Dispersal strategies in patchy environments

THEORETICAL

POPULATION

Dispersal

26,

165-l 91 (1984)

Strategies

in Patchy

BIOLOGY

Environments

SIMON A. LEVIN Section of Ecology and Systematic&

Cornell University,

Ithaca, New York

DAN COHEN Department

of Botany, Hebrew University of Jerusalem, Israel

AND ALAN HASTINGS Department of Mathematics,

University

of California,

Davis, California

ReceivedFebruary 19, 1984

1. INTRODUCTION The phenomenon of patchiness-variability in environmental quality both in time and in space-presents the biota a number of evolutionary problems, each with many possible solutions. That challenge, more than being a threat which species must confront to survive, is the salvation for the large number of species which would become extinct in constant environments. Indeed (Levin, 1976, 198 1; Hastings, 1980), patchiness can significantly increase the potential for coexistence, and patterns of unpredictability or disturbance which increase patchiness will in general increase diversity. Dispersal and dormancy, two of the most common mechanisms for dealing with patchiness, are to some extent analogues in that they provide the means for the evolutionary unit to average, respectively, over space and time. Our focus in this paper will be on the evolution of dispersal strategies, including situations in which some dormancy is assumed; in a sequel to this paper (Cohen and Levin, in preparation) we consider the evolution of dormancy and its interaction with dispersal.

2. DISPERSAL Any environment is subject to change, and the organisms which exploit it must either develop adaptations to deal with fluctuations in favorability or 165 0040~5809/84 $3.00 65312612.4

Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

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LEVIN, COHEN, AND HASTINGS

become extinct. Since over a large area many changes are not synchronous, dispersal of seeds and other propagules may guarantee that some fraction reach a suitable habitat every generation. Levin and Paine (1974) argue that in environments subject to disturbance, most species are locally ephemeral, but depend on disturbance over space to convert a locally unpredictable environment into one which is globally predictable. Indeed, as Harper (1977; see also Werner, 1976) points out, “All successional species are.doomed in their present habitats, and their continued survival depends on escape and establishment elsewhere.” Dispersal is escapein space.For a given pattern of environmental variability, how much dispersal is most advantageous? What fraction of one’s propagules should be dispersed, and over what distances? We do not treat directly the latter question in this paper (but seePalmer and Strathmann, 1981). The former, which is our focus, has also been considered in a number of related theoretical studies of the evolution of dispersal, including Den Boer (1971), Gadgil (1971), Reddingius (1971), Strathmann (1974) Roff (1975), Hamilton and May (1977), Comins et al. (1980), Comins (1982), Kuno (1981), Motro (1982a, b), and Metz et al. (1983). Harper (1977) states that “Attempts to determine the fittest dispersal strategies for specified environmental regimes have so far proved too complex to be handled other than through numerical experiments on digital computers.” The present paper, as well as others since 1977 (Hamilton and May, 1977; Comins et al., 1980; Comins, 1982; Motro, 1982a,b), although also depending to a large extent on numerical simulations, begins the development of the analytical results necessary to bring understanding of numerical results. Our approach is to develop a mathematical model which will allow the exploration of environmental effects on the evolution of dispersal. For purposes of clarity, we choose to begin from a somewhat restricted model and leave generalization until later. Thus we initially consider only organisms which do not have the potential for dormancy, and compare dispersal strategies. We restrict also the modes of dispersal we are willing to consider, since each form raises different problems. Basically, we assume that dispersal is an individual rather than a collective action. Moreover, it is an “event” in the life of an organism, undertaken at some cost. Thus we do not consider repeated short-range movements such as are associated with foraging. We further ignore the effects of geographical location and dispersal distance by treating dispersing individuals as forming a common pool rather than using the more complicated diffusion-reaction approach necessarywhen considering spatial patterns (Levin, 1976). Specifically, we shall devote attention to annual plants, which exploit opportunities in the community by dispersal of seedsfrom other patches and by long-term storage of seeds in the soil. However, other organisms (e.g., some marine invertebrates) which are sessile as adults but widely disperse their young fit the same general

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mode (Vance, 1973; Strathmann, 1974; Scheltema, 1977; Christiansen and Fenchel, 1979; Palmer and Strathmann, 1981). We leave for a later paper consideration of dispersal strategies in plants which are perennial and iteroparous, but do discuss the effects of dormancy; there are strong analogies (see Werner, 1979). We also here restrict attention to asexual reproduction; others authors (Hamilton and May, 1977; Motro, 1982a, b) have discussed some aspects of sexuality. Our focus will be on environments which are locally unpredictable, but where the local environmental variation is described by a stationary probability distribution which does not vary among sites. Local increase in population density decreasesthe growth rates of individuals which remain close to the parental site; on the other hand, only a fraction of the dispersing individuals reach a favorable habitat, and there may be other costs associated with dispersal if dispersing individuals differ morphologically from nondispersing ones. Dormant individuals also suffer from mortality in the soil. How does natural selection balance the trade-offs between these effects? To aid in the understanding of this problem, we develop a mathematical model incorporating these general characteristics. We assume that there are two forms of dispersal available to organisms, short-range and long-range. Short-range dispersers are those which remain close to the parent organism and are for our purposes termed nondispersers; long-range dispersers are simply termed dispersers. Within a species, individuals are allowed to differ genetically according to the fraction D of their offspring which are dispersers, and possibly with regard to the fraction G which germinate each year; the probability of germination is independent of whether a seed was a disperser or not. Organisms use a variety of different mechanisms for achieving mixed strategies of dispersal: dispersing individuals of a particular genotype may or may not differ morphologically from their congenotypic nondispersers. Some plants develop elaborate structures to aid dispersal, and may produce two types of seeds or simply depend on chance to divide the seedsinto dispersers and nondispersers. On the other hand (Paine, 1979), the annual brown alga Postelsia palmaeformis drips meiospores near the parent throughout the summer, and then depends on the floating capabilities of the adult sporophyte to transport remaining seedsto distant sites when the adult is torn loose in fall and winter storms. In the latter case, G equals 1 and 1 - D is the fraction of spores which are dripped during the summer season. Sites vary in quality both in response to differences in local densities of seeds,and according to density-independent random factors. Proximity (e.g., nearest-neighbor) effects of the type introduced, for example, in steppingstone models (Malecot, 1948; Wright, 1949; Kimura and Weiss, 1964) are ignored in our model. This is nearly identical to an assumption that dispersal is basically long-range; it is equivalent to assuming that at any point in time

