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8703557

Hatfield, Jeffrey Scott

DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE LOTTERY COMPETITION MODEL

The Ohio State University Ph.D. 1986

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University Microfilms International

DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION

OF THE LOTTERY COMPETITION MODEL

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

by

Jeffrey Scott Hatfield, B.A., M.S., M.A.S.

*****

Tin Ohio State University

1986

Dissertation Committee: Approved by

P.L. Chesson

J.D. Powers ______Adviser R.C. Srivastava Department of Statistics Copyright by Jeffrey Scott Hatfield 1986 ACKNOWLEDGMENTS

I would like to thank my graduate committee for their suggestions in the preparation of this document. In particular. I am grateful to my adviser, Peter L. Chesson, for his guidance during my graduate education. My deepest appreciation goes to my parents for their support throughout my education. Finally, I would like to give a special thanks to Bryan Knedler for his friendship and encouragement while I was in pursuit of this degree. VITA

October 4, 1957 ...... Bom, Cincinnati, Ohio

1979 ...... B .A , Miami University. Oxford, Ohio

1979 - 1981 ...... Teaching Assistant, Miami University, Department of Zoology, Oxford, Ohio

1981 ...... M.S., Miami University, Oxford, Ohio

1981 - 1982 ...... University Fellowship, The Ohio State University, Department of Statistics, Columbus, Ohio

1982 ...... M. A.S., The Ohio State University, Columbus, Ohio

1982 - 1985 ...... Teaching Assistant, The Ohio State University, Department of Statistics, Columbus, Ohio

1986 ...... Presidential Fellowship, The Ohio State University, Department of Statistics, Columbus, Ohio

PUBLICATIONS

Hatfield, J.S., T.E. Wissing, S.I. Guttman, and M.P. Farrell. 1982. Electrophoretic analysis of gizzard shad from the lower Mississippi River and Ohio, Trans. Am. Fish. Soc. I l l , 742-748.

FIELDS OF STUDY

Major Field: Biostatistics

iii TABLE OF CONTENTS

ACKNOWLEDGMENTS...... ii

VTTA...... iii

LIST OF FIGURES...... vi

INTRODUCTION...... 1

CHAPTER PAGE

I. DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE TWO-SPECIES LOTTERY COMPETITION MODEL...... 5

Introduction...... 5 The Stationary Distribution...... 8 Species Diversity Indices ...... 11 Comparison with other Models ...... 11 Conclusion...... 16

II. THE MULTISPECIES LOTTERY COMPETITION MODEL: A DIFFUSION ANALYSIS...... 18

Introduction ...... 18 The Stationary Distribution ...... 21 Species Diversity Indices ...... 23 Conclusion...... 24

III. DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE TWO-SPECIES LOTTERY COMPETITION MODEL WITH JUVENILE PRESETTLEMENT COMPETITION...... 26

Introduction ...... 26 The Stationary Distribution ...... 29 Conclusion ...... 30 General Conclusion...... 31

APPENDICES

A. DERIVATIONS FOR CHAPTER 1...... 35

Diffusion Coefficients ...... 35

iv Boundary Classification ...... 36 Comparison of Coexistence Criteria...... 40

B. DERIVATIONS FOR CHAPTER II...... 42

Diffusion Coefficients ...... 42

C. DERIVATIONS FOR CHAPTER III...... 44

Diffusion Coefficients ...... 44 Boundary Classification ...... 47

LIST OF REFERENCES...... 50

v LIST OF FIGURES

FIGURES PAGE

1. Comparison of the coexistence conditions of equation (1.9) (dashed lines) with the numerical calculations of Chesson and Warner (1981) (solid lines) for different values of the geometric mean death rate, d. Note that for d = 0 both conditions are identical. The coexistence region for a given d is the area above the line specified by that particular d ...... 12 INTRODUCTION

Competition among two or more species of organisms has been demonstrated time and again fee vsrious species of animals and plants (e.g. amphibians - Hairston, 1981; Morin, 1983; 1986; barnacles-Connell, 1961a; 1961b; birds-Gram, 1968; Dhoarifc, 1977; Dhondt and Eyckerman,

1980; Garcia, 1983; Grant and Schluter, 1984; fish-H ixon, 1980; Fauscle and White, 1981;

Werner, 1984; insects-Park, 1962;Rathcke, 1976; McAuliffe, 1984; plants-Handel. 1978;

Lobchenco, 1978; 1986; Platt and Weis, 198S; reptiles - Pacala and Rooghgarden, 1982;

Salzbutg, 1984; rodents - Sheppard, 1971; Gram, 1972; Brown, 1975; Glass and Slade, 1980;

Frye, 1983; and spiders-Spiller, 1984; Wise, 1984). Reviews by Connell (1983) and Schoener

( 1983)offield experiments attempting to find evidenceof interspecific competition estimate that

55% and 76% of the studies, respectively, successfully detected competition. Schoener (1985) and

Fenonet al. (1986) explain the discrepancy in the estimates, butthe point is that interspecific competition is an undeniablecompoaent of many natural communities.

Early in the study of interspecific competition, researchers formulated the "competitive exclusion principle" (Cole, 1960; Hardin, 1960). Briefly stated, the principle asserts that two or more species competiting for the same resource coexist together in a spatially homogeneous environment, although the principle has been extensively modified due to the results of numerous theoretical studies since its initial conception. Many of these studies have dealt with versions of the Lotka-Volterra competition equations in either deteminitticor stochastic environments and have rtstated the competitive exclusion principle in the form that k species (k * 2) cannot coexist on fewer than n resources (where n ^ k depends on the particular study involved and its associated set of assumptions).

1 2 Recent interest in the Lotka-Volterr* competition equations was stimulated by several studies which redefined the equations to make them more interpretable (Mac. Arthur and Levin?, 1964;

MacArthur, 1970; Case and Caste®, 1979). Wi»h their mathematical definitioncf competition coemdents, these studies developed the version of the competitive exclusion p

1974; Roughgarden, 1974; Abrams, 1975). Levin (1970) broadened the competitive exclusion prindple by staling that kspedes cannot coexist oafewer thanklimiting factors (which ran include predators, for instance).

It was not until Armstrong and McGefaee (1976,1980) relaxed the assumptions of coexistence at fixed densities and growth rates of the competing spedes as linear functions of resource densities thatitwas concluded thatkspedescan coexist onfewerthankresourcesinaspatiaily homogeneous environment (in fact, they showed that any number of spedes can coexist on four resources). Thus, Levins (1979) reformulated the competitive exclusion prindple as "thenumber of spedes coexisting an resources cannot exceed the number of resources plus the number of distinct nonlinearities."

Levins (1979) also stated that, for coexistence to occur, there must be asource of variance in the system1' and "this variation may be exogenous, some function of time, critm aybea consequence ofthe structure ofthe system itself." However, with regard to incorporating environmental fluctuations into the Lotba-Volterra competition equations, various authors have concluded that this type ofvarianceis detrimental to coexistence (MarAithur and Levins, 1967;

May and MacAithur, 1972; May, 1973; 1974). Abrams (1976) performed a simulation of

Lotka-Voiterra competition and found that May and the others may have overestimated the effect of environmental variability on competitive exclusion. Furthermore, the analysis of a stochastic version of these equations by Turelli (1978,1981a)andTurelli and Gillespie(1980) found thatlow to moderate levels ofenvironmental A u c tio n s had little effect on the coexistence ofthe spedes. 3 However, none of these studies found environmental fluctuations Co promote coexistence in the

Lotka-Volterra competition equations.

Contrary to thecomdusions above, theoretical studies of other models have found that environmental variability promotes coexistence (Koch, 1974; Wocdinand York, 1975; Cushing,

1980; Chesson and Warner, 1981; Chesson, 1982; 1983; 1984; 1985; 1986; 1987; Abrams, 1984;

ShmidaandEllner, 1984). Two of these studies (Koch, 1974; Cushing, 1980) incorporate deceministicvarMo!i{e.g. periodic functions) as the source of variability. WoodinandYorke

(1975) dontspecifywhether the variation in their non-rigorous model is deterministic or stochastic. Chesson and Warner (1981) were the first to show that stochastic variation can promote coexistence. Inrelanonto thecomperitiveexdusioapriaciple, Chessonfor example has shown that, with the sdditionof environmental fluctuations to the lottery competition model (Chesson and

Warner, 1981; Chesson, 1982; 1983; 1984; 1985; 1986; 1987), any number of species can coexist on only one resource (space). Moreover, the behavior of the lottery model has been shown to be characteristic of a variety of other stochastic models (Chesson, 1986). Thus, the remainder of this document will focus onthe effect of environmental fluctuations in the lottery competition model, anditsmodifications.

The lottery model is a stochastic competition model developed for two or more species of space-holding organisms and is believed to apply to coral reef fishes (Chesson and Warner, 1981), rainforests (Leigh, 1982; Warner and Chesson, 1985), and Eucalyptus forests (Comins and Noble,

1985). Other pertinent communities are marine benthic communities and many types of pi aat communities. Themathemarical details of themodel are described in thefollowing chapters but important aspects of the model are that comperitionoccurs at the juvenile stage and coexistence results from overlapping generations of the adults as well as variable birth rates of the juveniles and death rates (although variable death rates may also result in competitive exclusion).

