The Evolution of Patch Selection in Stochastic Environments

vol. 180, no. 1 the american naturalist july 2012

Sebastian J. Schreiber*

Department of Evolution and Ecology and the Center for Population Biology, University of California, Davis, California 95616 Submitted July 25, 2011; Accepted February 8, 2012; Electronically published May 17, 2012

1969; Sutherland 1983, 1996; Tregenza et al. 1996; Ksha- abstract: A null model for habitat patch selection in spatially triya and Cosner 2002; Cosner 2005; Cantrell et al. 2007, heterogeneous environments is the ideal free distribution (IFD), which assumes individuals have complete knowledge about the en- 2010; Morris 2011), the combined effects of temporal and vironment and can freely disperse. Under equilibrium conditions, spatial variation on the evolution of patch selection is less the IFD predicts that local population growth rates are zero in all understood (Holt and Barfield 2001; Morris 2011). Given occupied patches, sink patches are unoccupied, and the fraction of the ubiquity of temporal variation and its notable impacts the population selecting a patch is proportional to the patch’s car- on demography (Boyce et al. 2006) and the evolution of rying capacity. Individuals, however, often experience stochastic fluc- passive dispersal (Johnson and Gaines 1990; Ronce 2007), tuations in environmental conditions and cannot respond to these fluctuations instantaneously. An evolutionary stability analysis for I examine the evolution of habitat patch selection for freely fixed patch-selection strategies reveals that environmental uncer- dispersing individuals in a stochastic environment. tainty disrupts the classical IFD predictions: individuals playing the In their seminal article, Fretwell and Lucas (1969) in- evolutionarily stable strategy may occupy sink patches, local growth troduced the ideal free distribution (IFD), which describes rates are negative and typically unequal in all patches, and individuals the patch distribution of freely dispersing individuals that prefer higher-quality patches less than predicted by their carrying have complete knowledge about spatial variation in in- capacities. Spatial correlations in environmental fluctuations can enhance or marginalize these trends. The analysis predicts that contin- dividual fitness. For population exhibiting this distribu- ually increasing environmental variation first selects for range ex- tion, individual fitness is equal in all occupied habitat pansion, then selects for persisting coupled sink populations, and patches and individuals would decrease their fitness by ultimately leads to regional extinction. In contrast, continually in- moving into unoccupied patches. This distribution has creasing habitat degradation first selects for range contraction and been explored under a variety of conditions, including may select for persisting coupled sink populations before regional interference competition (Sutherland 1983, 1996; Tregenza extinction. These results highlight the combined roles of spatial and temporal heterogeneity on the evolution of habitat selection. et al. 1996), exploitative competition (Sutherland 1983; Milinski and Parker 1991; Oksanen et al. 1995; Krˇivan Keywords: habitat selection, evolutionarily stable strategies, environ- 2003), and multispecies interactions (van Baalen and Sa- mental stochasticity, spatial heterogeneity, source-sink dynamics, belis 1993, 1999; Krˇivan 1997; Cressman et al. 2004; Schrei- ideal free distribution. ber and Vejdani 2006). Surprisingly, while many of these studies assert that the ideal free distribution is an evolu- Introduction tionarily stable strategy (ESS), only recently has a delicate nonlinear analysis verified the ESS conditions (Cressman Habitat selection by individuals can profoundly influence et al. 2004; Cressman and Krˇivan 2006, 2010; Cantrell et population persistence in heterogenous landscapes (Pul- al. 2007, 2010). Under equilibrium conditions, the IFD liam and Danielson 1991; Greene 2003; Schmidt 2004), predicts that the fraction of individuals selecting a source stability of predator-prey interactions (van Baalen and Sa- patch is proportional to the carrying capacity of the source belis 1993; Krˇivan 1997; Schreiber and Vejdani 2006), and patch (Holt and Barfield 2001). Consequently, all sink geographical shifts in species distributions in response to patches are unoccupied as their carrying capacity is zero climate change (Morris 2011). While there are significant (Holt 1985). Neither of these predictions, however, agrees and extensive advances in the evolutionary theory of patch selection for populations living in spatially heterogeneous fully with empirical observations. environments (Morisita 1952, 1969; Fretwell and Lucas Observed patch distributions are frequently less extreme than predicted by the IFD: individuals underexploit

* E-mail: [email protected]. higher-quality patches (i.e., undermatching) than predicted by the IFD and overuse lower-quality patches (i.e., Am. Nat. 2012. Vol. 180, pp. 17–34. ᭧ 2012 by The University of Chicago. 0003-0147/2012/18001-53211$15.00. All rights reserved. overmatching; Milinski 1979, 1994; Abrahams 1986; Ken- DOI: 10.1086/665655 nedy and Gray 1993; Tregenza 1995). Explanations for