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each patch is exposed to the same pool of potential colonists; i.e., exchange is only through a bath. Again, this distinguishes the kind of movement herein considered from shorter-range diffusive-type motion. 3. THE BASIC MODEL

Based on the assumptions already stated, we consider an environment made up of L distinct sites, numbered i= l,..., L. Consider first a single genotype. Let si(t) be the number of seedsat site i in generation t following the dispersal phase; this includes both aboveground seeds which have just dispersed in and dormant seeds,as well as nondormant ones which were not dispersed. We make the simplification that a fraction G of all seeds will germinate, independent of whether they were dormant at t - 1 or dispersed in. The germinating seeds GS,(t) will then have a per capita yield Yi(t) = K,(f) F(GS,(t)), where K,(t) is a nonnegative random variable (uncorrelated over space and time) and F is a continuous function which measures the effects of density on decreasing yield. The distribution from which K,(t) is drawn is assumed constant over space and time; it is independent of Si(t). After yield, a fraction D disperse; a fraction (1 - a) are lost, but the residual fraction a find new sites and are uniformly spread over patches. (This implicitly assumes a large number of seeds; a more general treatment allowing for Poisson arrivals is considered by Comins et al. (1980), but for a somewhat different model.) For the present, D is taken to be a constant; this is relaxed in Section 7. The nondispersing seedsare assumedto undergo no additional mortality, but the nongerminating seeds have mortality 1 - v. Thus, after one year, the new value of Si is Si(t + 1) = S,(t)[GY,(t)(l

aDG + L

-0)

+ (1 - G)u]

L C Sj(t) Y,(t); j=l

i= l,..., L

(1)

(see Fig. 1). For a, v given, types are distinguished by their associated dispersal and dormancy fractions D, G. What type or types would natural selection choose? Cohen (1966, 1967), in discussing the optimization of reproduction in a density-independent, randomly-varying environment, postulates maximization of the long-term growth rate. However, populations cannot grow without bound, and thus we must consider both density-dependent and frequency-dependent effects. To understand what will be selected naturally, one must replace (1) by a system Sf(t + 1) = Sf(t)

G’Y;(t)(l -D’)

+ (1 - G’)v + qy

7 Sj(t) Y;(t), (2)

DISPERSAL STRATEGIES

169

FIG. 1. A generalized schematic for the basic model (1). Terms are explained in text. a, v are the survival probabilities of dispersing and dormant seeds,respectively. Si(f) Yi(f) denotes the spatial average (l/L) cf=, Sj(t) Yj(t).

where 1 denotes the particular type being considered, and (G’, 0’) defines its “strategy” in terms of germination and dispersal. For the remainder of this paper we assumethat, for all 1, G’ = G and Yf(t) = K,(t) F (G C Si(t)) = K,(t) F(z), I

(3)

where z = G 2, Sf(r). As before, the random variable K,(t) is assumed uncorrelated over space and time, but will be the same for all types within a particular patch. Simple-minded approaches to determining the winning strategy or strategies, for example, maximizing (without constraints) the long-term (geometric) growth rate, are not valid and do not lead to sensible results. For example, for most models of interest, the winning population(s) will eventually be “regulated,” fluctuating (in stochastic equilibrium) about a fixed finite level. The long-term geometric mean growth rate will be 1 (asymptotically). This is larger than the corresponding growth rates for any types which have been eliminated, but this is only descriptive of a particular historical interaction. Within limits, any type considered in isolation would regulate with a long-term growth rate of 1; therefore, it is useless to think about the winning type as that which maximizes the long-term growth rate. Further, Roff (1975) and Hamilton and May (1977) show that the winning type in such problems is not, in general, that which maximizes population size (see also Comins et al., 1980; Motro, 1982a); there is no reason why it should be. Since we are assuming asexual reproduction, a stable winning type (if one exists) in general should be one whose strategy is evolutionarily stable (Maynard Smith, 1976; see also Levin, 1978): once established, it cannot be

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invaded and displaced by mutant types. Its asymptotic growth rate will be maximal within the steady state environment defined by its own presence,in the sense that no other type can have a growth rate greater than 1. However, because of the stochastic nature of the problem at hand, this notion of an evolutionarily stable type must be revised before we can apply it. For a finite number of patches, there are two sources of variability: environmental stochasticity, reflecting the random nature of the environment of a particular patch; and sampling stochasticity, due to random sampling from a finite number of patches. There will not, in general, be any type which cannot be invaded: As in the real world, there is always the chance of a run of years which will favor an inferior type. We expect this problem to be minimized if the number of patches is large. In a similar model, Chesson (1981) has shown that as the number of patches increaseswithout bound, the average population densities converge to a deterministic process provided the initial conditions satisfy a technical exchangeability condition. In our experience with simulations, variation in winning type due to sampling stochasticity was of minor importance as long as the number of patches was 20 or larger. We use these facts as justification for computing evolutionarily stable strategies (ESS’s) using the limiting form of the model as the number of patches becomes infinite. The definition given above of an evolutionarily stable strategy is nonexclusive. It does not exclude the stable coexistenceof types, through frequency dependence. On the other hand, there may be several possible exclusive ESS’s (or combinations of them) which cannot be invaded once established, but cannot always invade from low densities. Furthermore, the definition of an evolutionarily stable strategy is complicated by the fact, which cannot be ruled out a priori, that a particular type may be superior to any other in head-to-head competition with particular combinations of other types. Finally, distinction must be made between local and global stability within the space of possible strategies; and this, and other constraints including the lack of evolutionary equilibrium, may prevent the ESS from ever being reached in the natural environment. Thus, the ESS notion, while a convenient place to begin as a working hypothesis, must be used with caution. In simulations, other problems arise. In any particular realization of (2), the winning type may not be the one which is in theory an ESS. Stochastic effects coupled with the influence of initial conditions which limit the types considered may lead to results which differ from theory, and this is especially true becausethe strength of selection is weak in the vicinity of the “optimal” strategy. We assumehenceforth that F(z) is a monotonically decreasing function of its argument z, and that zF(z) is a monotonically nondecreasing function of z. The latter assumption means simply that the total yield does not decrease

