2 ( t ) - ^ 2 i ^ L o , J O ( t ) + k ( k - D ^ 2 g 2 ! ( t ) m [ 0 ^ - i : i ( t ) - ^ 2 2 m £ . 1 > k + i ; ] ( t ) - k ( k - l ) ^ 2 2 m ^ k ; i ( t ) (2.26b) 58. k» Some special cases of equation (2.22^. I t i s of interest here to return f o r a moment t,o equation (2.22) and look at some special cases. The notation which w i l l he used i s stated c l e a r l y at the outset* Ms before, p x > y ( t ) 2 p (x(t)=x ft Y(t)=y} , and G x Y ( r , s ; t ) =; T 2L r x s y ' P (t) . x » x x=0 y=0 x ' y Now l e t P x - ( t ) =" P {x(t)=x"J , G x ( r ; t ) = r x P Ct) , x=0 " and G Y ( s ; t ) 3. SI s yP (t) . y=0 ( i ) Setting B=1 i n equation (2.22), a d i f f e r e n t i a l equation involving the p r o b a b i l i t y generating function of the marginal d i s t r i b u t i o n of X(t) i s obtained. For, G x > Y(r,s 5t ) j = £ ZA X f 7 ( t ) = I r \ ( t ) - G x #(-r;t) " I .LPx..^*) , *%|- ^ X . ^ * ) > and d r I s=l d r ^r| S a ( > 2G dr dS | S . j ISsI x * 21 x r * " 1 Z-yP x (t) x y x ' y = £ > & { Y ( t ) j X ( t ) = x ^ r x ^ 1 P x < ( t ) 59. Hence, equation (2.22) becomes sLG x.(r;t) m ft^Ct)* - f i ^ l (r-1) £j*x<(r;t) dt 1 S 3r - p 1 2 r ( r - * l ) ^ G x # ( p ; t ) - Z * ^ ^ 1 ) Z x & {Y(t) |x(t)=x} r ^ " ^ . (t) . ( i i ) S i m i l a r l y , s e t t i n g r = l i n equation (2o22), a d i f f e r e n t i a l equation involving the p r o b a b i l i t y generating function of the marginal d i s t r i b u t i o n of Y(t) i s obtained* QX,Y^ r» s ; t^ I x s G.yC8**)* da a 5 G Y ( B J t ) » and r=l d a B(B-1) f ^ L G x v ( f , s ; t ) d f l = B ( S - 1 ) ^ ^ y ( , Y - l ) s y - 2 P (t) P J * s 2 |r=l x y X + 1 -- E y(y-D € | | X ( t ) ^ J Y ( t ) = y p r - 2 P ^(t), So ^ y ( s ; t ) = p 2g 2(t)s - f* 2 1^ (s-1) |_G # Y(s;t) * 1*22 ^ y ( y - D i { f x ( t ) ^ |Y(t)=y} sy-"2P>Y(t) ( i i i ) Setting s=0 and r=0, G x > Y ( r , s ; t ) = P 0 > 0 ( t ) , *G = *lf0(*), M = * 0 > l ( t ) ' dr ^ = ^ O ^ ' and = p i f i ( * ) -£r 2 &r&a 60. Hence., equation (2.22) reduces to p'o,o(t) = .Pii*i,o<*> + fsA.x'*' • ( i v ) Setting s=0 and r=l, G x > Y ( r , B j t ) = 2 Z p x > Q ( t ) - P # Q ( t ) , "* * - 2L I j « V ^ y ( t ) - E A x l ( t ) ' = P x ( t ) . 3s x y x ' y x x ' i •* Therefore equation (2.22) reduces to *.<><*> = /*ap.i<*> ' (v) Setting r=0 and s=l, 0 x > Y ( r , s ; t ) = P Q > ( t ) , |a - E l ^ V p ^ t t ) - , ^ - Z. Zw?-1**'1** yCt) * P x Ct). t U ( t ) l x ( t ) = i l ^rds x y x ' y l o < 1 i Therefore equation (2.22) reduces to p,0,C*> - P l A . ^ ) + / 4 1 3 Plo ( t )' ^{Y(t)|x(t)=l} . ( v i ) Setting s=l and r = l , 0 x > y ( r , 8 ; t ) =Z.Z.Px>y(t) = 1 ; and equation (2.22) gives the t r i v i a l r e s u l t , dt - 0 # 61. G. CONCLUDING REMARKS:. The object of t h i s paper, as stated i n the introduction, was to develop a stochastic model f o r a. predator-prey problem, taking into consideration the s p a t i a l d i s t r i b u t i o n of the populations. Ih chapter I I , part A, postulates 3» k and 5» the s p a t i a l d i s t r i b u t i o n of the prey i s considered at some length* The underlying assumption here is, that of quasi-unif ormity of the prey population. However, as the development of the model proceeds i n postulate 11, i t becomes apparent that neither the s p e c i f i c d i s t r i b u t i o n of the prey over the habitat nor the s p e c i f i c d i s t r i b u t i o n of the prey w i t h i n a given c l u s t e r i s necessary to the formulation of the model. The assumption of complete independence between the number of indiv i d u a l s i n d i f f e r e n t clusters:, and the further assumption