This content downloaded from 169.237.045.004 on August 02, 2019 16:55:12 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 18 The American Naturalist these deviations include perceptual constraints (Abrahams lead to shifts in species distributions (Allen and Breshears 1986; Gray and Kennedy 1994), demographic stochasticity 1998), it is becoming increasingly important to understand (Regelmann 1984; Miller and Coll 2010), sampling vari- how environmental fluctuations influence species’ habitat ation (Regelmann 1984; Earn and Johnstone 1997), and choices. As a step toward this understanding, I present a environmental stochasticity (Hakoyama 2003; Jonzeń et general analysis of the evolution of patch selection strat- al. 2004). Of greatest relevance here, Hakoyama (2003) egies for freely dispersing populations living in spatially showed that if stochastic variation in resource availability and temporally heterogeneous landscapes. For these pop- in higher-quality patches is sufficiently greater than sto- ulations, I derive an analytic characterization for the ESS chastic variation in lower-quality patches, populations in these stochastic environments. Using this characteri- playing the ESS for patch selection undermatch the higher- zation, I examine how spatial correlations and temporal quality patches. This study, however, does not account for fluctuations inhibit or enhance the evolution of sink pop- population growth or resource depletion due to con- ulations, result in undermatching of higher-quality sumption. Consequently, ecological feedbacks on the evo- patches, determine local stochastic growth rates, and in- lution of patch selection are missing. In another study, fluence the evolution of species ranges in response to in- Jonzeń et al. (2004) examined the effects of environmental creased environmental variation or habitat destruction. stochasticity and population dynamics on patch distributions using a heuristic stochastic patch selection rule. However, Jonzeń et al. (2004, p. E110) readily admit that Model and Methods it “is not clear to what extent the stochastic habitat selec- Local and Regional Population Dynamics tion rule implemented here would also be [an] ESS.” Hence, it remains to be understand how environmental I model a population with overlapping generations living stochasticity and ecological feedbacks determine the ESS in a spatially heterogeneous environment consisting of k for patch selection. distinct patches. These patches may represent distinct hab- Contrary to the predictions of the classical IFD theory, itats, patches of the same habitat type, or combinations sink populations exist in a diversity of taxa, including vi- thereof. The abundance of the population in the ith patch i ruses (Sokurenko et al. 2006), plants (Anderson and Geber at time t isNt . Its rate of change is determined by a mixture 2010), birds (Tittler et al. 2006; Møller et al. 2006), fish of deterministic and stochastic environmental forces. More iip Ϫ i (Barson et al. 2009), and mammals (Naranjo and Bodmer specifically, the changeDNttN ϩDttN in small time 2007; Sulkava et al. 2007; Robinson et al. 2008). These stepDt satisfies population exhibit an extreme form of overmatching, as tϩDt sink patches cannot sustain population growth in the ab- ޅ D iiF p ͵ ޅ iiiF sence of immigration. Under equilibrium conditions, one [ NttN ] [fis(N )N sN t]ds explanation for the existence of these sink populations is t (1) maladaptive patch choice (Holt 1997). Alternatively, the- ≈ f (N ii)N Dt, ory suggests that sink populations may evolve when tem- it t poral variation is sufficiently great in source patches (Holt wherefi is the average per capita growth rate in patch i 1997; Jansen and Yoshimura 1998; Holt and Barfield 2001). andޅ[XFY] denotes the conditional expectation of a ran- When this occurs, numerical simulations by Holt and Bar- dom variable X given the random variable Y. To account field (2001) show that the geometric mean of fitness, a for intraspecific competition and population regulation natural measure of fitness in fluctuating environments, is within the patch, I assume thatfi is a decreasing function not equal in all occupied patches. In contrast, if all patches of local population abundance in patch i and is negative are sources, their simulations suggests that the geometric when the local population abundance is sufficiently large, ii! mean of fitness is approximately equal in all patches. These that is,fi(N ) 0 forN sufficiently large. contrasting outcomes raise the question of whether, if at To capture the role of stochastic forces, the covariance all, there is a simple rule of thumb analogous to the clas- in the growth of the populations in patches i and j over sical IFD for stochastically fluctuating environments. Em- a time interval of lengthDt satisfies phasizing this point, Holt and Barfield (2001, p. 94) assert Cov [DN ijij, DN FN , N ] ≈ j NN ijDt, that “it will be an important task in future work to assess ttttijtt the generality of these conclusions.” whereCov [X, YFZ, W] denotes the covariance between Given that global climate models predict increasing tem- random variables X and Y given the random variables Z p poral variability in environmental conditions in the next and W. The covariance matrixS (jij) captures the spa- century (Scha¨r et al. 2004; Tebaldi et al. 2006) and that tial dependence between the temporal fluctuations in increased variability in these conditions may have already patches. Taking the limit asDt gets infinitesimally small,

Table 1: Deﬁnitions of key terms Term Deﬁnition

i Et Environmental noise in patch i p p (p1,…,pk) Patch selection strategy with fraction ofpi individuals living in patch i i fit(N ) Per capita growth rate in patch i 2 p jjij(with ij ii ) Covariance in temporal fluctuations between patch i and patch j Ϫ 2 fii(0) j /2 Intrinsic stochastic growth rate in patch i Nˆ (p) Stochastic equilibrium for a population with patch selection strategy p ޅ ˆ Ϫ 2 [fii(pN(p))] j i/2 Local stochastic growth rate in patch i 2 p ͸ j(p) i, j ppijijj Total temporal variation experienced by a population playing the patch selection strategy p Ϫ 2 ! Sink patch Patch in whichfii(0) j /2 0 ; local population persists only due to immigration ! Deterministic sink patch Sink patch in which fi(0) 0 Ϫ 2 ! 1 Stochastic sink patch Sink patch in whichfii(0) j /2 0 but f i(0) 0 the spatially uncoupled local population dynamics are creases their fitness. When this intrinsic stochastic growth given by the Itoˆ stochastic differential equations (Gardiner rate is negative, patch i is a sink and, in the absence of 2009) immigration from other patches, populations restricted to this patch go extinct. I make a distinction between two dN iiip N (f (N )dt ϩ dE i), (2) ttitt types of sinks (table 1). If the deterministic intrinsic rate 1 k of growthfi(0) is negative, then patch i is a sink for all where(Ett,…,E ) is a multivariate Brownian motion with covariance matrix S. For convenience, I refer to the en- levels of environmental stochasticity. I call such patches 2 1 Ϫ “deterministic sinks.” In contrast, iffii(0) 0 and f (0) vironmental variancejjii in patch i as i . 2 ! When populations in the different patches are uncoup- ji /2 0, patch i is a stochastic sink, as stochastic fluc- led, Itoˆ’s formula (Itoˆ 1950; Gardiner 2009) implies that tuations drive the local population to extinction in the the rate of change of the log population abundance in absence of immigration. Unlike deterministic sinks, sto- patch i satisfies the Itoˆ stochastic differential equation chastic sinks exhibit periods of positive population growth and, consequently, can contribute to regional persistence. j 2 In contrast to sinks, patch i is called a source when iip Ϫ i ϩ i 2 d log Ntitf (N ) dt dE t. f (0) Ϫ j /2 is positive. Populations in source patches are ()2 ii i able to persist in the sense thatNt converges in probability i ˆ ii1 SincedEt is on average equal to zero, the mean change to a positive random variableNN whenever0 0 (Brau- in the log abundance over a small time intervalDt satisfies mann 1999). When the population persists in the absence of immigration, the local stochastic growth rate 2 ޅ ˆ i Ϫ 2 j [fii(N )] j /2 at this stochastic equilibrium is zero; on ޅ[D log (N ii)FN ] ≈ f (N i) Ϫ i Dt, tt() it 2 average the population is neither increasing nor decreasing. iip Ϫ i whereDlog(Ntt) log(N ϩDtt) log(N ) . Hence, unlike the To couple the population dynamics across patches, let i mean per capita changefit(N ) in abundance (conditioned pi be the fraction of the population selecting patch i. The p on the current abundance), the mean change in log abun- patch selection strategyp (p1,…,pk) may be a pure dance (conditioned on the current abundance) decreases type in which each individual occupies only one patch, or with the variance in environmental fluctuations. This re- a mixed type in which individuals spends a proportion duction plays a crucial role in determining population pi of their time in patch i (Morris 2011). With this no- i p p 1 ϩ ϩ k persistence. tation,NtitpN , where N tN t… N t is the total Ϫ The sign of the intrinsic stochastic growth rate fi(0) population abundance at time t. Summing equation (2) 2 ji /2 in patch i determines population persistence and in- across all patches yields the Itoˆ stochastic differential equa- vasion success (Lewontin and Cohen 1969; Gillespie 1977; tion for the total population abundance: Lande et al. 2003). This intrinsic stochastic growth rate is the continuous-time analog of the geometric mean of fit- k p ͸ ϩ i dNtitiittpN(f (pN)dt dE ). (3) ness discussed extensively in the bet-hedging literature ip1 (Gillespie 1977; Childs et al. 2010); by decreasing temporal ͸ i variation in their reproductive success, individuals can in- Sincei pEit is a Brownian motion with variance