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171

when the number of competing seeds is increased. We shall attempt to characterize analytically necessary conditions for a winning strategy, an evolutionarily stable strategy, in the case when the number of patches is effectively infinite. This characterization defines the strategy, but except in special cases it does not lead to easy computation of the winning value of D in (2). In general, it may not even be the case that (1) will admit a nontrivial stable stationary probability distribution: this will depend on the form of the function F, on the properties of the distribution of Ki, and on D and G. Henceforth, we assume that (at least for D (0, l)), a stable stationary distribution will be reached and, further, that long-time averages at a particular location will be identical with spatial averagesat a particular time. The existence and stability of a stationary distribution represent challenging mathematical problems, and Ellner (1981, 1984) has obtained results which bear on the question.

4. THE SATURATION MODEL, AND THE GENERAL CASE WITHOUT DORMANCY

Because of the problems associated with models using finitely many patches, and especially the difficulties of deriving analytical results, we will analyze our model in the limiting case that the number of patches is large. However, as Chesson (198 1) has noted for a similar model, stochastic effects may still be important even though the number of patches is large. Determination of the evolutionarily stable value D* of D (given a specified value of G) is difficult or impossible in general, but is feasible in some special cases. In particular, consider as a starting point the “saturating” yield function defined by F(z) = l/z.

(4)

With this function, the total yield in any patch in any given year will be K,(t), the random variable. Each patch is saturated each year: the total yield

is always achieved regardless of the number of seedscompeting for it. After yield and before dispersal, patch i will have Ki(t) seeds above ground and Ri(t) dormant seeds, where, in the absence of knowledge concerning the history of patch i, R,(t) may be thought of as a random variable selectedfrom the steady-state distribution for the dormant pool. For a single patch, Ri will be positively autocorrelated over time; however, for a given t, Ri and Ki are uncorrelated. After dispersal, the patch will have Ki(t)(l -0) + a Dk nondormant seeds,where 5 denotes the expectation of

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LEVIN, COHEN, AND HASTINGS

K,(t). Thus, allowing for the mortality of 1 -u of the dormant seeds,the number of seedsprior to germination and yield will be Si(t + 1) = K,(t)( 1 - D) + O[Do + URi(t).

(5)

From (2), R,(t) = (1 - G) si(t). Some care must be taken in the usage of averages.Let P(k) = Pr[Ki = k]. (For simplicity, we assume here that there are finitely many choices k; the extension to continuous distributions is straightforward.) Then k = c P(k)k.

(64

Define now, in the steady state,

Then i== (1 - G)S; further, the stationarity assumption implies that time averagesof Ki equal spatial averages so that

Using these facts and (5), we obtain S= /?(l -D + aD)/(l - ~(1 -G)).

WI

In the case of no dormancy (G = l), the complete prior history of a patch is obliterated after yield; the only thing which matters is the current yield Ki. After yield, every morph will be found in a patch of size k with probability kP(k)/k. The optimal strategy D* may be 0, 1, or an intermediate value. If the latter applies, then, as in Levin (1980), D* will be the value of D for which the one-year payoff to dispersers is the same as that which applies to nondispersers. If this equalization condition were violated, the “payoffs” from the two activities would not be equal, and a morph which placed a greater emphasis on the “superior” strategy would increase in frequency; that is, a type which does not equilibrate its expectations from the two activities is not pursuing an evolutionarily stable strategy. This argument rests on the observation that for a mutant type the payoff for each activity is the same as for the resident type, and that only the proportions dedicated to each vary. Hamilton and May (1977) consider a case where this assumption is not valid: all offspring produced at a particular site derive from a single individual, and so a mutant nondisperser faces a different average environment than a type with different dispersal fraction (see discussion in Levin, 1980). In this case,there is a higher premium placed on dispersal than

DISPERSAL

173

STRATEGIES

would otherwise be expected, primarily because of kin selection to reduce competition among siblings. Patch i will have K,( 1 - D*) + a D *k seedscompeting for the following year’s yield. Thus, by the stationarity assumption the expected yield Yi (at the same census point) for any seed which, after dispersal, is in a patch which experienced yield Ki the previous year is E,& = I;l(K,( 1 - o*> + a D*k).

(8)

Therefore, the expected yield of any dispersing seed is the average ED = a c P(k) E,, = ak c P(k)/(k( 1 - D*) + a D*k).

(9)

Becauseof the assumption of stationarity, the average yield per seed,over all seeds,is 1. Because of the equilibration principle discussed in the previous paragraph, the expectation ED must be the same as the average taken over all seeds,and hence must equal unity. Thus, the ESS is defined implicitly by l=akC

(10)

P(k)/(k(l -D*)+aD*&).

The same formula could also be derived using END,the expected yield for nondispersers,which also must equal unity. In computation of END,one must weight appropriately to take into account the nonuniform distribution of non-dispersed seeds;that is, more of them occur in patches which had high yields the previous year. Thus 1 = END= c kP(k) E,,/x

kP(k) = 2 kP(k)/(k(l - D*) + a D*k)

= 1 -ID*

IX (k(l-D*)fTk))/(k(l

-D*)+aD*l;)

= 1 -ID*

(1 - C(W)

-D*)

aD*k)/(k(l

(11)

+ a D*L)).

Formula (10) now follows by rearrangement of terms. Simulations with finite numbers of patches confirm that, to a first approximation, the strategy defined by (10) does in fact outcompete any other, and is the ESS. We expect that approximation to become increasingly accurate as the number of patches is increased. The optimal value D* as defined by (10) will tend to 1 as a tends to 1 (if dispersal is without risk, disperse everyone); however, as long as a ( 1 (there is some risk), D* is also less than 1. Provided the harmonic mean of k is positive, D* is an increasing function of a for all a, and tends to 0 as a is decreasedto the critical value a,, = (2 P(k)(k-I))-l/(x

P(k)k) = k/E,

(12)

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LEVIN, COHEN, AND HASTINGS

where k^is the harmonic mean of k and k is the arithmetic mean; (12) can be obtained simply by setting D* = 0 in (10). For a < aCr, D* = 0 is the ESS. As a is increased beyond a,,, D* increases, and at a rate given by the slope (13) where pilk is the relativized variance of l/k; i.e., P flk = I&;,,

= L2 1 P(k) k-2 -

1.