that a l l the clusters have the same d i s t r i b u t i o n , r e s u l t i n equation (2.h.3), which states that £ [w ±(t) ) X(t)=x, U(t)=u^ = x/u. I t turns out that t h i s i s a l l that i t i s necessary to know about the s p a t i a l d i s t r i b u t i o n of the prey. Discussion of the formulation of an expression f o r the p r o b a b i l i t y of death by predation i n the prey population concludes with the rather unexpected res u l t that t h i s expression d i f f e r s from the corresponding expression f o r unclustered populations only i n the constant: c o e f f i c i e n t f - ^ J hence c l u s t e r i n g of the prey population of the type described i n postulates 3 to 7 does not a f f e c t 62. the; model from the mathematical point of view. This i s probably the most important conclusion reached i n part A of chapter I I . The basic assumptions made i n chapter I I , part A, determine expressions f o r the f i ^ , A^, and j u y of Chiang's general stochastic model, and by subs t i t u t i n g these expressions i n equation (1 . 2 ) , the d i f f e r e n t i a l equation (2.17) f o r ^ ^x»y^^ i s obtained. Prom t h i s equation, d i f f e r e n t i a l equations (2.18a) and (2.18b) f o r d m,, n 1 ( t ) and d m r n n ( t ) are e a s i l y obtained. These d t l 1 ' 0 1 dt L 0 ' 1 ] equations bear the expected resemblance to the deterministic equations on which the model i s based. The analogy which has been observed between the d i f f e r e n t i a l equations of _a deterministic model and the d i f f e r e n t i a l equations of the corresponding stochastic model i s as follows: Deterministic model. Stochastic model. x I [ X(t)} = m ^ t ) = m [ 1 > Q ] ( t ) y & { Y(t)} = n\) f l(t) = m [ 0 , 1 3 ( t ) xy 4{x(t)Y(t)}«mL>1(t) = m £ 1 > 1 1 ( t ) 4 { { X ( t ) } h f Y ( t ) \ k j - m ^ O O x W , W 6 ^ { x ( t ) l ^ V ( t ) } I k J -»[h,kl<*> • Although the equations f o r d m r l m ( t ) and dt l--L»uJ d m r n T,(t) are e a s i l y obtainable without the introduction dt l u , J - J of the p r o b a b i l i t y generating function y ( r , s ; t ) , the 63. introduction of t h i s function enables the i n f i n i t e system of equations represented by equation (2.17) to be replaced by the equation (2.22) f o r ^ ; and s i m p l i f i e s the calculations necessary i n f i n d i n g a general equation f o r d_mr. v-,(t) • dt L n' KJ The r e s u l t s of the special cases considered i n the f i n a l section of part B of chapter I I , could probably have been obtained d i r e c t l y from equation (2.17), but neither of the two methods seensto have any p a r t i c u l a r advantage over the other. No attempt i s made i n t h i s paper to solve the many d i f f e r e n t i a l equations which have been obtained. Although i t may t h e o r e t i c a l l y be possible to solve these equations, possibly by l e t t i n g the function being considered be an i n f i n i t e power series i n the variables concerned, d i f f e r e n t i a t i n g t h i s series as indicated, and comparing c o e f f i c i e n t s , t h i s process would be lengthy and time consuming, and has not been investigated i n any d e t a i l i n the preparation of t h i s paper. The development of stochastic models of the type described i n t h i s paper seems, i n the cases of many authors and papers, to come to a stop with the a c q u i s i t i o n of d i f f e r e n t i a l equations l i k e those obtained here. A closer study of these d i f f e r e n t i a l equations may prove p r o f i t a b l e . 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