This content downloaded from 169.237.045.004 on August 02, 2019 16:55:12 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 20 The American Naturalist ͸ t i, j ppijijj at time t, the dynamics of equation (3) are ary dynamic on the strategy space in which small muta- equivalent to tions occurring at a rate m randomly shufﬂe the “inﬁni- tesimal” weights of the patch selection strategy: p ͸ ϩ dNttiiitNpf(pN)dt j(p)dB t,(4) i dp p j () ikjkp ޅ ˆ ϩ ͸ Ϫ mpiii[f (pN(p))] p jj ij.(7) 2 p ͸ dt {()}j, k 2 wherej(p) i, j ppijijj andB t is a standard Brownian motion (i.e., variance at time t equals t). As in the case An explicit representation of this dynamic is possible of the single-patch model, if the regional intrinsic sto- i p Ϫ for linear per capita growth rates: fii(N ) b (1 2 ͸ Ϫ 1 i chastic growth ratepfii(0) j(p)/2 0 , then the pop- Ϫ i N /K ii) dbdK, where iii , , and are the per capita birth ulation persists andNt converges in probability to a unique rate, per capita death rate, and the “reproductive carrying ˆ 1 positive random variableN(p) wheneverN0 0 . In con- capacity” of patch i. This representation of the logistic trast, the population becomes extinct with probability 1 equation allows for deterministic sink patches (i.e., when ͸ Ϫ 2 ! wheneverpfii(0) j(p)/2 0 . Table 1 summarizes the ! i biid ) as well as source patches and stochastic sink key terms presented here. patches. For the numerical results, equation (7) with the

linearfi was simulated with the deSolve package of R (R Invasion Rates and the ESS for Patch Selection Core Devlopment Team 2008) for 1,000 time steps. Ex- tensive simulations suggest that these evolutionary dy- Given the uncertainty in local growth rates, one can ask namics always converge to a globally stable equilibrium from an evolutionary perspective, what is the optimal corresponding to the ESS for patch selection after the 1,000 patch selection strategy? To identify this “optimal” strategy, time steps. I use the noninvasibilty criterion for an ESS (Maynard Smith and Price 1973). More specifically, assume there is p a resident population playing strategy p (p1,…,pk) Results competing with a “mutant” population playing strategy p q (q1,…,qktt). LetNM and denote the total popula- To Persist or Not to Persist? tion abundance of the resident strategy and the mutant In order for there to be an ESS, the population needs to strategy, respectively. If these populations only differ in persist: there is a patch selection strategy p for which the their patch selection strategy, then the population dynam- regional intrinsic stochastic growth rate ͸ pf(0) Ϫ ics of these competing strategies are given by the following i ii j(p)/22 is positive. Evans et al. (2012) developed an al- system of Itoˆ stochastic differential equations: gebraic solution to maximizing this intrinsic stochastic p ͸ ϩ ϩ dNttiiitittNpf(pN qM)dt N j(p)dB t,(5)growth rate with respect to the patch selection strategy. i Appendix B uses this algebraic solution to provide an explicit, analytical expression for when this intrinsic sto- p ͸ ϩ ϩ dMttiiitittMqf(pN qM)dt M j(q)dB t.(6)chastic growth rate is positive for some p. While this con- i dition is quite unwieldy, it provides some useful insights. When the resident population is at its stochastic equilib- Not surprisingly, persistence is always possible when rium, the intrinsic stochastic growth rate of the mutant there is at least one source patch and impossible if all population equals patches are deterministic sinks. Unlike deterministic sinks, stochastic sinks ensure that there are opportunities for 1 I p ͸ ޅ ˆ Ϫ 2 population growth. Consequently, consistent with earlier (p, q) qiii[f (pN(p))] j(q), i 2 work on discrete time models (Holt 1997; Jansen and Yoshimura 1998; Bascompte et al. 2002), persistence is whereNˆ (p) is the stochastic equilibrium associated with possible in environments with only sink patches, provided the resident strategy. WhenI(p, q) 1 0 , the mutant can that at least one of them is a stochastic sink. For example, invade. Conversely ifI(p, q) ! 0 , the mutant population if all patches have similar intrinsic rate of growth (i.e., fails to invade. A sufficient condition for p to be an ESS f (0) p r for all i) and experience similar environmental is that all mutant strategies cannot invade, that is, i variation (i.e.,j p j 2 ), then persistence requires I(p, q) ! 0 for allq ( p . ii While I will show that there is a unique ESS and provide j 2 r 1 [1 ϩ r(k Ϫ 1)], (8) an analytic characterization of it, one cannot in general 2k write down an explicit formula for the ESS. To solve for the ESS numerically, I derive in appendix A an evolution- where r is the correlation in environmental fluctuations