(14)

Recall that the harmonic mean 6 is the reciprocal of the mean of l/k. As a approaches 1, the distribution of I/k is no longer of direct importance (except to the extent that it is related to the distribution of k). As is easily shown (e.g., by expanding in 1 - a), D* approaches 1 as a tends to 1 at a rate given by its slope $$

(a) Ia=, = l/p: = K’/a:,

(15)

where pi is the relativized variance of k; i.e., p; = a;/k* = (r P(k) k*/k*) - 1.

(16)

If l= 0, there is a positive probability that k = 0 and thus a positive probability of local extinction in the absenceof dispersal. In this case, from (1% 1 = ark x

P(k)/(k(l

-D*)

+ a D*k)) + P(O)/D*.

(17)

k#O

Thus, as a + 0, the optimal D tends to w = P(O), and hence is positive. In the special case that there are only two possible values of k (k, and 0), (10) reduces to ((l-(1-~w)a)D*-w)(D*-1)=0.

(18)

Thus, the optimal solution in this case is D” = w/(1 - (1 - w)a)

(19)

as given in Levin (1980; see also Hamilton and May, 1977; Motro,

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175

1982a,b). More generally, if the two values of k are k, and k,, with probabilities w and 1 - w, then k,k, - a(wk, + (1 - w) k,)(wk, + (1 - w) k,) D* = (k, - a(wk, + (1 - w) k2))(k2 - a(wk, + (1 - w) k2))

(20)

= w/( 1 - c&/k,) + (1 - w)/( 1 - d/k,), provided a > a,, ; and D* = 0 otherwise. Note that D* is strictly increasing with a, as is to’ be expected from the fact that decreasing dispersal’s costs increases its desirability. It is instructive to rewrite (20) by setting k,=k-(1

-~)a,

k2=&wc,

(21)

in which we assume (without loss of generality) k, -k, = E > 0. Then the variance in environmental types is given by +w(1-w)&*;

(22)

and it is easily shown that D*=1+a(l-a)k2/(k2(1-a)2+(2~-1)~k(l-a)-u~).

(23)

From this it is clear that in accordance with (22), if w is held fixed and E varied, D is an increasing function of environmental variance for E sufficiently large; but this is not necessarily the case for small variance. In particular, if w > 4 (bad years are more common than good years) and E is small, then the linear term in E in the denominator of D* dominates and increased variance actually selects for reduced dispersal. Some insight into the meaning of (12) can be obtained by examining the analogous formula for a continuous case, a lognormally distributed k for which the mean and variance of log k are respectively ~1and u*. In this case, it is easily shown that the harmonic mean of k is exp@ - a*/2) and the arithmetic mean is exp(u + a*/2); thus (12) becomesa,, = exp(-u*). If u is small, so that environmental variability is slight, dispersal is completely selected against as soon as there is any substantial cost to it. However, as environmental variability (as measured by the variance of log k) increases, the critical value of a decreasestowards 0; that is, there will be selection for dispersal even when dispersal is very risky. The possible confusion of interpretation, depending on the measure chosen, of even the qualitative effects of environmental variability makes clear the care which must be exercised in stating conclusions concerning the effects of variability.

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LEVIN,COHEN,

AND HASTINGS

5. THE GENERAL MODEL WITHOUT DORMANCY

Still keeping G = 1, we may by simulation examine optimal dispersal strategies for nonsaturating models and compare with the analytical formulae for the saturating case. In particular, we consider the hyperbolic yield function defined by setting F(z) = l/(b + z).

(24)

There is experimental evidence justifying this choice (Watkinson, 1980; see discussion in Ellner, 1981). Note that the saturation function (4) is the limiting form as b tends to zero. In the simulation program, k is a serially noncorrelated pseudo-random variable. The maximum per capita yield in any given year (the yield at vanishing low density) will be Y,,,,, = k/b and, in general, the per capita yield given z competing seeds is Y,,,/(l + z/b). In practice, Y,,,,, values (rather than k values) were determined at each iteration by a pseudo-random number generated by the computer. The evolutionary stability of the “optimal” type was tested by examining its stability against invasion by rare types (frequency -10e4) with I) values 10.02 from the putative optimal type. Our method was to iterate for 60 generations, after which a search procedure then changed the D value of the conjectured “best” genotype and the process was repeated until a value was found which could not be invaded. The small number of patches (12-20) in the simulations reported here introduced an additional stochastic effect. Repeated simulations with the same parameters and initial values could vary considerably, and average values were utilized. Such a stochastic effect, because it makes the environment less predictable, will introduce a consistent bias in favor of increased averagedispersal. In general, as may be shown analytically for any choice of the yield function, D * = 1 is always the optimal choice if a = 1. In Fig. 2, the simulated optimal strategy is shown for a high variance situation, and with a value of b equal to 10. The cases considered involved two equally probable (w = f) environments with Y,,,,, values which differ by a factor of 100 (case 1: Y,,,,, = 2, 200; case 2: Y,,,,, = 1, 100). This notation, which we shall use repeatedly, means that Y,,,,, = 1 with probability w and Ymax= 100 with probability 1 - w. In Fig. 2, the simulation results are compared with the formula D” = (100 - a(50.5)*)/(100 - a(50.5))(1 - ~~(50.5))

(25)

given by (20) for the case b = 0, and also with the formula D* = l/(2 - a)

(26)

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177

FIG. 2. Computed and simulated D* (optimal D) curves assuming G = I (full germination). Upper computed curve (x) assumes a saturating yield function and a possible zero yield year (Y,, = 0); lower computed curve (. ). also for saturating yield, assumes YL = Y,/lOO. In the notation of the text, Y,,, = Y, , Y,,. Simulated cases are for nonsaturating yield functions, with b = 10 and Y,,, = I. 100 (medium saturation) (0) and Y max= 2. 200 (high saturation) (0). In all cases, w = 0.5.

