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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Stochastic processes in population studies Barrett, Marguerite Elaine 1962
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Title | Stochastic processes in population studies |
Creator |
Barrett, Marguerite Elaine |
Publisher | University of British Columbia |
Date Issued | 1962 |
Description | This paper develops a stochastic model for the growth of two interacting populations: when one species preys upon the other. The spatial distribution of the populations is considered, that of the prey being assumed to be clustered and quasi-uniform. This latter distribution is discussed in some detail, and it is found that, although it has been suggested that clustering of the prey may be a protective device against predators, any differences in the stochastic models for clustered and unclustered populations lie only in the constant coefficients involved in the formulation of the model. The approach used in developing the proposed model is that of Chiang. The size of the prey and predator populations; are assumed to be random variables X(t) and Y(t) respectively, and certain assumptions are made concerning the birth-rate and death-rate In either population. These assumptions are based on the deterministic equations (formula omitted). These equations are modifications of equations published by Leslie in 1958. Differential equations are developed for the rate of change of the probability that X(t)=x and Y(t)=y, by giving transition probabilities in some small interval of time and letting the interval shrink to a point. Hence differential equations, for the rate of change of the joint probability generating function of X and Y, and for the rate of change of the joint factorial moments of X and Y are obtained. Because of the complicated nature of these equations, however, no attempt is made to solve them. |
Subject |
Population |v Statistics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-11-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080610 |
URI | http://hdl.handle.net/2429/39254 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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