p 2 ( between any pair of patches (i.e.,jij rj for alli j ) when the environmental fluctuations are perfectly nega- p Ϫ and k is the number of patches. tively correlated in these two patches (i.e.,jijjj i j ), Equation (8) implies that persistence occurs provided inequality (9) requires that the local stochastic growth rate Ϫ ϩ 2 there are many uncorrelated patches even if all are sto- in the sink patch exceed(jijj )/2 . Therefore, provided chastic sinks. With many uncorrelated stochastic sinks, that environmental fluctuations are sufficiently strong in positive growth is likely in at least one of the patches at either patch, strong negative correlations ensure that pop- any point in time. Hence, by utilizing many patches, populations playing the ESS always select the sink patch. When ulations experience less environmental variation and may environmental fluctuations between the source and sink 2 p p persist; the variance termj(p) in the regional intrinsic patches covary perfectly (i.e.,jijjj i j and j ij j for all ͸ Ϫ 2 stochastic growth ratei pfii(0) j(p)/2 becomes smaller. i, j), the population playing the ESS does not select the In contrast, equation (8) implies persistence in a landscape sink patch, as spending time in both patches does not of sink patches is impossible whenever environmental fluc- reduce the environmental variation experienced by an in- tuations in patches are highly correlated (i.e.,r ≈ 1 ). With dividual (or its offspring). strong spatial correlations, individuals experience the same Provided all patches experience environmental fluctu- environmental conditions in all patches, and consequently, ations and these fluctuations do not perfectly covary across selecting multiple patches does not reduce the environ- the landscape (i.e., S is positive definite), appendix C mental variation experienced by the population. shows that all local stochastic growth rates ޅ ˆ Ϫ 2 The symmetry in the preceding example implies that a [fii(pN(p))] j i/2 are negative. Consequently, unlike the population playing the ESS is uniformly distributed across deterministic ideal free distribution where the per capita patches. To get a sense of how spatial heterogeneity in growth rates in all patches equal zero (Fretwell and Lucas local demography generates nonuniform distributions re- 1969; Holt and Barfield 2001; Cantrell et al. 2007), pop- quires a general characterization of the ESS for patch ulations playing the ESS in stochastic environments are selection. living in a landscape composed of sink patches and pseudo-sink patches (Watkinson and Sutherland 1995). Pseudo-sinks are source patches in which density-depen- What Types of Patches Are Occupied by Populations dent feedbacks cause the local stochastic growth to be ޅ ˆ Ϫ 2 ! Playing the ESS? negative, that is, patches where[fii(pN)] j i/2 0 and the intrinsic stochastic growth ratef (0) Ϫ j 2/2 is positive. Most of the analytic results involve an ESS for which mul- ii tiple patches are selected (app. C). There are two ways in which this occurs. As in the case in the deterministic theory (Fretwell and Lucas 1969; Holt and Barfield 2001), all What Is Balanced by Populations Playing the ESS? source patches are occupied by populations playing the ESS. Hence, provided there are multiple source patches, a Under the assumption of multiple patches being selected, population playing the ESS occupies multiple patches. appendix C shows that the ESS p for patch selection sat- Unlike the deterministic theory, even if there is only isfies one source patch, call it patch i, populations playing the 2 Ϫ j(p) p ޅ ˆ Ϫ ͸ ESS may also select a sink patch, say patch j, provided that [fii(pN(p))] p jjij (10) Ϫ 2 2 j intrinsic stochastic growth ratefjj(0) j /2 in this sink patch is not too negative (see app. C): 1 for all occupied patches (i.e., i such thatpi 0 ). Equation jj22 j 2 (10) implies that the differences between two local de- f (0) Ϫ jj1 Ϫϩj Ϫ i.(9) jij22 2 mographic quantities are equal in all occupied patches. In patch i, these demographic quantities are the average per Sincej corresponds to the covariation in fluctuations capita growth rateޅ[f (pNˆ (p))] in patch i, and the co- ij ͸ ii between the source patch i and sink patch j, inequality (9) variancej pjjij between the environmental fluctuations i implies that selection of a sink patch is more likely as jij Et within patch i and the environmental fluctuations ! ͸ i decreases, and especially ifjij 0 . Intuitively, strong neg- j pEjtexperienced by the “average” individual. All other ative correlations imply that at any point in time, envi- things being equal, preferred patches are more “out of ronmental fluctuations increase population growth in one sync” with the environmental fluctuations experienced in patch and decrease population growth in the other patch. the other patches, considered collectively. The difference ޅ ˆ Ϫ ͸ Spending time in both patches (or distributing offspring [fii(pN(p))] j p jjij need not be equal to the local sto- ޅ ˆ Ϫ 2 to both patches) reduces the temporal variation experi- chastic growth rate[fii(pN(p))] j i/2 . Consequently, in enced by an individual (or by its offspring). More precisely, contrast to the deterministic theory, one does not generally

expect the local stochastic growth rates to be equal in all To further explore these trends, I ran the evolutionary occupied patches for populations playing the ESS. dynamic in equation (7) to determine how the ESS re- sponds to spatial variation in the per capita birth rates i p Ϫ biiiiand environmental variationj where f (N ) b (1 Patch Selection in Spatially Uncorrelated Environments i Ϫ N /K ii) d . In both scenarios, the spatial variation is cho- To gain more detailed insights into the effects of environ- sen such that the intrinsic stochastic growth rates decrease mental stochasticity on patch selection, it is useful to ex- in a Gaussian manner with respect to the patch index i, Ϫ Ϫ 22p Ϫ ϩ amine the simpler case of spatially uncorrelated temporal that is,biiid j /2 a exp ( bi ) g for n constants p ( fluctuations, that is,jij 0 fori j . In the absence of a, b, and g. these correlations, appendix C shows that all stochastic Figures 1 and 2A report the numerical results for spatial sinks are occupied by populations playing the ESS. To variation in per capita birth rates. When there is little understand why this occurs, consider a population that environmental stochasticity, all stochastic sink patches are p doesn’t occupy a stochastic sink, saypj 0 , where patch occupied but no deterministic sinks are occupied (fig. 1A, j is a stochastic sink. If an individual, unlike the rest of 1B). At higher levels of environmental stochasticity, a pop- the population, spends a small fraction ␧ of time in patch ulation playing the ESS occupies all deterministic sinks j, then the deterministic component of this individual’s (fig. 1C,1D). More generally, continually increasing en- ␧ stochastic growth rate increases approximately byfj(0) , vironmental stochasticity selects for range expansion, then which is positive as patch j is a stochastic sink. On the selects for persistence of coupled sink populations and other hand, the environmental variance experienced by ultimately leads to extinction (fig. 2A). Alternatively, con- ␧22 this individual only increases at most byjj . Since the tinually increasing the per capita death rates d selects for ␧ added benefitfj(0) of entering patch j exceeds the cost range contraction, then selects for persistence of coupled ␧22 ␧ jj /2 of entering patch j for small , individuals selecting sink populations (provided environmental stochasticity is patch j would increase their stochastic growth rate and initially sufficiently large) and ultimately leads to extinc- could invade the resident population. Hence, any ESS for tion (fig. 2B). patch selection has individuals occupying all stochastic Along gradients of environmental stochasticity, individ- sinks. uals preferentially select patches with less environmental Deterministic sinks may or may not be occupied by stochasticity (fig. 3A; analytical predictions in appendix populations playing the ESS. Expansion into deterministic C). However, higher per capita birth rates ameliorate this ! sink patches, say patch j withfj(0) 0 , is selectively ad- bias by reducing the spatial variation in intrinsic stochastic vantageous whenever the rate of population decline growth rates. Surprisingly, the highest within-patch abun- ͸ 22 (1/2) i piij due to environmental stochasticity exceeds the dances occur at the lowest per capita birth rates (lower Ϫ deterministic ratefj(0) of decline in the sink patch. left in fig. 3A); strong preferences for patches with less Hence, sufficiently large stochastic fluctuations in source environmental stochasticity increases local population patches or stochastic sink patches select for the occupancy abundances within these patches, despite the low regional of deterministic sink patches. Since the additional envi- population abundanceޅ[Nˆ (p)] . ronmental variation experienced by an individual using As predicted by the analytic results, the local stochastic