given by (19) for the case b = 0, Y,,,,, = 0, 100. Clearly, the singular effect of having a possible zero year (as opposed to simply a very low yield year) becomes important only for low values of a. Second, either of the simulated cases is well approximated (except at low a) by the saturating model, with the better Iit being obtained for the case of the higher yields. Note that increasing yields by a constant factor increases the mean population size and thereby decreases the importance of b. In other words, the case (2, 200) should be more closely approximated by the saturation model than is (1, loo), and the simulations show this. From (9), S= k(1 -D + aD). Thus for a- l, one has S- I;. For b = 10, k= lou,,,,, and c= 1010 in the first case and 505 in the second. Thus, for z = S setting b = 10 instead of b = 0 accounts for about a 1 % increase in F(z) in the first case and 2 % in the second. For smaller values of a, the mean population size is decreased, and the discrepancy from the saturating curve becomes generally more pronounced; further, the stochastic effects of a small number of patches also become more important. In Fig. 3, simulations for Y,,,,, = 10, 100 and b = 10 are compared with the computed values for the saturating model (20). Three cases are considered: w(=P( Y,,,,, = 10)) = 0.1, 0.5, and 0.9. Again, the Iit to the saturation model is excellent for large a (it would not be as good for higher values of b). As a is reduced, D* drops, variability increases, and the tit becomes less good. Moreover, reducing the factor for variability to 10 selects for much lower dispersal levels than observed in Fig. 2. Among those cases

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COHEN,

AND

HASTINGS

FIG. 3. Computed and simulated D* (optimal D) curves assuming G = 1 (full germination). Computed curves (x, ., +) assume a saturating yield function, Y,~= Y,/lO, and w values respectively of (0.9, 0.5, 0.1). Simulated cases are for nonsaturating yield functions, with b= 10, YL = 10, Y,= 100, and w values of 0.9 (@I),0.5 (o), and 0.1 (@I).

considered, w = 0.5 corresponds to the greatest intrinsic variability in the environment and this selects for the highest levels of dispersal for a small. However, increasing w also decreases the mean value of k and this selects for higher values of D (see formula (20)); this effect is most pronounced for a large, for which the selected values of D* when w = 0.9 are higher than with w = 0.5. This is because, as observed earlier for the saturating model, it is the relativized variance which controls the behavior near a = 1. a:/k’ = 1.96 when w = 0.9; 0.67 when w = 0.5; and 0.09 when w = 0.1; and the slopes at a = 1 are approximately equal to k2/ui, as predicted from the saturating model approximation. It is easily shown for the saturating model, and using (23) or (15), that for a close to 1 the minimum slope of D*(a) and hence the largest values of D*(a) will occur for w = k,/(k, + k,); thus in the present case w = 0.9 is very close to the maximum (0.909). In Fig. 4, similar effects of varying w are shown again for b = 10 for Y,,, ratios of 100, but with several different absolute values of Y,,,. Thus, the simulated curves confound two effects. Y,,,,X values were 1, 100 with w = 0.1; 1, 100 with w = 0.5; and 10, 1000 with w = 0.9. The computed curves in the saturation case depend only upon the Y,,,,, ratios rather than absolute values. Again, the tits are excellent. Note because of the high variability in Y,,,,X, the low mean k case (w = 0.9) results in higher dispersal than the case w = 0.5 for most values of a. The case w = 0.1, which results in high population size and low environmental variability, consistently leads to selection for lower dispersal than the other two cases. The simulations shown do not demonstrate the effects of increasing b; in general, these will be to decrease dispersal by reducing the effective variance. As we shall see in the next section, dormancy has a similar effect.

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179

FIG. 4. Computed and simulated D* (optimal D) curves, assuming G = 1 (full germination). Computed curves (x, ., +) assume a saturating yield function, YL = Y,/lOO, and w values respectively of (0.9, 0.5, 0.1). Simulated cases are for nonsaturating yield functions, with 6 = 10, and (Y,, Y,, w) values of (10, 1000, 0.9) (@I), (1, 100, 0.5) (o), (1, 100, 0.1) (0).

6. THE EFFECTS OF DORMANCY Because dormancy and dispersal are alternative avenues to the solution of the same problem, one intuitively expects dormancy to reduce the optimal level of dispersal. Dormancy may be thought of as reducing the effective variability of the environment; but we have already observed the ambiguous nature of that influence and the need for precision in discussing environmental variability. The effect of dormancy in reducing the optimal dispersal level is apparent in simulations, as may be seen in Figs. 5, 6, and 7. Each of these figures compares the optimal dispersal fraction under three different dormancy regimes: G= 1; G=OS, v=O.5; and G=0.2, v=O.8. For each, b= 10. While the germinating fraction is determined entirely by G, the dormant pool is ~(1 - G) times the previous density. Thus both v and G are important in considering the effects of dormancy. Similar studies of optimal dormancy fractions (Ellner, 1981; Cohen and Levin, in preparation) show a positive relationship between optimal dormancy (1 - G*) and u which parallels the relationship discussed in this paper between D* and cz.Thus, to most closely match the positive relationship we would expect to find in nature, we have chosen u = 1 - G in the simulations. In Fig. 5, w = 0.1; in Fig. 6, w = 0.5; in Fig. 7, w = 0.9. In each, Y,,,,, = 1, 100; that is, as before Y,,,,, = 1 with probability w and Y,,,,, = 100 with

180

LEVIN, COHEN, AND HASTINGS 10 optlmal dispersal D*

I

a

FIG. 5. Effects of dormancy on lowering optimal dispersal. Simulated D* (optimal D) curves as a function of a at two levels of effective dormancy; one (. ) with G = 0.5, v = 0.5, and the second (x) with G = 0.2, v = 0.8. These are compared (x) with simulated values of D* in the absence of dormancy (G = 1). In all cases Y,,, = 1, 100; b = 10; and w = (probability Y,,,., = 1) = 0.1.

probability 1 - W. Note that in general, increasing w (the probability of a bad year) increases selection for dispersal at any level of dormancy. As is intuitively obvious and as the simulations show, if a = 1 the optimal strategy is to disperse all seeds (D* = 1); this was already noted for the cases without dormancy. On the other hand, D* < 1 if a < 1. This can be seen by observing that D = 1 may always be invaded by a rare type for which D < 1.