Thelotterymodel predictsthatwith greater amounts of environmental variability, coexistence is more likely and, in addition, more species can coexist together. Many natural marine systems 4 demonstrate this relationship of i ncreasi ng species diversity with increasing levels of disturbance or variability (Dayton, 1971; 1984; Abele, 1976; Jackson, 1977; Lubehenco, 1978; 1986; Sousa,

1979; Ayting, 1981; Mook, 1981; Miller, 1982; Hixonand Brostoff, 1983). The importance of disturbance to themaintenance of many forest communities has also been verified (Loucks, 1970;

Grubb, 1977; Brokaw, 1982; Canham and Loocks, 1984; Canham, 1985; Veblen, 1986). That variability in recruitment and/or birth rates is present in fish, marine benthic, and forest communities is well established (Paine, 1974; Cushing, 1977; 1981; Sale, 1977; 1979; 1984;

Hubbell, 1980; Williams, 1980; 1983; Butler and Keough, 1981; Fowler and Antonovics, 1981;

Williams and Sale, 1981; Doherty, 1983; Victor, 1983; 1986; Eckert, 1984; Saleet. al., 1984;

Keough, 1984; Caffey, 1985; Gaines and Rougbgarden, 1985). Thus, a model such as the lottery model which incorporates these mechanisms should have broad applications to a variety of natural communities.

Other researchers have attempted tomodel this type of space-limited community inavari able environment. These include verbal (Connell, 1978; 1979), simulation (Sale, 1982), and analytical models (Hubbell, 1979). Nevertheless, these models are over-simplified and do not result in useful predictiansin a general setting. In contrast, the lottery model has been generalized to a variety of different situations (e.g. more than two species, vacant space, and patchiness).

However, though the lottery model is analytical, the predictions have tended to be qualitative in form, usually lacking easily interpreted quantitative formulae for the ccnditions of coexistence and the magnitude of the population fluctuations which the adult organisms will experience (i.e. stationary distribution). The remaining chapters of this document attempt to remedy this problem byperforming a diffusion analysis (Karlin and Taylor, 1981) on different versions of the lottery model. Thus, Chapter I deals with the two-spedes model, Chapter II with the k-spedes model, and Chapter III with the two-spedesmodel incorporating presettlemem competition of the juveniles. Simple, interpretable quantitative formulae for the conditions of coexistence and stationary distribution are the result. CHAPTERI

Diffusion Analysis and Stationary Distribution of the Two-Species

Lottery Competition Model

h te te ia ® Severalmodelshave been devdopedto investigate thequestianaf coexistence of competing spedes in a variable environment, Turdli(1978,1981a) andTurdli and Gillespie (1980) studied a stocbasticversion ofthe Lotka-Vdterra equations and found environmental flnctnations to have little effect. In contrast, coexistence of spedes in the lottery modd (Chesson and Warner, 1981;

Chesson, 1982) was found to be enhanced by environmental variability, provided the environmental variability affected birth rates and generations were overlapping. Since then a variety of other stochastic models have predicted coexistence as a result of environmental flnctnations (Abrams, 1984; Chesson, 1983,1984; Shmida and Ellner, 1984).

Chesson (1986,1987) shows how these different models and their predictions can be understood within a common framework. However, labile these models are understood qualilativdy, easily interpretable quantitative information is generally not available. While simple and interpretable conditionsfor coexistence have been produced, ingeneral they have been snffldent, bntnotnecessary. Altemativdy, they have applied onlyin situations where unlikdy symmetry assumptions have beenmade, or have been stated vagudy in the fonn "coexistence occurs with snffldent variability." Exceptions arefonnd inTnrdli and Gillespie (1980) andTurelli

(1981a), and the numerical calculations of Chesson and Wamer (1981). Also, little information is available far the population fluctuations that the competing spedes will experience (i.e. the stationary distribution). Results applicable to spedal cases (extreme environmental

5 6 variability or long-lived species), however, appear in Chesson (1982,1984,1985). In this paper, the two-species lottery equations are approximated with a diffusion process (Kariin and Taylor,

1981)and the stationaty distribution is determined. Simple explicit conditions for the coexistence ofthe two spedes result.

The lottery model was designed specifically for space-holding organisms and has been suggested as applying to coral reef fishes (Chesson and Warner, 1981), rainforests (Leigh, 1982;

Warner and Chesson, 1985), and Eucalyptus forests (Comins and Noble, 1985). However, the

stoffi important featere of the lottery model is that many of its properties have provengeneralizable to other models (Chesson, 1986; 1987). The lottery model provides simple illustrations of the

effects of a variable environment on coexistence and competitive exclusion. Thus, detailed study of this model is important. In the lotteqr model, we define Pj(t) as the total proportion of space occupied by spedes i at time t, pj(t) as the per capita reproduction and Sj(t) as the adult death rate of spedes i during the interval (t, t+1]. Then the equation for the proportion of adults at time t+1 in the two-spedes lottery model is

P i W ) Pi(t+i ) = [l - + ftW P tft)+ «2(‘W h ------~n ( I D p ^ o + ptft^t) fori *> 1,2. In (l.l), [1 - 8j(t)]Pj(t) is the proportion of adults cf spedes i surviving from time t to time t+1. The proportion of new site3 available for settling by juveniles is

[6j(t)Pj(t) + &2 (t)?2 (t)]. which is the proportion of space given up by adult death. This space is divided among the juveniles of different spedes it proportion to their abundance. Thus

Pj(t)Pj(t)/ [p j(t)P |(t) + P 2 (t)P2 (t)l is the proportion of available space that is captured by juveniles of spedes i. This is the lottery aspect of the model. Chesson and Warner (1981) show

how (1.1) can be modified to achieve abiased lottery in which one spedes is ^rperior at securing

space than the other. However, this modification is equivalent to assigning Afferent birth rates

to the spedes. Thus itissuffidenttostudy equation (1.1). to this model an important quantity is

Pl(t)/5i(t)

p ( t ) = — r - • which compares the ratios of birth rates and death rates for the two species. Far coexistence, p must fluctuate on either side of 1 by a sufficient amount (Chesson and Warner, 1981; Chesson,

1982). Coexistence is favored by increasing thevariability of p (orlnp). This can be accomplished by increasing the variability ia the birth rates. However, increasingtbe variability in the death rates need not favor coexistence. Only if 8j is positively correlated with Pj I pj will an increase in death rate variability make coexistence more likely, Ifthe correlation is negative, variable death rates are associated with competitive exclusion. By using a diffusion approximation, we obtainformulae that precisely define these conditions for coexistence. In this situation wetreat the existenceofthestationary distribution as being equivalent to coexistence. Note that this implies coexistence in the sense of stochastic boundedness (Chesson, 1982).

A diffusion process is a continuous-time process and is characterized by its infinitesimal mean and variance, defined as

H*) - S o C E P lfr + e> - p l « I PlW - *1 0-3) and

o2^) =S() cE[{pi(t + e)-Pi(t)}21Pj(t) = x], (1.4) respectively (Karlin and Taylor, 1981). Diffusion approGrimstion of the lottery model involves converting it to continuous time. Todothis, the time scale is accelerated by increasing the number of breeding seasons per unittime while decreasing the amount of change taking place in a season.

In the lottery model, this is done by letting X(t) = In p(t) and assuming E[X(t)] =ep and

Var[X(t)] = C02, C>0. This slows down both deterministic and stochastic change. In addition, we define 5j(t) = d je ^ 1), and assume E[Yj(t)] = 0 (djis the geometric mean of 8j(t)), 8

CovfY^t), X(t)] = i andCov[Y 2 (t), -X(t)] = ed2 . Higher order central moments are assumed to be o(e). These definitions decrease the amount of change taking place from one season to the next as € is reduced. To change the number of seasons per unit time, P j(t+ £) is substituted for

Pj(t+ l)intheLHSofeqviation(l.l).

In Appendix A, regularity conditicns are imposed on the distribution of (X(t), Yj(t), Y 2 (t» and (1.3) and (1.4) gfs evaluated for the discrete-time lottery model, where C adjusts the time scale and other effects as described above. It follows from Stroock and Varadhan (1979, theorem

11.2.3) that the lottery model converges weakly to a diffusion process with the infinitesimal mean and variance given by

(1.5) and

[d1d2ox(l-x)]2

The Stationary PiaBfrmwa The diffusion process determined by (1.5) and (1.6) has state space (0,1). In order to show the existence of a stationary distribution, it is necessary to investigate the boundary behavior of the process. In Appendix A it is shown that the boundaries 0 and 1 are natural boundaries, in Feller's terminology (Karlin and Taylor, 1981), which means that the process can reach neither boundary in finite mean time, nor can it be started from either boundary. The conditions for a process with natural boundaries to have a stationary distribution are given by Mandl (1968, Chapter IV, theorem

7). These conditions are evaluated for the lottery model in Appendix A. To state the results we introduce the quantities

2(|l + 01) 2(-jl + 02) Yj ------— and Y2 = (1.7) 9 Then the conditions for a stationary distribution canbe stated as

dj < Vj + 1 and • (1-8)

The quantities Yi and y2 1:2(1 be considered as measuring for each species its mean relative competitive ability divided by the environmental variance. Tosee this note that |i = E[lnp] =

E[ln B i /5 j - in . *cd so it reflects mean demographic differences between the species. In the

Yj, these mean tendencies are adjusted by the covariance of lnp with the adult death rate. Condition