patch j only increases quadratically withpj , large stochastic growth rates are not equal in all occupied patches, even fluctuations in deterministic sinks do not select against when they are all source patches (figs. 1B,1D,3B). The their occupancy. These large stochastic fluctuations, how- differences in local stochastic growth rates are most pro- ever, may select for fewer individuals residing in deter- nounced between source patches and sink patches and at

ministic sinks, that is, pj is positive but small. higher levels of environmental stochasticity. Sufficiently When multiple patches are selected, equation (10) im- many independent patches, however, can marginalize these ޅ ˆ Ϫ 2 plies that the quantities[fii(pN)] p iij are equal to differences and result in the average per capita growth rates Ϫ ͸ 22 ޅ ˆ Ϫ ޅ ˆ i piij in all occupied patches. While [f i(pN i)] [fii(pN)] being approximately equal in all occupied 2 piij is determined by the local demography, these quan- patches (fig. D1 in app. D). tities do not generally equal the local stochastic growth ratesޅ[f (pNˆ )] Ϫ j 2/2 . However, when there are suffi- ii i Patch Selection in Spatially Correlated Environments ciently many patches selected by the ESS and thepi are small (i.e., the population selects many patches), the terms With perfect environmental covariation in all patches (i.e., 222͸ p 2 piij andi p ij i are approximately zero and one recovers jij j for alli, j ), the characterization of the ESS in a classical prediction of IFD that the average per capita equation (10) implies that the local stochastic growth rates ޅ ˆ p growth rates[fii(pN)] are (approximately) equal to zero are zero in all occupied patches, that is, 0 ޅ ˆ Ϫ 2 1 in all occupied patches. [fii(pN(p))] j /2 for all i such thatp i 0 . In particular,

ޅ ˆ Figure 1: Spatially uncorrelated environmental stochasticity favors selection of sink patches. A, C, Mean abundance[pNi (p)] in each patch is plotted along a gradient of patches where per capita birth rates decrease in a Gaussian manner from patch 1 to patch 30. B, D, Intrinsic Ϫ 2 ޅ ˆ Ϫ 2 stochastic growth ratefii(0) j /2 , the local stochastic growth rate[f iii(pN(p))] j /2 , and the “equalizing” stochastic growth rate ޅ ˆ Ϫ 2 p Ϫ Ϫ p Ϫ 2 ϩ p p [fii(pN(p))] p iij are plotted for each patch. Parameters:f i(N i) b i(1 N i/K i) db i with i3 exp ( i /300) 1 ,d i1.5 ,K i 200 , p 2 p ( andjiijj i as shown and ij 0 fori j .

Ϫ 1 2 ˜ i p no sink patches are occupied. Intuitively, when environ- wheneverbiid j /2 , andN 0 otherwise. The ESS mental fluctuations are perfectly correlated, spending time for patch selection is in sink patches does not reduce the environmental variance experienced by an individual. Figure 4 illustrates how par- N˜ i tial spatial correlations influence the ESS along a spatial p p . i ͸ ˜ j gradient of varying per capita birth rates. Spatial corre- j N lations marginalize the degree to which selecting multiple patches reduces the environmental variation experienced Since the local equilibrium abundancesN˜ i decrease with by an individual. Consequently, increasing spatial corre- increasing environmental variationj 2 , patch preferences lations selects for range contraction and reduces the equi- pi become less differentiated at higher levels of environ- librium population abundance in all patches (fig. 4A). mental variation. i p Ϫ For linear per capita growth rates fii(N ) b (1 To illustrate the role of partial spatial correlations on i Ϫ N /K ii) d and perfectly covarying environmental fluc- overmatching or undermatching along a spatial gradient tuations, the ESS for patch selection corresponds to the of varying per capita birth rates, figure 4B plots the dif- classical “patch matching rule”: the fraction of individuals ference between the ESS patch selection preferencespi and in patch i is proportional to the local population abun- ˜˜ij͸ classical patch matching ruleN / j N . Partial spatial cor- dance at equilibrium. More specifically, the equilibrium relations results in overmatching of the lower-quality abundance for populations restricted to patch i equals patches (i.e., greater fraction of individuals selecting the lower-quality patches than predicted by the local equilibrium abundances) and undermatching of the higher-qual- d ϩ j 2/2 N˜ i p K 1 Ϫ i ity patches. Somewhat unexpectedly, the greatest over- i()b i matching occurs at strong spatial correlations despite