FIG. 6. Effects of dormancy on lowering optimal dispersal. Simulated D* (optimal D) curves as a function of a at two levels of effective dormancy; one (. ) with G = 0.5, v = 0.5, and the second (x) with G = 0.2, v = 0.8. These are compared (X) with simulated values of D* in the absence of dormancy (G = 1). In all cases Y,,,,, = 1, 100; b = 10; and w = (probability Y,,,,, = 1) = 0.5.

181

DISPERSAL STRATEGIES I.0 optimal dtspersal D”

05

a

IO

FIG. 7. Effects of dormancy on lowering optimal dispersal. Simulated D* (optimal D) curves as a function of a at two levels of effective dormancy; one (. ) with G = 0.5, c = 0.5, and the second (x) with G = 0.2, c = 0.8. These are compared (x) with simulated values of D” in the absence of dormancy (G = I ). In all cases YrnaY = 1. 100: b = 10; and ~1’= (probability I’,,,,, = 1 ) = 0.9.

Unfortunately, we have not yet succeeded in deriving a satisfactory analytic approximation for the ESS in the case with dormancy. Dormancy introduces a “memory” for a patch, and renders our earlier arguments invalid. It is not the case anymore that one year yields will be equilibrated for dispersal and nondispersal. We have conjectured that what are equilibrated are the asymptotic expected proportions of the population derived from seeds committed to particular functions. However, we have neither been able to prove nor to disprove this conjecture. This remains a problem of considerable interest.

7. DENSITY-DEPENDENT

DISPERSAL

Allowing dispersal strategies to be cued by local densities or by patch stage (age since last disturbance) at first may seem to lead to a much harder mathematical problem. In truth, it does not, because it “localizes” the control problem. For each stage, the species can “choose” a D value appropriate to that stage, without “considering” the long-range implications. In this section, we discuss, in a preliminary way, density-dependent dispersal, in order to illustrate the differences which occur. As earlier, we consider the general yield function (24), and without dormancy. We assume, as in the earlier sections, that a stationary distribution exists, and that within it spatial averages and temporal averages are identical. Note that for such a distribution to exist, the conditional expectation E(Y,(t) / Si(t)) must exceed 1 for at least some values of Si.

653/26/2-5

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LEVIN, COHEN, AND HASTINGS

In this section, we shall argue heuristically and not attempt formal proof of our claims. Let S be the population size prior to yield, and n = SY be the value after yield but before dispersal. For a given level of dispersal, and hence a given input of dispersers per patch, we may compute E(Y,(t + 1) 1ni(t)). If nJt) is such that

E(Yi(C+ l) 1ni(c)>> l,

(274

then there is a premium to not dispersing: it pays to wait a year, given that the option to disperse will be presented again then. Thus, at densities so low that (27a) holds, the only possibly evolutionarily stable strategy is D = 0. This argument depends on stationarity. On the other hand, if the inequality in (27a) were reversed, an opposite argument would apply, and any strategy for which D < 1 would be unstable. Care must be exercised in interpreting this, however; it does not follow that D = 1 is an ESS. Rather, as D is increased (and assuming a < l), a point will be reached at which

E(Yi(f + l) / ni(t)) = l;

(27b)

and we conjecture that this defines the evolutionarily Now

stable D.

E( Y~(C+ 1) 1ni(t)) = kF(ni(t)( 1 - D) + 1)

(28)

Z=;

(29)

in which $ SiYiD(SiYi)=; I-1

t n,D(rq) I-I

is the (constant) input to each patch. Note that in the steady state, as already mentioned, E( Y,(t) 1Si(t)) must exceed 1 for at least some values of si(t); and since, by (l), Si > I, it follows by (27b) and monotonicity that 1 < @(I) and hence that F-‘((k)-‘) > I. From (28), it follows that D(~,)= I

ni+Z-F-‘((k)-‘) ni

= 1 + Z-F-‘((k)-‘1

< 1

9

(30)

ni

where n, = Si Yi ; note that this only implicitly determines D(n,), since Z involves D(q). What is more important, however, is that for n ,< ncr = F-‘(k-l) -I, the evolutionarily stable D is zero; whereas, for n beyond this threshold, the ESS choice of D is intermediate between 0 and 1 and chosen so that (27b) applies. Were ki = k, a constant for all i, this would mean that both D and n

DISPERSAL STRATEGIES

183

would remain fixed at constant levels once a threshold density ncr was exceeded; and that, except for an initial “adjustment” year of intermediate dispersal just prior to the period of constant dispersal, D = 0 prior to the achievement of the threshold. Dispersal is first initiated when a patch exceedsthe density ncr.

8.

THE CASE OF COMPLETE RESETTING; A CONTINUOUS APPROXIMATION

Because of the difficulty of finding an analytic representation for the evolutionarily stable strategy, we seek a continuum approximation in the hope that the special methods available for such approximations will allow new insights. The model given in this section applies to only a very special case. Years of low yield, while disasters for the resident individuals, create opportunities for new colonists by reducing competition in subsequentyears. An extreme case is that of total resetting of the environment through a year or several years of zero yield. We have already discussed some of the effects of this. The underlying habitat pattern is, as before, environmentally determined, although modified by population change. The model considered here differs from that treated in the earlier sections (but see Cohen and Levin, in preparation) in that a particular pattern of autocorrelation of yields is assumed. The basic paradigm is essentially that of Clementsian succession (see, e.g., Miles, 1979, Fig. 1.2); patches experience a temporal sequenceof probabilistically determined states, controlled in part by a clock which is repeatedly being reset (not necessarily with fixed period) to a point which we designate as the zero or initial point. That is to say, the probability distribution for yields is conditioned on the time since last disturbance. Equivalently, new patches are always being created to replace those which disappear through encroachment by other species or through other mechanisms. If the clock is strictly periodic, as with a successional microcycle, the choice of “initial” point may be arbitrary, but this is not a problem. In general, each patch is assigned an “age” a, the time since the initial point was last passed (e.g., time since last major disturbance), and this age determines the basic local background environment. Denote by p((a, t) the density of “age-class” a patches at time t (since a is a continuous variable, &a, t) is a density function on a and does not actually represent the number of age a). Our approach ignores explicit consideration of age structure within the dispersing population, basically assuming a stable seed-agedistribution and

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LEVIN, COHEN, AND HASTINGS

focusing attention upon the density function n(a, t) for the number of individuals in a patch of age a at time t. The population size at time t is

N(l) = j" n(u,t) F(u,t) da.