(1.8) thus shows that if the dj are less than 1. coexistence necessarily occurs as 0, provided fy2j is sufficiently small. The latter situation can occur with large enough

0 2. The condirionsfor a stationary distribution (1.8) can be stated in the alternative form

, -01 + 01) -(-11 + 02) O2 > 2max[ . -] , (1.9) (1-di) (l-d2) which also exhibits how increases in ri2 must inevitably lead to coexistence provided that the dj's are less than 1. When the stationary distribution exists, the methods of Karlin and Taylor (1981) yield the following formula for the density:

¥

(Yi + 1) (Y2 + 1) t)j = ------1 , 1 , (1.11) “l “2 and c is a constant such that Jw(x)dx = 1. Note that this density is the product of a quadratic and a beta density. The first and second moments of the stationary distribution are

H dl 20>l + 2X^1 + 1) + 2d1d2(V1 + 1XV2) + d22(t>2 + 1X^)1 E(X)= , ------—— ------— (1-12) (bj + \>2 + 2)[ dj2(u1 + lXbj) + 2d1d2(l>1XV2) + <»2 0*2 + M ^)] and 10

E(X2) =

H <*l2(^l + 3X^1 + 2X1>1 + 1) + I d id j^ ! + 2X1)! + 1X1>2) + <*220>1 + *Xl>2 + lXl>2)] (1.13) (Dj + Dj + 3XDi + 1>2 + 2)[ di2(l)j + lXDi) + 2d1d2(V1Xi)2) +

The shape of the density is important because it yields information about the type of population fluctuations which the two species will experience. Far example, if dj = d j= d , then the density is a beta distribution, far which the shape can easily be determined based on the parameters Dj and Dj (Devore, 1982). WhenDj > 1 and 1>2 > 1, the distribution is zero si 0 sad

1 and rises to a unique mode between 0 and 1. For this case, the proportion of space occupied by either species is unlikely to be very small or large and thus stable coexistence of the two species occurs. WhenDj > 1 and D 2 < 1 o rl)i < 1 and Dj > 1, then eitherthe distribution increases from zero to <* or decreases from °° to zero, respectively, as the proportion of space occupied by one spedesvcriesfromOto 1. This isthe case of the infinite mode and is not a very stable type of coexistence becauseit is likely that the proportion of space occupied by agiven spedes will either be small orlarge relative to the other species at any given time. The beta distribution can also have a U-shape between 0 and land this occurs if i>i < landi> 2 < 1. Because of reasons similar to those given above, this results in the most unstable form of coexistence. Whendj * t^ .th e essential character of these findings remains unchanged exceptthat an additional interior mode can be introduced if the death rates are markedly different.

Itis important to note that when the stationary distribution exists its shape is strongly controlled by thegeometric mean adult death rates, thedj. If the dj are small, the vj will necessarily be large and the distribution will be concentrated about themean, leading to highly stable coexistence. Furthermore, ifo2 is large, the Yj will be small and Dj»dj~^-1. Thus for latgeo2, the only features of the biology of the organisms that matters are the adult death rates. Therefore the adultdeath rates are themost critical parameters determining the nature of population fluctuations of species coexisting in the lottery model. 11 SpeciesDiversity Indices Another paint of interest in relation to the distribution of the two-species lottery model is the ability to determine information about species diversity indices such as the Simpson index,

S = 1 - 2 (Simpson, 1949). and the Shannon index, H = - 2 Pj In Pj (Shannon and Weaver, 1962). Because we are dealing with a mixture of beta distributions, the mean and variance of these indices are not difficult to calculate. The mean and variance ofthe Simpson index involve computing expected values of several higher order moments. For example, E(S)= 1 - E(Pj^) -

E( Pj^), where E( P j^) is given by (1.13) and E( is analogous to (1.13). Calculation of

Var(S) is similar.

To determine the mean and variance of the Shannon index, power series expansions must be employed. Now, E(H) = -E(Pj In P j) - E O ^ lr^ )- Using the power series expansion for In P],

E ^I1®*!)- jig (1.14)

Computation a f E C P ^ ^ b is straight-forward and E(P 2 InPjJis analogous to (1.14). To determine Var(H), it is necessary to find expected values of squared terms meh as(Pj InPj)^.

Again making use of a power series expansion, we have

» 1 1 k 1 „ . „ E[(PilnPJ)2]=2k| ) — +X -JE ^Pj^2) . (1.15) k+2 k+1 j

Comparison with other Models

Our results can be compared with those of Chesson and Warner (1981) for the discrete-time lottery model. Their figure 1, whose results are represented in figure 1 of this paper, represents conditions for coexistence derived numerically under the assumption of equal, nonrandom adult death rates with p having a lognormal distribution. Corresponding to that situation, our results give the condition for coexistence: 2||l|/(l-d). This condition is represented by dashed lines in figure 1. The close correspondence attests to the accuracy of the diffusion approximation. Figure I. Comparison of the eoejdstesuee conditions of equation (1.9) (dashed lines) with the

numerical calculations of Chesson and Warner (1981) (solid lines) for different values

of the geometric mean death rate, i . Note that for d = 0 both conditions are identical.

The coexistence regionfor a given d is the area above the line specified by that particular

d.

12 0.2 0.4 0.6 0.8 1.0 MEAN ADVANTAGE (l/Al) 14 Chesson and Wamer also derived the conditions for coexistence for the case of small adult death rates, but arbitrary distributions for birth rates and death races. The conditions are:

E[5j(t){p(t) -1}] > 0 and E[$ 2 (t){p‘*(t) -1}] > 0. If X(t) and Y^(t) are normal random variables, thenthese conditions become 2 max [-{p + 8 j), -{-p + 02>] (Appendix A), which agree precisely with the resalts of the diffusion approximation for small dj's.

Finally, Chesson (1982) derived shape criteria for the model for large (effectively infinite) 0^.

Wenoted above thatin that caseVj * dj"* -1, which then determines the shape of the distribution.

The conditions on the shape derived from this are in em t accordance with the conditions derived by Chesson for the discrete-time model. Chesson also derived the shape in situations where both p and are possibly large, but comparison with that caseis complicated. In ail cases where comparisons are available, agreement with the diffusion approximation is good.

To obtain a diffosian process from the lottery model (which is a discrete-time Markov process) we speed up the time scale by increasing the number of breeding seasons per unit time, but decrease the amount of change that can take place for each season. There are several ways of decreasing the amount of change that can take place over a season, of which just one method is used here. The primary technique here is to let the quantity p approach 1. This means that the birth-rate, death-rate ratios (Pj 18j) of the species approach equality so that the competitive advantage that either species has over the other at any time is in fact quite small. The geometric mean adult death rates are kept constant in time, which means that the meanlifetime of an adult remains constant when measured in units of seasons, but in fact decreases in units of time.

An alternative method used by Chesson (1982,1984,1985) instead does not alter the value of p but decreases the adult death rate to 0, thus making the organisms more long-lived in terms of breeding seasons butnot interns ofanits of time. However, the biologically important quantity is the number of breeding seasons that an organism lives. Adjustment of the timescale is merely a mathematical device toget adiffusian approximation. Therefore this alternative method isgenerally regarded as an approximation applicable to long-lived organisms. In this approximation it is also 15 necessary to magnify tic population variables P^(t) about their mean values because as the death rates approach 0 the stationary distribution converges on its mean. In essence, the variance of the stationary distributionis proportional to the adult death rates, which are approaching 0. The stationary distributions derived from this method are always normal but take accounted the actual distributionsof the birth rates and death rates, notjust the first two moments. In contrast, the approximation in this paper depends only on the first and second order moments of the random variables. As noted above, it tends to agree with results for long-lived organisms when the adult death rates are small. In particular, the beta distributioacoaverges on the normal distributionasthe dj approach 0, in agreement with Chesson's approximation for long-lived organisms.

The most important feature of the new approximation is that it is not restricted in application to long-lived organisms and does not require altering the scale of Pj(t) about its mean to obtain a stationary distribution. Thus a distribution on the interval (0. l)isobtained. Moreover, while the derivation assumes certain means and variances become small, comparison of the conditions for coexistence derived from the discrete-time model (above) indicates that these assumptions do not restrict the applicability of the results. The approximationis best when the various population parameters involved have lognormal distributions. Comparison of the results on the stationary distributionfor laigevariability, also suggests thatfeatures of the stationary distribution, inaddition to conditions for coexistence, can be trusted far outside the assumptions of their derivations. Whend} = d j, our diffusion approximation is the same foem as that found by Gillespie (1977,

1980) for a model of natural selection of genetic alleles in a random environment. Although the diffusion processes are quite similar, the underlying assumptions that we make are somewhat different than those made in the genetics literature. Consider the case of constant adult death rates.

To be consistent with Gillespie's assumptions, we would define (lj(t)/dj = 1 + Xj(t). E[Xj(t)] = qLj, Var[Xj(t)] =zo^, and Cov[X j(t), X j^)] = e0j2- These assumptions lead to a diffusion process with the infinitesimal coefficients 16 djd2x(l-x) , , [L(x) = — ------{dixllli + 0 12l + d2(1'x)ftl l ' l l2 + 02 0 12l) (116) [djx + d 2 (l-x )r and

[did2x(l-x)l2[ 0 12 + O2 2 - 2012]

Although (1.16) and (1.17) appear to be different than die Mioitesimal coefficients gives is (1.5) asd(l. 6). they are infact identical when one considers thatwe used thelog scale to derive (l.S).