Figure 2: Range shifts, persistence of coupled sink populations, and extinction in response to increasing environmental variation (A)or ޅ ˆ patch degradation (B). Contour plots of the mean patch abundancespi [N(p)] for populations playing the evolutionarily stable strategy in a landscape where per capita birth rates decrease in a Gaussian manner. Dashed contours corresponds to where the deterministic and Ϫ 2 stochastic intrinsic growth rates,fiii(0) andf (0) j /2 , equal zero. Labels indicate patch and parameter combinations that correspond to 1 22! ! ! p Ϫ Ϫ p sources (fii(0) j /2 ), stochastic sinks ( 0 f ii(0) j /2 ), and deterministic sinks (f i(0) 0 ). Parameters:f iiiiiii(N ) b (1 N /K ) dK , p ( p Ϫ 22ϩ p p Ϫ ϩ p 200 for all i,jij0 fori jb ; i3 exp ( i /300) 1 andd i1.5 in A;b i3exp( i /150) 1 andj ii 2.5 in B. perfect matching occurring in perfectly covarying consequently, the local stochastic growth rates in these environments. regions are more negative. Spatial correlations, in and of themselves, can influence A surprising consequence of these analytical results is the evolution of patch selection. To see why, consider a that patch destruction within a region can select for in- landscape where there are a finite number of distinct dividuals exhibiting greater preference for the remaining regions or habitats. Patches within different regions only patches in the perturbed region. To illustrate this coun- differ to the extent the temporal fluctuations are corre- terintuitive prediction, I simulated a landscape consisting p 22p lated, that is,fijf andj ij for alli, j butj ij for of two regions with 20 patches in region 1 (before patch ( i jkmay vary. Leti be the number of patches in the destruction) and 10 patches in region 2. Figure 5 illustrates p ϩ ϩ ith region andk k 1 … k n be the total number of that patch destruction in region 1 always results in a lower patches. In the absence of any variation in the spatial fraction of individuals selecting region 1 and a lower mean correlations, individuals select all patches with equal like- total population abundance. Despite this regional trend, p p p p lihood, that is,p12p … pk 1/k . In contrast, as- individuals may exhibit a greater preference for patches sume the correlation between patches within region i is within region 1 following patch destruction. For instance, ri and patches from different regions are uncorrelated. if spatial correlations within both regions are equal p p Appendix D shows individuals playing the ESS preferen- (r12r 0.5 ), then as predicted by the analytic results, Ϫ tially select patches in regions for whichrii(k 1) is smaller individuals exhibit greater preference for patches in region and show no preference for patches within a given region. 2 before patch destruction as there are fewer patches in All else being equal, individuals preferentially select region 2. However, if patch destruction results in region patches within regions with fewer patches or lower spatial 1 having fewer patches, then there is selection for the correlations; both of these preferences reduce the envi- opposite preference. Intuitively, this phenomena occurs ronmental variation experienced by an individual and, because patches in each region offer special opportunities thereby, increases their fitness. Patches within regions with to buffer fluctuations in fitness across the entire ensemble Ϫ lower values ofrii(k 1) support larger populations, and of patches. So, as patches in one region get scarce, they

ޅ ˆ Figure 3: Selection for patches with lower environmental uncertainty. Contour plots of the mean patch abundancespi [N(p)] in A and ޅ ˆ Ϫ 2 p local stochastic growth rates[fii(pN)] j i/2 in B for populations playing the ESS. Dashed contours as in figure 2. Parameters: fii(N ) Ϫ Ϫ p p 22p Ϫ Ϫ p ( biiiii(1 N /K ) ddwith0 ,K i200 ,j i8 8 exp ( i /600) andj ij0 fori j . offer an increasingly precious commodity—a habitat with theory predicts that the ESS for patch selection is an ideal out-of-sync fluctuations in quality. When patches in region free distribution for which the per capita growth rates of 1 are more correlated than patches in region 2, greater individuals in occupied patches equal zero, sink patches patch destruction is required for an evolutionary reversal are unoccupied, and the fraction of individuals selecting in patch preferences. Alternatively, when patches within a patch is proportional to the “carrying capacity” of the region 1 are sufficiently less correlated than patches within patch (Holt and Barfield 2001; Cressman et al. 2004; Cress- region 2, no amount of patch destruction reverses patch man and Krˇivan 2006; Morris 2011). Morris (2011, p. preferences; patches within region 2 are preferred at all 2408) concludes with a call to arms as “there is much work levels of patch destruction. Intuitively, weaker correlations that needs our attention” especially stochastic effects as amongst patches in region 1 increases their value as patches they “produce results that can depart rather dramatically with out-of-sync fluctuations in quality. from deterministic solutions.” In response to this call, I analyzed the evolution of patch selection for a general class of stochastic models accounting for any number of Discussion patches, general patterns of spatial-temporal covariance, Global climate models predict that increasing temporal and general forms of density-dependent population variability in temperature, precipitation, and storms in the growth. The analysis of these models shows that all of the next hundred years (Scha¨r et al. 2004; Tebaldi et al. 2006). classical IFD predictions can break down in the face of For example, Scha¨r et al. (2004) predict temperature var- environmental stochasticity. iability in Europe may increase up to 100% with maximum A powerful prediction of the classical deterministic the- changes in central and eastern Europe. Changes in mean ory is that evolution of patch selection equalizes local per temperature and increased variability in rainfall have al- capita growth rates across all occupied patches. For the ready led to shifts in species distributions (Allen and Bre- empiricist, this prediction is testable (Abrahams 1986; Mi- shears 1998; Parmesan and Yohe 2003; Kelly and Goulden linski and Parker 1991; Tregenza 1995; Oksanen et al. 2008). Evolution of patch selection may play an important 1995). For the theoretician, this prediction often reduces role in these shifts (Parmesan 2006; Morris 2011). Morris solving the ESS to an algebraic problem (Cressman and (2011) recently reviewed how the deterministic theory of Krˇivan 2006). My analysis of models accounting for en- patch selection serves as a powerful tool for making pre- vironmental stochasticity reveals that the evolution of dictions about future shifts in patch use. This deterministic patch selection generally does not equalize any local de-