(31)

0

This does not ignore the possibility that dispersal may be restricted to particular life stages, e.g., seedsor larvae; but in the latter case, n(u, t) will be most appropriately measured in terms of some basic unit (e.g., “seed equivalents”) which reflects the true reproductive potential, and so “number of individuals” really means number of such units. Let r( .) represent the rate at which the nondispersing local population is growing, D( .) the dispersal probability per individual per unit time, and p(r) the density function on T for the fraction of dispersing individuals which successfully settle in a patch at time T after dispersal. The expression r( .) indicates that r depends upon some set of arguments. In the simplest case, r is constant; more realistically r depends at least upon population density. Similarly D(a) indicates dependenceupon some set of parameters. If an individual has a constant probability of dispersal per unit time, D is constant; if it utilizes cues based upon patch age, D depends upon a; or, as in the last section, D might depend upon local population density. For the remainder of this section, we assume D depends at most upon n and write D = D(n).

Finally, it is important to note that a = Ir /?(T, a, t) dt, the chance a disperser from a patch of age a at time t will eventually find a haven, is positive and less than 1. Indeed, 1 - a is a measure of the cost of dispersal, the fraction of dispersers which never find a new home. T = (l/a) (r /3(T)Tdt, the mean time spent in transit, is an indicator of dispersal delay. Under these assumptions, and if we ignore habitat selection, the dynamics of n(a, t) will be governed by the first-order partial differential integral equation

$+g=(r(. >- D(n))n +jam jomP( T,

U, ~)P(u,

t -

T)D(n)n(a,

t - T)dTda (32)

in which P(u, t) = &a, t)/jF F(u, t) da. The operator on the left is analogous to that which occurs in the McKendrick or von Foerster model for continuous age-structured populations (see,-e.g.,Hoppensteadt, 19751 If all dispersers take an identical time T to settle, so p(T) = a&T), (32) becomes

DISPERSAL an

185

STRATEGIES

an

at + aa = (r( . ) - D(n))n + Irn D(n) a(u, t - F) P(a, t - Fj n(a, t - T>da 0

(33) in which a(a, t) is the mean probability of successfor a dispersing individual at time t from a patch of age a. For simplicity, we assume henceforth that a = a(u, t - r> is a constant. The appropriate boundary conditions to impose depend upon the nature of the resetting process. For the remainder of this section, we assume that resetting is equivalent to a catastrophic event which kills all individuals. This restriction would be relaxed with dormancy and would similarly be altered if destruction were less than total, as in the examples considered earlier. The assumption corresponds to the condition n(0, t) = 0.

(34)

We concentrate henceforth on the problem (33~(34) and assume that with minimal restrictions on the functional forms (including that I(. ) = r(n, a, t)), a well-posed problem is formed by specifying appropriate initial conditions on an interval of length ?;. The exact statement of such a theorem is not undertaken here, but the general correctness is assumed.The result has been established for certain special cases by Elderkin (1982), under more general boundary conditions than (34). A feasible solution to the initial value problem consists of a function n(u, t) which is everywhere nonnegative and for which the integral in (33) is finite; obviously the initial values must also be so constrained for the problem to have a feasible solution. Elderkin (1982) derives several other results for the cases considered, including monotonic dependence of the solution on initial and boundary data and conditions for the existence and stability of a nontrivial steady-statesolution. The latter is analogous to results of the type obtained by Ellner (198 1) for the existence of a stationary distribution in the discrete stochastic model. Except in the case that there is an absolute maximum attainable age for a patch, there will be arbitrarily old patches (P(u, t) > 0 for a arbitrarily large), and further it may be that n(u, t) tends to infinity with a, although crowding effects will usually prevent this. However, in general, P(u, t) will tend to zero as a becomes infinite, and it is the product n(u, t) P(u, t) which is of primary importance; for feasibility nP must tend to 0 sufficiently rapidly as a becomes infinite (i.e., the integral (31) must exist). As mentioned earlier, we are ignoring the possibility for habitat selection. This can be introduced easily by multiplication of the integral in (32) or (33) by an age-dependentfrequency function representing settlement preferences. However, mathematical extensions of models are cheap: results are more

186

LEVIN, COHEN, AND HASTINGS

expensive to come by, and hence we see no point in introducing more complications than we have already done. To summarize, we have described the global growth and distribution of a single species for which the local growth is density dependent, and which possesses a dispersal capability. The appropriate description, under the conditions specified, is given by the system an

an

5 + aa = (r(n, a, t) - D(n))n + a lrn D(n) P(a, t - ?;>n(u, t - T) da 0

(35)

n(0, t) = 0

in which n(u, t) is the density of organisms in patches of age a, P(u, t) is the specified frequency distribution of those patches, r(n, a, t) is the local per capita growth rate of n, D(n) is the per capita dispersal rate from patches of age a, and f is the mean time spent in transit. For given functions r(n, a, t), P(u, t), and D(n) a number of problems present themselves. (i) If one specifies initial values on an interval of length T, what are the further restrictions necessary to form a well-posed problem? (ii) Assuming a well-posed initial value problem, and assuming initially n(u, t) > 0 and finite population size [J^Fn(u, t) P(u, t) da < co], do those constraints remain true for any finite time? (iii) Under the restrictions of (ii), what asymptotic behaviors are possible for solutions? In particular, is there a nontrivial symptotic solution? Are there multiple such solutions? What are their stability properties? What is the stability character (for given D(a)) of the trivial solution n = O? In particular, if n = 0 is globally asymptotically stable, no population may persist indefinitely. A related set of problems arise from consideration of competition or other local interactions between species, the necessary formulation for defining ESS’s. Corresponding to (35) is then the system 2