Note that q t = E[ln p(t)J = eQl j - |l£ + (Oj2 - 0^V 2] + o(C) andEO2 = V#r[ln p(t)] =

E(Cj2 + Oj2 - 20 j j ] + o(e). Letting p. = |lj - [tj + (Cj2 - Oj2)/2 and a2 = Oj2 + Oj2 - 20 j j , we get the infinitesimal coefficients given is (1. S) and (1.6). A similar argument canbeputfotthfor the case of variableadult death rates.

Conclusion

The diffusion approximation that we have derived for the lottery model appears to provide a good approximation to the behavior of the original discrete-time lottery model over abroad range of conditions, but performs best when the population parameters in the model have lognormal distributions. This approximation has allowed us to obtain simple formulae giving conditions for coexistence and the stationary density, which might thenbeused in testing themodel. Especially impartantisthe observation that the adult death rates are the primary determinants of the population fluctuations ofthe coexisting species. While the environmental variance is critical to coexistence, its effect on population fluctuations, once coexistence has been achieved, is much less important than the adult death rates. In some ways this is not surprising because environmental variation has a dual role in the model. As shown in other work (e.g. Chesson and Warner, 1981), environmental variation leads to positive mean instantaneous growth rates at low density, which 17 means that positive growth away from the boundaries toward the mean population density is promoted by environmental variability. But it is also clear that environmental variation should tend to cause & population near the mean, to be pushed away from the mean. Thus environmental variability has two opposing tendencies in this model, and the results thatwe have obtained here suggest that these two tendencies of environmental variation very nearly cancel out.

As emphasized by Chesson (1986,1987), coexistence in a stochastic c^v-j-onmear results from an interactionbetween environmental variability and competitive factors. This interaction is caused hereby the particular kinds of life-Mstory traits that the organisms have. The life-history traits of relevance in the lottery model are the fact that the organisms have overlapping generation?, and that competition occurs at the juvenile stage, which is also a stage that is sensitive to environmental flnctuations(through variabilityinthefi' s). The overlap in generations can be measured quantitatively as 1/dj, whichis essentially the mean longevity of adults. We have seen that thisquantity involving life-history traits is most important in determining the stationary distribution and therefote the magnitude of the population fluctuations. Given that many properties of the lottery model have proved generalizable to other models (Chesson, 1986), it seems a reasonable conjecturethat life-history features maywellbethe major determinantsof population fluctuations in general stochastic environment models in which environmental variability promotes coexistence. CHAPTER II

The Muldspedes Lottery Competition Model: A Diffuaon Analysis

Introduction

Analysis of models of competitioa among more than two species of organisms inhabiting a stochastic eawoamenthM beesmcommoaiatheliterature of theoretical ecology. Tureili(1978,

1981 a)investigated a stochastic version of the muicispeties Lctka-Volterra equations and found that low to moderate levels of random environmental fluctuations had little effect on the coexistence of species, although previous authors believedastochasdcenviroamenttobedetrimental to coexistence (MacAithurand Levins, 1967; May andMacArthur, 1972; May, 1973; May, 1974).

ChessQn(1983,1984) found environmental variance to have apositive effect on the coexistence of species in a multispecies version of the lottery competition model. Chesson (1986) discusses the effect of afluctuating environment on coexistence in ageneral setting and explains the differing results of these models in terms of differe at implicit assumptions about the biology of the species.

In the aforementioned studies, the conditions for coexistence tend to bequalitative inform, with exception of Turelli and others studying Lotka-Volterracompetition. Simpleformulaefor conditions of coexistence in the multispecies lottery competitionmodel of Chesson do not exist.

Furthermore, most of the previous work on multispecies competition does not address the problem

of the type and magnitude of the population fluctuations which the organisms will experience,

although some results can be found in Chesson (1984).

In Chapter I, we performed a diffusion analysis on the two-species lottery model and derived

explicitformulaeforthe conditions of coexistence and type of population fluctuations (i.e.

stationary distribution). The obvious extension of that workinto the multispecies situation is to

18 19 appratim^? the k-species lottery model (k i 2) with a I t- 1 dimensional diffusion process and obtain similar results.

Inthek-sperieslotterymodel, the equatianfortijeith species, i = 1,2,k, is given by

Pi(t)Pi(t) + g m m ] — ------(2.1) f f 2Pj(t)Pj(t) where Pj(t) is the proportion of space occupied by species i at time t , Pj(t) is the per capita reproduction of species i, and 8j(t) is the adult death rate daring the time interval(t,t+ 1], In

(2.1), (1 - 6j(t)| Pj(t) is the proportion of adults of species i surviving from time t to t +1 and

2 6j(t) Pj(t) is the proportion of space given up by adult death. The empty space is colonized by juveniles of species i in proportion to their abundance in relation to all juveniles, given by

Pj(t)Pj(t)/Z pj(t) Pj(t). Multiplying this term by the term for empty space yields the proportion of space colonized by speciesi from time tto t+ 1. Adding thisco adult survival produces (2.1).

As in the two-species model, Chesson (1983,1984) found coexistence of k species to be possibleif the generations are overlapping and sufficient variability exists in the birthrates, although predsequantitativeresuits giving the amount of variability necessary for coexistence are lacking. Theseformulae,asweUastheappraximatestatiQnarydistributionoftheprocess,will result from the diffusion analysis.

To approximate the k-specieslottery model with a diffusion process, wemust rescale the lottery model for continuous time. A description of this is given in Chapter I and amounts to repladngPj(t + 1) by Pj(t + €) in the LHS of (2.1) and letting e-»0. Furthermore, the amount of change occurring per season must be decreased and is accomplished by defining Xj(t) = ln[pj(t) / Sj(t)j, and assuming E[X^(t)] = q ij and Var[Xj(t)] = GOj2. In addition, let Cov[Xj(t),

Xj(t)] = C0ij( i x j, ^ (t) = d ^ V 1), E[Y„(t)] = 0, and Cov[Y„(t), Xtft) - Xft)] = £ 0 ^ Higher order moments are assumed to be o(e).

T h ek -1 dimensional diffusion approximation to the k-spedes lottery model is determined by its infinitesimal mean, variance, and covariance coefficients (Karlin and Taylor, 1981), defined as 20

W = S o c E[Pj(t + e)-Pi(t)|Ei(t)=x=(x1,i2 xk) ] . (2.2)

o f t ) - S o e E[{Pi(t + e) - Pi(t)}2 1 E

and

o^GO-So c E[{Pi(t + e) - Pi(t)>{Pj

i, j = 1,2 k .i* j. Substituting the definitions and assumptions above into (2.2), (2.3), and (2.4) and letting e-*0, we get the diffusion coefficients fo» the k-spedes lottery model (Appendix B):

d jij cInXn d ^ I4

d _ i_ + (------1/2X 0^+ ou2 -20iu)]

1 - + 2 2 4,dm*uK®i ' ®iu" ®im + ^mw)+ ®iim' ®uind J 1 (2-5) n d“*a

dj2 Xj2 O^OO = ^2 2^ d^ dm Xq xm{Of - 0jG - 9jm + Qtujj) , (2 6)

and

di di xi xi dm xu xm (0ij “ e ju - eim+ eum) ■ (2-7> 21 i, j = 1.2,..., k,i*j.

The Stationary Distribution

The stationary distribution, f(x), of the diffusion process represented by (2.5), (2.6), and

(2.7) must satisfy the Kolmogorov forward equation (Kielson, 1965):

k-i a t i a o=z — [2 — (-£ ------)-m(x)w(x)] (2 8) ,==1 axj J=1 axj 2 where

Oj^-o2, Ojj-po2. i*j, 0*|p|£ 1, (2.9)

®nin = To2- (2 10) and Sjjjjj, = 0, u * m, u * i, m * i . (2.11)

The assumption concerning the variances implies that each species experiences a similar amount of variatian(i.e. environmental fluctuatjons)intheXp The assumption concerning the demeans that the correlation between any Xj and Xj is identical. The assumptions concerning the 0 ^ Q, 0^u , and would result from a dependency between Yn and Xu, in addition to the independence of

Yu and Xj (m * i). Defining (p = 0^(1 -p)and

^ ^n ^n *n ^ ^n2 xn2 ^ *i m(x) = di xi {pi ------+ «p'[------3 ------] }. (2.12) 2 dflxfl ( 2 dflx n>2 I dflxn

■) 7 7 16a xa 2di xi Oi2(X) = di2(pxi2 { l + ------}, (2.13) (2dnxnr 2dnxn and 22 2dn *n dixi

Equation (2.8) is still too difficult to solve for (2.12), (2.13), and (2.14). Kdlson(196S) a aw a aw proved thtt if the process is irrotatianal (i.e. — ^ ) then a unique solution exists to the inner expression of (2.8):

k-1 d Oi/x)qr(i) 0 = 2 _(_£Z )-|ti(S)*(S),i = l,2,...,k. (2-15) H dxj 2

For the process determined by 12), (2.13), and (2.14), extensive algebra shows that the process willbeirrotationalif andanlyifdj =d 2 = ... = dj. = d. Adding tins assumption to (2.12), (2.13),

and (2.14) yields the coefficients

m(X) = dxi{m-2|l11xn+(p,[Ix112-xi], (2.16)

GjpOO = d2?*!2 {1 + 2 X jj2 -2ii}, (2.17)

OipO^d2?^^^!,!2-^-^}. (2.18)

In this case, these equations are identical to those far a diffusion process from the SAS-CFF model

in population genetics (Gillespie, 1980) and thus the solution, f(g),to(2.15)isD irichlet,

k 2 WOO=c.n*jVi " 1 , whereOj = ------[k( |ij - |L) +

c is the constant of integration which allows y (l) to be a proto ability density.