ޅ ˆ Figure 4: Spatial correlations select for range contraction. A, Contour plots of the mean patch abundancespi [N(p)] for populations playing Ϫ ˜˜**͸ an ESS as a function of the patch location and the spatial correlation r. B, Contour plots of the differencepiiN / j N j between the evolutionarily stable strategy preferences and classical patch matching rule are shown. Positive values correspond to overmatching and the p Ϫ Ϫ p Ϫ Ϫ negative values correspond to undermatching. Dashed contours as in figure 2. Parameters:fii(N ) b i(1 N ii/K ) db ii , 3 exp [ (15 2 ϩ p p p p ( i) /75] 1,di1.5 ,K i100 ,j ii2.1 , andj ij 2.1r fori j . mographic measure of fitness across all occupied patches, patches in simulations of discrete-time models with two a finding consistent with discrete-time, two-patch models patches. (Schmidt et al. 2000; Holt and Barfield 2001). Rather, cor- At a demographic equilibrium, populations conforming relations in temporal fluctuations between patches result to the ideal free distribution never enter sink patches and in a nonlocal measure of fitness being equal in all occupied have balanced birth and death rates in all source patches. patches: the difference between the expected per capita In sharp contrast, my analysis shows that environmental growth rate within a patch and the covariance between stochasticity results in local stochastic growth rates being environmental fluctuations within this patch and the av- negative in all occupied patches. Hence, all local popu- erage environmental fluctuations experienced by the pop- lations are either sink populations or pseudo-sink popu- ulation across all patches. For two special cases, however, lations that only exhibit a negative local stochastic growth local demography determines the ESS for patch prefer- rate due to density-dependent feedbacks (Watkinson and ences. First, if environmental fluctuations tightly covary Sutherland 1995). Intuitively, across all patches, the pop- across the landscape, then one recovers a stochastic analog ulation maintains a larger, stable size in the long term by of the deterministic theory for which the local stochastic reducing the temporal variance. In particular, this implies ޅ ˆ Ϫ 2 growth rates[fii(pN)] j i/2 are equal to zero in all oc- that any particular patch represents some sort of a sink, cupied patches. Second, if environmental fluctuations are because the isolated population cannot realize this vari- uncorrelated between patches, then a nonstandard but lo- ance-reducing effect. ޅ ˆ Ϫ 2 cally determined stochastic growth rate [fii(pN)] p iij When correlations in environmental fluctuations be- is equal in all occupied patches. When there are only two tween patches are weak, populations playing the ESS oc- patches and there is not a strong skew in patch preferences, cupy all stochastic sink patches. In addition, if environ- ޅ ˆ Ϫ 2 this nonstandard growth rate[fii(pN)] p iij is approx- mental variations are sufficiently great in source patches imately equal to the local stochastic growth rate. This may and stochastic sink patches, populations playing the ESS explain why Holt and Barfield (2001) observed the geo- also occupy deterministic sink patches. This latter result metric mean of fitness (a discrete time analog of the local is consistent with earlier studies using discrete-time, two- stochastic growth rate) being equal in occupied source patch models (Holt 1997; Jansen and Yoshimura 1998;

Figure 5: Evolutionary responses of patch selection to patch destruction. Prior to destruction, there are two regions with 20 patch patches in the ﬁrst region and 10 patch patches in the second region. The temporal correlation between patches within region i isri . Patches from different regions are uncorrelated. In A, the total fraction of individuals selecting patches in region 1 is plotted as a function of the number of patches lost in region 1. The dashed curve represents the fraction of patches found in region 1. Points lying above the dashed curve correspond to individuals preferring patches within region 1 and points lying below this curve correspond to individuals preferring patches ޅ ˆ iip Ϫ Ϫ p within region 2. In B, the mean abundance[N(p)] of the total population is plotted. Parameters:fi(N ) b(1 N /K) db with2 , K p 200, andd p 0.5 .

Holt and Barfield 2001). However, these earlier studies did rates. Even if local demography is statistically similar in not predict the evolution of stochastic sink populations all patches, I also show that correlations in the environ- for arbitrarily small Gaussian fluctuations in source mental fluctuations, in and of themselves, lead to under- patches. My analysis also reveals that strong correlations matching. For example, all else being equal, individuals between patches prevent the evolution of sink populations are less likely to select patches in regions experiencing and lead to local stochastic growth rates being close to stronger correlated environmental fluctuations than zero in occupied patches. patches in regions experiencing weaker correlations. Under equilibrium conditions, the IFD predicts the frac- Climate change have resulted in shifts in species geo- tion of individuals selecting a patch is proportional to the graphical distribution in response to changes in mean tem- carrying capacity of the patch (i.e., the equilibrium abun- perature (Parmesan and Yohe 2003; Parmesan 2006). For dance supported by the patch in isolation of other example, in a meta-analysis involving more than 1,700 patches). When environmental fluctuations are synchro- species, Parmesan and Yohe (2003) categorized 434 species nous across all patches and per capita growth rates are as exhibiting a range shift in the past 17–1,000 years (me- linear, this prediction still holds with respect to the mean dian 66 years). Of these, 80% shifted as predicted by population abundance supported by a patch. However, changes in mean temperature with range expansions as when environmental fluctuations are not perfectly co- species colonized previously cool regions or range con- varying, populations playing the ESS overmatch their us- tractions in the case of some arctic species. Alternatively age of lower-quality patches and undermatch their usage and consistent with prior work (Holt and Barfield 2001; of higher-quality patches as predicted by the mean pop- Holt 2003), my analysis reveals that increasing environ- ulation abundances. These predictions are consistent with mental variation can select for species range expansions if results of Hakoyama (2003) on optimal patch selection this variability is weakly or negatively correlated across for landscapes with stochastic variation in resource supply space or range contractions if this variability is highly cor-