+ 2

= (ri(n,

ni + CL,lrn D,(n) P(u, t - ?;>n,(u, t - 73 da; 0 (36) i= 1,..., k; ni(0, t) = 0

U, t) - Di(n))

in which n, is the mean density of the ith speciesin patches of age a and n is the vector (n i,..., nJ of local densities. In other words, (36) is a vectorial analogue of (35), and a set of questions (i’)--(iii’) analogous to (i)-(iii) may be posed; in particular, the question of coexistence may be addressedin part by looking for nontrivial steady state solutions of (36), and some insights

DISPERSAL STRATEGIES

187

may also be gained from studying the stability of trivial solutions (in which only one speciesis present). Consider the case D, = constant and ri = r(n, + .. . + nk, a, t). Our primary objective is to find evolutionarily stable strategies (ESS’s). An ESS is a choice of Di which leads to a feasible steady-state solution n of (35), and for which n, = n, n, = n, = aem= 0 is an asymptotically stable solution of (36). At this point, we are unable to derive the necessary and sufficient conditions for the existence of such ESS solutions; however, the mathematical problems are precisely stated and well defined. An ESS, if it exists, will be a steady-state solution n of (35) (with D(n) = D*) which has the property that any nonnegative solution ~(a, t) to DP(a,t-flm(a,t--da

$+$=[r(n+m,a,r)-D]m+a!.m

(37)

0

m(0, t) = 0

will tend pointwise to 0 as t tends to co unless D = D*. We consider the development of this section very preliminary, but a promising direction for future work. 9. SUMMARY Considering an environment composed of a number of uncorrelated and randomly-varying patches, we have developed a model as an aid to explaining observed levels of long-range dispersal in patchy environments for annual, asexual plants and other sessile organisms, and to identifying the environmental characteristics which select for dispersal. The notion of evolutionarily stable strategy is introduced and computed for the case of a “saturating” yield function and in relation to parameters controlling environmental variability and the hazardousness of dispersal. Simulations were performed for more general yield functions and in the presenceof dormancy, and the results compared with the saturating case. In general, the optimal level of dispersal is an increasing function of a, the probability that a propagule will successfully attain a new site, and tends to 1 as (x tends to 1. It is tautological that most seeds are in sites relatively more crowded than the average, and dispersal always favors improvement in that respect. If a = 1, dispersal is always advantageous. For CI< 1, the evolutionarily stable D is always less than 1, and we conjecture that it is determined by the equilibration of long-term expectations from dispersing and nondispersing units. Except for (r very small, the general model can be approximated well by one in which every local patch is filled each year; that is, one in which the input of seeds is always sufficient to

188

LEVIN, COHEN, AND HASTINGS

saturate. In this model, the local environment is characterized by the distribution of total yield k. Near 01= 1, the primary determinant of the optimal dispersal level is the relativized variance of k. For a < 1, the distribution of l/k assumesimportance; and if P[k = 0] = 0, optimal D tends to 0 as a tends to a critical level a,,, the harmonic mean of k divided by the arithmetic mean. Near this critical value, the optimal value of D increases at a rate equal to the reciprocal of the product of a,, and the relativized variance of l/k. These results are presented for the saturating model, but the simulations for the more general case show remarkable agreement with the saturating model provided a is not too small. Dormancy serves to reduces the optimal level of seed dispersal, as intuition would suggest since dispersal and dormancy are alternative mechanisms for escaping environmental unpredictability. The effects of cueing are discussedin a preliminary way, and results given. The model presented assumesfull knowledge of local density, and it would be of interest to investigate the effects of incomplete information. Future work will also examine optimal dormancy strategies (see Venable and Lawlor, 1980; Ellner, 1981, for related models) and the interaction of selection for dormancy and dispersal; optimal dispersal strategies in temporally and spatially correlated environments (Cohen and Levin, in preparation); sexual species and the maintenance of dispersal polymorphisms; and models for perennial species. When disturbance can be severe enough to completely obliterate the local population, it is convenient to characterize patches according to age since last disturbance. In this case, a continuous approximation is suggestedand discussed. Although to date results are few for this model, a number of challenging open mathematical problems are posed, and some preliminary results presented. In the long run, in all real situations there is a nonzero probability for a zero yield and complete obliteration locally, due either to some extreme event or to a permanent environmental change. Thus zero optimal dispersal is never to be expected to be found in nature, except under artificial conditions (as in agriculture). Under natural conditions, a plant without the capacity for dispersal is doomed to extinction. It is important to recognize that the positive correlation of optimal dispersal with some measure of environmental variability is not the same as correlation with environmental harshness.Little dispersal is to be expected in those harsh environments in which localized disturbance is rare, whereas high dispersal may be expected in relatively favorable environments with frequent localized disturbances. For example, seeddispersal is very limited in annual plants of extremely dry deserts and semi-deserts(van der Pijl, 1972; Elmer and Shmida, 1981), while plants occupying disturbed habitats and

DISPERSAL

STRATEGIES

189

early successional stages often have, very good seed dispersal mechanisms. Ellner and Shmida argue that temporal variations in deserts, although extreme, have a very high spatial correlation over the range of possible seed dispersal values, and that this nullifies the benefits of spatial averaging by dispersal. In a sequel to this paper, we shall explore the consequencesof such averaging. Obviously, the spatial scale of the environment has important consequences for dispersal strategies (Palmer and Strathmann, 1981). Since the advantage of dispersing seedscomes from the averaging over many patches, the optimal dispersal distance must be large enough for adequate averaging, and to a first approximation should correlate positively with patch size and inter-patch distance. Thus, weeds and early successionalplants in forest gaps should have relatively long-distance dispersal, while plants which colonize small-scale disturbances such as mole hills, worm casings, animal burrows, etc., should have a more restricted dispersal range. The spatial scale and the distance distribution of seeds around the parent will influence seed survivability (a), and hence the optimal dispersal level.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of National Science Foundation Grants MC?-7701076 and MCS-8001618 (to SAL) and DEB-8002593 (to AH), Hatch Project 183414 (to SAL), and US-Israel Binational Grant 2523 (to DC and SAL). Numerous colleagues tried to help in improving this paper, and special thanks are due Peter Chesson, Steve Ellner, and Yoh Iwasa. That does not imply that they would be satisfied with the present version.

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