The conditions for the existence of (2.19) as a proto ability distribution are Dj > 0, which implies 23

f-kmndl-m) (2.20)

kmax(|t-|q) > (2.21) [(l-dXl-p)-T]

Tims, (2.20) and (2.21) are the conditions of coexistence for the k species in this system. It should he mentioned that we have not proved that (2.20) ami (2.21) are the conditions for the existence of a stationary distribution, only that (2.20) and (2.21) guarantee (2.19) is a probability distribution.

However, Seno and Shiga (1984) show that in fact (2.20) and (2.21) are the conditions for the existence of the stationary distribution and thus the conditions for the coexistence of the It species.

Species Diversity Indices

Knowing that the stationary distribution of the k-spedes lottery model is Dirichlet (2.19), it is possible to calculate the mean and variance (and highermoments) of species diversity indices: the

Simpson index, S = 1 - 2 (Simpson, 1949) and the Shannon index, H = -Z PjlnPj (Shannon and Weaver, 1962). These indices are frequently used by ecologists in the measurement of spedes diversity in natural populations. The mean and variance of the Simpson index require calculation of higher order moments. For example, E(S) = 1 - 2 E( P^), where

O + W i) E(Pj2) = (2.22) (l + ZdnX 2V

Similarly, E(S^) = 1 - 21 E( Pj^) + E(Z Pp)^, where! E( ?^) is computed from (2.22), and E(Z Pj2)2 contains the terms

(3 + ui)(2 + '0i)(l+ u i)eJi) E(Pi4) = (2.23) (3 +2 V P + 2 + 2 vnM2 V 24 and

(1 + t>i) (1 + Dj) (Wj) (up E( Pj2 Ph ------J- 1 . i * j • (2.24) 1 (3 + 2 Vn) (2 + 2 Dfl) (1 + 2 t)n) (2 Dfl)

To find the mean and variance of the Shannon index, power series expansions are utilized.

Now, E(H) = • 2 E(PjlnPj), and employing a power series expansion for In Pj, we get

oo m+1 m + , (-l)°E(Pi “ + 1 ) E(P:InP:) = - 2 I n ( ) ------(2-25) 1 1 m=0 n=0 v n ; (m + 1)

(n+Vj) (n -1 + t>j) ••• (Vj) E(Pi n + 1 ) = — ------. (2.26) (n + 2\>m)(n-l + 2\>m)- ( 2 V

Computation of E(H2) is mare complicated because it is necessary to find expected values of the terms (Pj In Pj)2 and (Pj Pj In Pj In Pp, i * j. Using a power series expansion for (InPj)2, we get

oo 1 1 m 1 m+2 m+2 E(pitaP->2-2^ — ; x-'rwr2) aw and

(n+ 1 +bj)(n + Vj) " (bj) E(Pjn + 2) = ------1------i------i------. (2.23) (n + 1 + 2um) (n + 2um) • • • (2um)

Calculation of E(Pj Pj In Pj In Ppis similar.

gaflSla&flB By converting the k-spedes lottery model to continuous time, it has been possible to derive a k -1 dimensional diffusion approximation. To accomplish this, the number of breeding seasons per unit time is accelerated while decreasing the amount of change taking place in aseason. The result is a diffusion process having infinitesimal coefficients (2.5), (2.6), and (2.7). Although no simplifying assumptions are needed for thatderivation (other than imposition of certain regularity 25 conditions), several are necessary in order to solve for the stationary distribution and the conditions for coexistence. In particular, assumptions (2.9), (2.10), and (2.11) concerning the variances and covariances are utilized, as well as the assumption of equal geometric mean acMt death rates. It should be noted that assumptions (2.9), (2.10), and (2.11) may be relaxed somewhat as was done by Turelli (198 lb) for the SAS-CFF model of genetic alleles in a random environment. Future work might proceed in that direction. However, the assumpdonof equal geometric mean adult death rates was absolutely necessary in order to reduce the Kolmogorov forward equation (2.8) into the much simpler form (2. 15) so that the stationary distribution could be found. If the geometric mean adult death rates are unequal, acomplicatedsetofpartial differential equations results. Although it may be possible to solve this system, it is beyond our means at the present time to do so. By obtaining the stationary distribution and conditions of coexistence for the k-spedesiottery model, it is now possible to determine haw various life-history parameters affect coexistence and population fluctuations. As in Chapter I, it is easily seen from (2.20) and (2.21) that small d and la r g e facvor coexistence. Furthermore, small dimply large V ; iorhe stationary distribution

(2.19) and thus stable population fluctuations and coesistence are inferred If is large, then

2(1 - p-t) 2 Uj«------(2.29) dk(l-p) k

and again the geometric mean adult death rate dean be seen to have an important effect on the type

of coexistence. Therefore, we come to the same conclusion as in Chapter I, that the adult death

rates are the most crucial life-hi&cty parameters in determining the type of population fluctuations

which the coexisting species experience in a fluctuating environment. CHAPTER III

Diffusion Analysis and Stationary Distribution of the Two-Species Lottery

Competition Model with Juvenile Presettlement Competition

Introduction In Chapters I and II, competition among two or more species of space-holding organisms in a fluctuating environment was investigated. However, the version of the lottery model used in those chapters assumed that competitionwas occuring at the time the juveniles settled out of the pool of juveniles onto the vacant space. In other words, from the time of birth until settling, a juvenile cf species i is affected similarly by additional juveniles of either species i or species j (i. e. interspecific competition equals intraspecific competition). Conceivably (and more realistically) the juveniles may compete both in an interspecific and intraspedfic maimer before the settling stage. Competition at the time of settling is still important, but clearly, competitionat other times of the lifecycle may also have amajoreffect. For example, larval fish may compete for food or other resources while in the plaofctonic, free-swimming stage but the effect of the juveniles of species i to a juvenile of species j may be quantitatively different than the effect to a juvenile of species i (e.g. different resource use).

The modelling of presettlement competition for space-holding organisms in a fluctuating

environment is absent in the literature, except for the work of Chesson (1983) on the two-species

lottery model. However, Chesson1 s results are againqualitative and of the form "competitionmust

always occur if birth rates vary sufficiently." There is no information an the magnitude of the

oopulation fluctuations which the two species will experience. In this chapter, we perform a

diffusion approximation of the two-sped es lottery model with presettlement competition and obtain

26 27 precise quantitative conditions of coexistence and the stationary distribution.

Incorporating presettlement competition into the two-species lottery model involves making the P j(t) in equation (1.1) density dependent (Chesson, 1983). Thus, in (l.l)le t

Pj(t) = Bj(t) fjlB^t), Bjd^t)], (3.1) where Bj(t) is the biith rate of species i, Bj(t)Pj(t) represents the number of juveniles before presettlementcompetition.andfj is the fractional reduction of Bj(t)due to presettlement competition. So, p j(t) represents the per capita net reproduction of species i after competition among the juveniles. Note that (3. i) canbe easily modified into competition among the adults

(affecting the birth rates)by having fjbea function of Pj(t) and Pj(t) instead of Bj(t)Pj(t) and

BjMPjW. Analysis ofthismodelisfacilitatedbythedeflnitian of generalized competition coefficients

(Chesson, 1983):

= ■ jj] ^ (*i» *j) (3-2) and

« iJ(si. sj ) = - | : 1flfi ( si*sj) • <3-3)

Because of the presence affj in the RHS of (3.2), (^represents the effect of juveniles of species j on theprob ability of survival from birth to settling of an additional juvenile of species i. Similarly, due to the presence of f- in the RHS of (3.3), represents the effect of juveniles of species i on the probability of survival from birth to settling of an additional juvenile of species i.

Assuming that fj(0,0) = 1, equations (3.2) and (3.3) imply

W i . Bjpj] The diffusion approximation of the two-spedes lottery model with presettlement competition involves the rescaling procedure utilized in Chapters I and II. This amounts to replacing Pj(t + 1) with Pj(t + C)in the LHS of (1.1), allowing £ to approach 0, and defining

Bi (Q/Stft) P(t) = — — . (3.6) B2(t)/«2(t) where B^(t)isthe birth rate and 8j(t)is the death rate of spedesi during time [t, t+ 1).

Furthermore, define X(t) * In p(t), and assume E[X(t)J= q t, Var{X(t)] = eo^, 8j(t) = dje^iW,

EfY^t)] = 0 (djis the geometric mean of S|(t», CovfY j(t), X(t)] = c9j. CovlYjlt), - X(t)] = cSj, lim sad In bj == £ E[ln BjJ. These assumptions imply that bj/dj = l^/dj. Higher order central moments are assumed to be o(e). In addition, we assume that thegeneraiized competition coefficients are of the form ea^ and cotjj.

The diffusion apprrarimationis characterized by its infinitesimal mean and variance, defined as

K * )= S o C ^ 1 ^ + e) - p l(t)l p i(t) = *] (3 7) and

= gTo e E[{p l(l + e) - p l(1)}2 I p i(t) = x ], (3.8) respectively (Karlin and Taylor, 1981). Toevaluate(3.7)and(3.8)farthetwo-spedeslotteiy model with presettlement competition, it is convenient to define 29 «l(si. *2) = - otii(s 1.0) + Q2i(si. ®2) (3.9) and

0t2(sj, S2 ) = - 0 2 2 (0 , S2 ) + a !2(s l* *2) • (3.10)

If we define

b jx M M ) A(x) = J0 a j(z , b 2 (l - x)) dz - j0 a2(b1x. z) dz, (3.11)

(3.7) and (3.8) are evaluated for this version of the lottery model (Appendix C, with the imposition of regularity conditions in addition to those imposed in Appendix A), and it follows from Stroock and Varadhan(1979, theorem 11.2.3) that the two-species lottery model with presettlement competition converges weakly to a diffusion process with the infinitesimal mean and variance given by

+ [d1x + d2(l-x)]A(x) } (3.12) and

[d1d2ox(l-x)]2 o ^ x ) = (3.13) [djx + <* 2 (1 -x)]2

The Stationary Distribution As in the two-species lottery model without presettlement competition, the diffusion process specified by (3.12) and (3.13) has state space (0,1). Existence of a stationary distribution is determined by the boundary behavior of the process. In Appendix C, it is shown that the boundaries are natural boundaries and, as such, the conditions of Mandl (1968)for the existence of 30 the stationary distribution are

b j bj -(11 + 0 1 -J0 a2(0,z)dz) *('li + ®2"^0 «i(*.0)dz) 0 > 2 max [ ------— — =------, ------— ------] , (3.14) 1 -d j l-d2 assuming that a^ S j, 3 2 ) is continuous for G [0,1]. If (3.14)is satisfied, then the stationary distribution must have the following density (Kariin and Taylor, 1981):

x 2 A(y) 1 1 f(x) = c[d*x + ttyl-x)]2 x ^ l'1 ( l - x ) ^ '1 exp I — — [ — + ------] dy. (3.15) o 2 dl7 djd-y) where b j and are definedby equation (1.11), A(y)is defined by (3.11), and c is the constant such that i v(x) dx = 1.

Condunon

By using the technique of diffusion approximation set forth in Chapter I, the conditions of coexistence and the stationary distribution have been derived for the two-spedes lottery model with presettlement (as well as settlememt) competition among the juveniles. By converting this version of thelotterymodd to continuous time, accelerating thenumber of breeding seasons per unittime, and decreasing the amount of change occurring per unittime, the results of Chapter I have been extended to indude presettlement competition. Note that if a j = a 2=0, presettlement interspecific competition equals intraspedfic competition, and the results of this chapter reduce to those of Chapter I. Thus, the condition for coexistence (3.14) becomes identical to (1.9) and the stationary distribution (3.15) becomes (1.10).

If presettlement interspecific competition is not equal to intraspedfic competition, then

(3.14) dearly illustrates the effect on coexistence. The situation a j < 0 and a2<0 (intraspedfic competition exceeds interspecific competition) is the most favorable to coexistence while the situation a j > 0 and a 2 > 0 (interspecific competition exceeds intraspedfic competition) favors competitive exdusion. If 0 (j > 0andG(j< 0(spedesi is the better competitor) then spedesi is 31 favored, although exclusion of species j need not occur depending on other life-histoty parameters.

As in Chapterl.it is easilyvisible feom (3.14) that a large environmental variance, o^, and small geometric mean adult death rates, dj, favor coexistence of the two species. Similarly, inspection of the stationary distribution (3. IS) shows that as

Qq w fd CdqplM dfl The lottery model has been converted to a continuous-time version and approximated with a diffusianprocess. The lottery model is converted to continuous time by defining the breeding season as e units instead of one unit of time and allowing £ to approach zero., Inother words, the number of breeding seasons per unit time is increased. Simultaneously, the magnitude of change occuring in a season must be decreased; otherwise the diffusion coefficients of the approximation escape to infinity. The decrease in changeis accomplished by letting certain life-history parameters approach zero at a specific rate. In so doing, a diffusion process is obtained for the two-species case having diffusion coefficients given by (1.5) and (1.6). A similar procedure is followed for the k-spedesmodel and results in the diffusion coefficients (2.5), (2.6), and (2.7), assuming equal geometric mean death rates among the species and certain assumptions about the covariance structure (i.e. (2.9), (2.10), and (2.11)). Far the two-species model with presettlement competition an additional assumptionis required to derive the diffusion approximation. That assumption involves the generalized competition coefficients Qtjj and amounts to assuming 32 continuity and allowing these coefficients to approach zero. The resulting diffusion coefficients,

(3.12) and (3.13), reduce to (1.5) and (1.6) in the absence of interspecific presettlement competition.

These diffusion coefficients are not of interest by themselves, except perhaps to students interested in the appficatian of diffusion processes to biological systems. However, the stationary distribution of the process, analogous to a frequency distribution for the proportion of space occupied by adults of species i as t approaches infinity, can be derived under certain conditions once the diffusion coefficients are known. Furthermore, the conditions few the existence of the stationary distribution can be interpreted as the conditions for coexistence of the two or more species in the system. Based on the means, variances, and covariances of certain life-history parameters we get simple formulae for the coexistence of the species and the stationary distribution.

These types of results have been uncommon in the literature of theoretical competition models, although some results appearinTurelli (1978,1981a), Turelli and Gillespie (1980), Chesson

(1982,1984,1985), Abrams (1984), and Shmida and Ellner (1984). However, in terms of the stationary distribution, none of these findings art as extensive as those given here for the lottery model.

The conditions of coexistence for the three versions of the lottery model considered in this document are presented in equations (1.8), (1.9), (2.20), (2.21), and (3.14). Theusefullnessof these equations lies in their simplicity and interpretability, and are based on the means, variances,

and covariances of various life-history parameters which incorporate the birth and death rates of the

adult organisms. The contention that environmental variability promotes coexistence is apparent from the position of

is more likely for larger values of 0^. The role of the death rates is evident from theposition of the

geometricmeandeathrates.dj, in these equations. Clearly, smaller dj favor coexistence. The

competition coefficients in the presettlement model represent an additional complexity, but

the effect of interspecific competition (and intraspecific) is easily discemablein (3.14) due 33 to the simple way in which the a's are incorporated into the equation. As noted ab ove, the formulae for the coexistence criteria are easily understood. Similarly, the stationary distributions are explicit and easilyinterpreted. Furthermore, they are based on the means, variances, and covariances of the same life-history parameters. The densities obtained, given by(l. 10), (2.19), and (3. IS), are mixtures of beta, Dirichlet, or more complicated distributions, respectively. Using these equations, it is possible to determine the effect of the various life-history parameters onthe magnitude and severity of the population fluctuations of the

adults. In this manner, the djare concluded tobecritical to stablepopulationfluctuations, espedallyif is large. Intuitively, this is reasonable. If only a small proportion of the adults are dying each season, then large population fluctuations should be rare and coexistence will be stable.

This is called the storage effect by Chesson(1983).

The roles of environmental variability and the storage effect in relation to coexistence in the lottery model were formulated by Chesson and Warner (1981) and in subsequent papers. The diffusion analysis is in general agreement with these earlier results, as emphasized by the comparison to Chesson and Wamer's coexistence criteria, figure 1. The close correspondence,

especially forsmail ||t|, provides the evidence. Furthermore, for small geometric mean death rates,

Chesson and Warner's analytical results are identical to those presented here.

The lottery model was the firs stochastic competition model for which envifonmentai variability was found to promote coexistence of species. Analyses of earlier models, such as the

analysis of the stochastic Lotka-Volterra equations by Turelli, did not conclude such a favorable

effect of a variable environment. Even earlier analyses of deterministic models (e.g. May and

MacArthur, 1972) found that a fluctuating environment would favor competitive exclusion. Since the appearance of the lottery model, though, otherstochasticmodelshavecondudedthat

environmental variability is favorable to coexistence (Abrams, 1984;ShmidaandEllner, 1984).

The results presented in this paper support the contention that environmental variability favors

coexistence, and the effect of 0 ^ is easily construed from the formulae presented. 34 Nowthat these explicitformulaeareavailablefor the lottery model, the next step is to attempt to validate them with data from real populations. The lottery model may well provide a good description of population dynamics for many types of natural communities. Space-limited communities are found evetywhere in nature and field studies are necessary to estimate the life-history parameters of these types of communities. Measurements taken over timein field studies could be used to estimate appropriate means, variances, and covariances to test whether the cnexiswoce criteria are valid for coexisting species. Such a test, deri ved from the coexistence conditions given in this paper, would be the type of aatistical tea known as a null model (Connor andSimberloff, 1986). It would not test for the underlying mechanisms of lottery competition, but whether the natural populations satisfy the predictions put forth in this paper. A similar test could be formulated for comparing the actual frequency distribution of the species with ine stationary distribution given hoe. The advantage of this null model approach is the ease of calculating estimatesfortheappropriate life-history parameters, given sufficient data. Clearly, datashouldbe collected over time for various space-limited communities in order to attempt to corroborate the lottery model. APPENDIX A

Derivations For Chapter I

Diffusion Coefficients

We have defined p(t) = $ j(t)%(t) / and assumed E[X(t)] = qi, Var[X(t)] = co2, 8 j(t) = djeYiW, E[Yi(t>] = 0. i = 1.2, Cov[Yj(t). X(t)] = c8 j , and Ccrv[Y2(t), -X(t)] = %

Note that X, Yj, and Y 2 are of the form X = cp + Ve V, Yj = %/c Vj, and Y 2 = ■Jz V2, with the

Vs having mean zero and appropriate variances and covariances (e.g. Var(V)=02 ). In the ft,'lowing derivation, itwill be necessary tohave convergence in mean to o(e) of functions of X,

Yj, and Y2 which are Op(C3/2) terms, and thus we must assume regularity conditions which allow us to conclude that EfCyc^)] = 0(f?ty = o(e). For example, assuming that X, Yj, and Y2are bounded random variables is one such set of conditions. To show this, suppose f(X) y V ) where ^(V ) is a finite random variable. Now, to approximate f(X), note thatf(X) = 2^ (e *^2)11 ^ (V ) + Op(c(m+1)/2) andE[0p(£/2) unless V is a bounded random variable. We are not seeking the most general conditions under which the moment approximations hold, but only to verify that they at least apply to a broad class of situations. Thus, assume that E[Op(t^m+^ ) ] = 0(^in+^ wth m = 3 in the derivation which follows. Substitution of the above assumptions and definitions into the lottery model yields

PlfltyO-p^OSjW APj= P,(t+e) - Pj(t) = Pj(t )P2(t){------} pj(t)Pj(t) + p 2(t)P2(t)

exp[X(t)] -1 = d1d2 P1(t)P2(t){------— ------} . (Al) djP j(t )exp[X(t) - Y2(t)J + d2P2(t )exp[-Yj(t)J

35 36 Using power series expansions for e^W, e'^lW, e '^ 2 ^ and taking expected values gives

Ed1d2x(l-x) E[AP! | Pjd) = x] ------{d2[|t+0 1+O2/2] {d1x+d 2(l-x)]i

+[(d|-d2)ji-d i82 -d20 i-(dj+d2)O^/2]x} + o(e)

edjd2x(l-x) {djxQi - 02 - o 2/2] + d2(l-x)fji + 8j+ o 2/2]} + o(e) (A2) [djx+d2(l-x)]‘‘

and

eldjt^oxO-x)]2 E[AP12 |P 1(t) = x] = — ------+ o(e) . (A3) [d,x+d 2(l-x )]2

Therefore,

Wx) = S o E Efpl(t+G) - m I pt(£) = X1

d jd ^ l-x ) {djx^ - 02 - 02/2] r d2(l-x)[H + 0j+ a2/2]} (A4) [djx+ttyl-x)] and

, h i , [did20x(1-x)]z ° (x) = clo eE[{pi(t+e) - pi

Boundarv aassificarion

In order to classify the boundaries, it is necessary to determine the conditions under w hich

certainintegrals are finite (Karlin and Taylor, 1981). Letting = -(Yi+ 1) / d j , tt 2 =

-0/2+ l)/d 2 , Vj = 2(|l + 0 j) /o2, y2 = 2(-[t + 02)/o 2, cj and c 2 are positive constants,

^6 (0 , l)isarbtirary, then for the left boundary these integrals are sfM -So £*7)47 •

where s(y) = exp[ -|][ { 2[l(x) / C^x)} dx]

scj/*! (l-y )a 2 (forthe lottery model),

M(0.x] = S 5 laXm(y)dy . (A8 )

•where m(y) = [02(y) s(y)]'^

= O lttty + ^(l-y )!2 y< ai + 2) (l-y)'(a2+ 2) (forthe lottery model), (A9)

2 (0) = fo { £ m(y)dy }s(z)dz. and (A10)

m = lXQ{lXz s(y)dy }m(z)dz . (All)

Now, S(0, x]< °c when Otj >-1, which implies dj >Yj + 1- Otherwise, S(0, x] = «>. Similariy,

M(0, x]<«> when-(a j+ 2) >-1, which implies dj < Yj+ 1. Otherwise, M(0,x]=°°. According to Karlin andTay'or (1981), S(0, x] = => 2(0) = 00 ^ M(0, x]=«> N(0) = <*>. Thus,

2(0) = 00 when dj s y<+ 1 and N(0) = «> when dj s Yi+ 1- It remains to be determined if

2(0) =00 when dj > Yj+ 1 and N(0) = <» when dj < Yj+ 1-

Assuming that dj >Yj+ 1 (=> Ctj > -1 and-(ctj+2)< -1), from (A10) we have

2 (0) = Jo {£ c2[(dj-d2)2y-«l +2 d2(dj-d2)y'(0tl'f l ) + (d2)2y-(a l +2>] (1- y)-(«2+2> dy}

x Cjz“ l(l-z)a 2 dz . Three situations are possible:

(l)Suppose d j * l andaj * 0 (by assumption, a j *-1). Then for x and z near 38

2 (0) * c, 0 2 Jj {(dr d2)2[y(-a l +1)]/(-cv1+l) - 2d2(d1-d2)|y-0{ll/(o1)

-(d^^l+ ^ai+ l) |* }z“ldz

cl®2 (finite since a j * -1, fj(x) < <», andxG(0,1)) (a ,+l)

ci®2 (di (finitesince a j * 1 ) 2(-Ctj + l)

2cjC2 d2(dj-d2)x + ------(finite since a j * 0 ) «1

x -i ln z | 0 ( +oo since cjcj (^/(Ctj+l) > 0 ) (a 1+l)

= +~ • (A13)

(2) Suppose a j = 1. H en far x and z near zero,

2 (0) *> {<) [(dt-dj) 2^ y ' 2d2(d1-d2)y' 1 - (d2) V 2 |* ] z dz

= clc2 ^2(x) x2^ ( fy(x) < 00 xG(0, I))

- Cjt>2 (d j^ )^ |q zlnzdz ( finite since 0 <-zlnz <-inzfor

zG(0, x)and Jg -inz= 1)

+ 2cjC2U2(dj-d2)x ( finite since xg(0, 1 )) 39

+ (c1c2d22lnzV2 |J ( + 00 since > 0 )

= + 0 0 . (A14) (3) Suppose a j =0. Thenforx and z near zero,

2 (0) * CjC2 fJ [ Q i- ty b + 2d2(d1-d2)ln y - (d2)2y 1 1*1 dz

= cjC2f3(x)x ( finite since f3(x) < °° and x£(0, 1))

- cl92 (dl'd 2)2 ( finite since x£( 0, 1))

- Icjt^d^dj-dj) Jg In z dz ( finite since lg*-lnz dz = 1)

+ (c1c?d22lnz) jg (+00 since CjC2d2 > 0 )

= + 0 0 . (A15)

Therefore,2(0) = +°° when dj > Yi+ 1. Assuming that dj < Yi+ 1 (=> Ofj < -1 and (Oj+2) > -1), from (All) we have

N(0) = Io{lzCiy«l(l-y)«2dy}

x c jR d ^ jV ® ! +2d2(d1-d2)z-(°!l+1) + (d2)2z-(0!l +2)](l-z)'(a 2+2)dz . (A16)

For x and z near zero,

N(0) * Jg {{* c j/^ l dy }c2f(dj-d2 )^z'ttl + 2 d2 (dj-d2 )z‘(a l +^ + (d2)^z'(Q£1 dz 40

cxOL f4(x) ^ ( d ^ ^ z ^ l + 2d2(dr d2)z-(a l+1) + (d2)V (a t+2>] dz (otj+1)

(finite since ft j < - 1, f4(x) < andxE(0,1))

c1c2 (dr d2)2x2 (finite since 0!j < - 1, andx£( 0, 1)) 2(ftj+l)

2CJ02 d ^ - d jjx (finite since ft j < - 1, andx£(0,1)) (ft1+l)

in z |g (+oo since cjt^ d 2z/(Otj+l) < 0 ) (Ot1+l)

= -w o . (A17) Therefore, N(0) = -h» when d j < Yi + 1 ■ Using the boundary classification scheme in Karlin and

Taylor (1981), the left boundary, 0, is a natural boundary because 2(0) = N(0) = +w for all d j£ (0 ,1). Due to the symmetry of the process, the right boundary, 1, is also anatural boundary.

Comparison of Coexistence Criteria

The conditions given by Chesson and Warner (1981) are E[5j(t){p(t) -1}] > 0 and

E[82 (t){p‘*(t) -1}] > 0, where p(t) = P jf t ^ t ) / p 2(t)8 j(t). Making the definitions that p(t) = e^W, E[X(t)j = [t, Var[X(t)] = a2, 8 j(t) = d^e^1), whereE[Yj(t)] = 0,

Cov[Y j(t), X(t)] = 0 j , and Cov[Y 2(t), -X(t)] = 02, and assuming that Yj(t) and X(t) have normal

distributions, then E[ 8 j(t){p(t) - 1}] = 41 = E[51(t)ex

= d1E[ex(t)+Yl(0] - dtE[eYl]

= dieXp{E[X(t) + Y^t)] + Var[X(t) + Yj(t)] 12} - d1exp{E[Y 1(t)] + VarfY^t)] / 2}

= djexp{(l + 8j+C^/2}exp{VarfYj(t)]/2} - d j 3xp{Var(Yj(t)]/ 2 }. (A 18) A Now, E[8 j(t){p(t)-1}] > 0 implies exp{}l + 0j+ 0 12} > I, giving the condition

<3^ > -2(p + 0j) Similarily, for the condition E ^ O fP '^ O - 1}] > 0, we get 0^ > -2(-|i + 0-.).

Therefore, the coexistence criteria from Chesson and Wamer (1981) become

<£■ > 2 max[-(|l + 0 j), -(-|i + Oj)]. APPENDIX B

Derivations For Chapter II

Diffusion Coefficients We have defined X^t) = ln[pj(t) / 5j(t)] and assumed E[Xj(t)] = q 4 , V ar^ t)] = eOj2,

Cov[Xj(t), Xj

AP^PjCt + Q-P^t)

[2 8 a(t) Pn(t)]pi(t) = Pi(t){ 5------HD) ZPfl

Pn(t)8i I h m . fl— i It ' 1 W a 8a(t)Pi(t) S ^ p ^ t) 6n(t) Pn(t) 8j(t) I t - [r - r - i p«(t) a «i(t) 8fl(t) Pj(t)

Yn(t) [Xn(t) - X|(t)] [Xj(t) - Xn(t)] 2dn e e [e - l]Pfl(t) (Bl) (Yfl(t) - Yj(t)] [Xn(t) - Xj(t)] 2(dfl/dj) e e P„(t)

± Xj(t) ± Yj(t) Employing power series expansions of e and e , and taking expected values, we get

42 43

OJj Hi H|j E[APj|E(t) = x] ------{ S d^ x^ O li-^ + a )8uiu+ ( ------)0uU I V » | dA

+ (— l/2XOi2 + Ou2 - 20iu)] a d»x®

1 + 1 m X,J xm K^i ' ®iu * ®im + ®iun^ + ®iim" ®uinJ J + ’ (B2) y/i y ** m £ wfi*ju

cd^xP E[APi2 | &t) =x] = - - ^ p I, l ‘‘u dm xu % (°i ' 0iu ' 9im + 9W + •

and

edi djXiXj E[APjAPj|EXt)=x]= ^ | ^ d^dm3caxmC0ij -ejll-0im + 0^ ) + 0(e) (B4) APPENDIX C Derivations For Chapter III

Diffusion Coefficients

If we define Rj(t) as the per capita recruitment rate of species i during [t, t + 1), then

Pi(t) y t ) = [^(tJP^t) + %(t)P2(t)] ------(Cl) P l W O + f c W ) and the formula for the two-species lottery model (1. l)is

Pi(t + 1) = [1 - 8 ^ ( 1 ) + Rj(t)Pj(t). (C2)

To make (Cl) and (C2) density dependent, substitute (3. 1) for p^(t) in (Cl), where Bj(t) is the birth rate of species i during [t, t + l)*ndfj[ 8 jPj, BjPp is given by (3.4). For species 1, we obtain

B2P2 . BjP i Rl(0 = [8iPi + Bjexp {-[Iq «12(Blp i- S2)dS2 + ^0 °11 ( S1 • 0 ) dsl 1)

r®2P2 r ®l? 1 - { B1P1exp {-[ J 0 a i 2 (B1P1>s2)ds2 + J0 a n ( sj. 0)dsj ]}

®1^1 r^2^2 + B2P2exp{-[Io a 2 i (Sj, B2P2)dsi + Jq a 22 (0 , S 2 ) ds2 ]} }. (C3)

Substituting (C3) into (C2) and using the a notation of (3.9) and (3.10), with manipulation we get

APi =P1(t + e)-P1(t)=

^2®1 ,^ 1^1 f ®2^2 = Plp2 { exp(J0 a 1(s1,B2P2)ds1 - 10 Of2 (BjPi, S 2 ) ] -1 } 8 j B2

44 45

Pi SjBi (Blp t i ®2P2 + { _ ------exp[ 10 a 1(sl>B2P 2 )ds1 - 10 oe2 (B1P1. s ^ d ^ ] °2 ^iB j

p2 + — } (04) 8,

If we assume that the a's of the form cos , then

Blpl B2p2 esp[cl0 Oi(s1,B2P2)ds1 - d 0 a2(B1P1,s2 )ds2 ]

Bjpi f% p2 = 1+£[I0 «i (sj. B2P2)dsi ] - e [ 10 a 2 (B1P1,s2 )ds2 ] + op(e) . (C5)

Note that if Oq (sj, S2)iscontiatious(msj) forsj G [0, lj.then

liin BjPi Zi0 E{[I0 «1 (»l. W d s ! + op(C)/t]|P1(t) = x }

tblx = Iq a i (si • t y 1 - x» dsi (C6) and

lim ®2P2 e*0EUl() ^ (B jP j. 92>«i92 + Op(£)/e]iPi(t) = x }

M l -x) = Jq Ot2 (b1x.s2)ds2 • (C7)

Proof of (C6):

qi« + yfV« DefineBj=bje (C8)

eji2 + ycv2 and = t > 2 e (C9) 46 Then,

B P liiO E{[Jo! «i (sj. B2P2)ds1 + op(e)/e]|P 1(t) = x }

lini b|P | BjP i = £i0 ( E[ I0 (s,. BjPj) ds, ] + E[ L r a , (sj. B2P2) dSl ] + o(£)/£ } (CIO)

Clearly, the third part of (CIO) converges to 0. The first part converges to (C 6), if a j (s j, s2) is continuous (in sj) for S| e [0,1] (because of the bounded convergence theorem). The middle part of (CIO) converges to 0 (if Vj is a bounded random variable) because

B P E{ [ ^ 1()ti (si* B2p2 )dsi = x }

s EI faj| dsj i EI (constant)dsj (since a j is continuous)

qii + v t Vi = bjxcE[e - 1]

qtj AlVjl ibjxc[e Ee - 1]

-* Oas d-0 (bythebounded convergence theorem). (Cl 1)

Result (C7) follows by symmetry.

Therefore, substitute (C5) into (C4), assume p(t) is defined by (3.6), define X(t) = In p(t),

E[X(t)] =C|l, Var(X(t)] 8j(t) = djeXi^, E[Yj(t)] = 0 (dj is the geometric mean of 8j(t)),

CovfY^t), X(t)] = €0j, Cov(Y2(t), - X(t)] = e8 2, lim and In bj = c|q E[ln Bj], Making use of (C 6) and (C7), using power series expansions of e^, e^l, e^2, assuming regularity conditions as in Appendix A about E[Op(C^)] = o(£), and defining A(x) by (3.11), we get 47

E[AP1|Pl(t) = x] =

Ed1d2x(l-x) {dlX[p. - 02 - o2/2] + d2(l-x)[[i + 0j+ o2/2] [djx + <* 2(1_3E)1

+ [dtx + d 2(l-x)] A(x) } + o(e) (C12) and

e[did2 ox(l-x )]2 EjAPj2 1 Pj(t) = x]= — + o( 0) . (C13) [djx + d ^ l -x )]2

BsufldagjHassificatioa

Classification of the boundaries follows the same procedure as in Appendix A. As such, itis necessary to determine underwhat conditions equations (A 6), (A8 ), (A10), and (Ail) are finite.

Defining A(z) as in (3.11),

bjz W l-z) A(z) = j0 ajfy, b^l -z)) dy - 10 a2(bjz, y)dy , (C14) and

2[}i + 0j + O^il] a o = ------= . (C15) dj C r

al = —— o[di(|i-02-° 2/2 )-d2(^ + 0l + °2/2)] • (Cl6) d f a a 1

2 *2 = o ' md

2 1 1 8 3 = _ ( ------) . (C18)

Then for |L(x) an 0^(x) gives by (3.12) and (3.13), (A 6) becomes

S M = S r o la C1 y'®0 (1 - y)*0 + al

I «2 a2 + a3 X exp{ L A(z)[— + ------]dz } dy (C19) 7 z 1 -z

Now, for z near 0 (and assuming a j(sj, S2) is continuous for Sj e [0,1]),

A(z) * aj(0, t>2)bjz + a 2(0, b2)b2Z - Iq2 a 2(0, y) dy . (C20)

For znearO, the term

a j *2 + A'(z) = A(z)[— + ------]* z 1 -z

^2 «2 lo * * 2 1 a l(°- ^ 1 + a 2(0, ------

+ (&l + 83 )! a t(0, b ^ j + a 2(0, b 2)b2] z

^2 - (dq +a3) I0 ot2 (°, y) dy . (C21)

Therefore, Jy A’(z)dz*

^2 « (constant)- a 2 [ Otj<0, b2)bi + Ot 2(0, b2)b2] + a2 [ !q 0£2(0, x) dx ] Iny 49

+ (*l + a3> t 1q a 2(0,x)dx]y . (C22)

Therefore, foraandxnearO,

nm f- 1 S(0, x] ® atoia (constant) y dy (C23)

andso^O, x]<<»iff[-aQ + ^ ( o C*2(Q, z)dzl > Otherwise,S(0, x] = 00. bj Sixnilariy, forM(0, xj given by (A8), wegetM(0, x] < <»iff[aQ - 2 - a< |q Ot2(0, z)dzl >

Otherwise, M(0,x]=«>. From Karlin and Taylor (1981), S(0, x] =°° => 2(0) = 0 0 and M(0, x] ^2 = 00 => N(0) = 00, Thus, 2(0) = 0 0 when [-bq + a2 Iq 0t2(0, z) dz ] s -1 and N(0) = <* when fb2 [sq - 2 - aj 1q Ct2(0, z)dz]s-l. Following the methods of Appendix A, it can be shown that

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