This content downloaded from 169.237.045.004 on August 02, 2019 16:55:12 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 28 The American Naturalist related across space. Since there is evidence that some dicate that optimal patch selection behavior may involve species ranges have been altered due to increasing envi- a willingness to settle in low-quality patches, even when ronmental variability (Allen and Breshears 1998; Mc- higher-quality patches are accessible (Ward 1987; Stamps Laughlin et al. 2002; Parmesan 2006), it would be inter- et al. 2005). In light of the results presented here, it seems esting to see how often these changes can be explained by likely that environmental stochasticity in conjunction with the theory presented here. search costs will further increase preferences for low-qual- When environmental variation is too great, population ity patches. Finally, populations may experience inverse models predict populations are extinction prone (Lande density dependence at low densities due to cooperative et al. 2003). For example, McLaughlin et al.’s (2002) anal- behavior, mate limitation, or predator saturation (Cour- ysis on long-term time series of two checkerspot butterfly champ et al. 1999). For populations exhibiting an IFD, populations in California concludes their extinctions were this inverse density dependence can generates alternative hastened by increasing variability in precipitation, a phe- stable states corresponding to all possible combinations of nomenon predicted by global climate models. If this en- occupied patches and unoccupied patches (Fretwell and vironmental variation increases sufficiently slowly to allow Lucas 1969; Greene 2003). Since environmental stochas- a population’s patch selection strategy to evolve, my anal- ticity can promote the evolution of sink populations, it ysis suggests that extinction is preceded by populations seems reasonable to conjecture that inclusion of environ- persisting solely in sink patches. Hence, persistence of cou- mental stochasticity in these models with inverse density pled sink populations, which to some extent have been a dependence will reduce the number of alternative states. fascinating theoretical curiosity (Jansen and Yoshimura To conclude, patch selection strategies that evolve in 1998; Roy et al. 2005; Matthews and Gonzalez 2007; stochastically varying environments can differ significantly Schreiber 2010), may be an inescapable reality for pop- from ideal free distributions under equilibrium conditions. ulations subject to increasing levels of environmental In particular, undermatching of higher-quality patches, variation. avoidance of clusters of highly correlated patches, and se- To keep the models general yet tractable, I made several lection of sink patches can be optimal in stochastically simplifying assumptions that could be relaxed in future fluctuating environments. These results highlight that the studies. First, the focal species does not interact with any spatial covariance structure of environmental stochasticity other species. It is well known that in deterministic patch plays a crucial role on the evolution of patch selection and selection theory, species interactions can substantially alter deserve further empirical and theoretical attention. patterns of patch use (van Baalen and Sabelis 1993; Schmidt et al. 2000; Schreiber et al. 2000; Cressman et al. 2004; Schreiber and Vejdani 2006). For example, prey may select resource poor patches to escape predation and pred- Acknowledgments ators may select resource-rich patches to capture more nutritious prey (Fox and Eisenbach 1992; Schreiber et al. Many thanks to B. Alger, M. Baskett, C. Boetigger, N. 2000; Schreiber and Vejdani 2006). It remains to be un- Fabina, H. Hakoyama, B. Meeker, M. Meisner, J. Moore, derstood whether stochasticity in the environment exag- S. Patel, J. Rosenheim, J. Stamps, and an anonymous re- gerates or ameliorates these contrary choices exhibited by viewer for providing many useful comments on previous prey and their predators. Second, the analysis assumes that drafts of this manuscript. This work was supported by individuals have unlimited, cost-free access to potential National Science Foundation grants EF-0928987 and patches. However models accounting for search costs in- DMS-1022639.

APPENDIX A

An Evolutionary Dynamic

In general, getting an explicit tractable form of the evolutionarily stable strategy (ESS) is difﬁcult. Hence, when this is not possible, it is useful to introduce an evolutionary dynamic that can be used to solve for the ESS numerically. The basic idea behind the evolutionary dynamic is that mutations arise that randomly reallocate time spent in one patch with time spent in another patch. This reassignment is only “adopted” by the population if it increases I ␧ 1 (p, q). More formally, if time is discretized into units of length0 andei denotes the standard unit vector whose

ϩ ␧ Ϫ ␧ ith component is 1 and remaining components are 0, then a mutation from p top eije occurs with probability mppijand establishes with probability I ϩ ␧ Ϫ ␧ ϩ (p, p eije ), wherex ϩ p max {0, x} denotes the positive part of a real number. The establishment probability is consistent with standard branching process approximations. The mutation probability is a choice of convenience. More general forms of mutation probabilities can be used. Discretizing mutation sizes as ␧ is done out of notational convenience. More general scalings proportional to ␧ simply correspond to rescaling time. Under these assumptions, ϩ ␧ Ϫ ≈ ͸ I ϩ ␧ Ϫ ϩϩϪ I Ϫ ␧ Ϫ piiijij(t ) p (t) mpp{ (p, p (e e )) (p, p (e ije )) } j

ѨI(p, q) ѨI(p, q) ≈ mpp͸ FFϪ ␧. ijѨ Ѩ j ()qiqppjqq pp Hence, taking the limit as␧ r 0 yields

dp ѨI(p, q) ѨI(p, q) i p mpp͸ FFϪ ijѨ Ѩ dt j ()qiqppjqq pp

ѨI(p, q) ѨI(p, q) p mp FFϪ ͸ p . ijѨ Ѩ ()qiqppjqj q pp ≥ ͸ p ≥ ≥ ͸ p ≥ I claim that ifpiii(0) 0 for all i andi p (0) 1 , thenp (t) 0 for all i witht 0 andi p i(t) 1 for allt 0 . This ͸ p claim follows from two observations. First, the system of equations are conservative asi dpi/dt 0 . Hence, if ͸ p ͸ p ≥ p p ≥ iipii(0) 1, thenp (t) 1 for allt 0 . Second, sincep i(t) 0 wheneverp ii(t) 0 ,p (t) 0 for t provided that ≥ pi(0) 0. Since ѨI (p, p) p ޅ[f (pNˆ (p))] Ϫ ͸ p j , Ѩ ii jij qi j this system of differential equations becomes

dp i p ޅ ˆˆϪ ͸͸Ϫ ޅ Ϫ ͸ mpiii[f (pN(p))] p jijjjjj p [f (pN(p))] p kjkj , dt ({})jj k

p ޅ ˆ Ϫ ͸͸ϩ 1 mpiii[f (pN(p))] p jijj pp jkjkj , ()jj2 , k ͸ ޅ ˆ Ϫ ͸ p where the second line follows fromjjpjjj[f (pN(p))] (1/2) , k pp jkjj , k 0 ; that is, the average stochastic growth rate equals zero at the stationary distribution for the resident population. iip Ϫ Ϫ When the per capita growth rates are linear, that is,fii(N ) b (1 N /K ii) d , a more explicit representation of the dynamic is possible as

p ޅ[Nˆ (p)] ޅ[f (pNˆ (p))] p b 1 ϪϪi d , ii i()K i i andޅ[Nˆ (p)] must satisfy the zero stochastic growth rate equation

ޅ ˆ 2 p ͸͸ϪϪ[N(p)] Ϫ j(p) 0 pbii1 pd i i . ii()K 2 i Solving forޅ[Nˆ (p)] yields

͸ Ϫ Ϫ 2 i pii(b d i) j(p)/2 ޅ[Nˆ (p)] p ͸ , i pbii/K i which can be substituted into the evolutionary dynamic equation to yield an explicit system of differential equations.

APPENDIX B

Persistence

In order for an evolutionarily stable strategy (ESS) analysis to be meaningful, there needs to be a patch selection ͸ Ϫ 2 1 strategy p such thati pfii(0) j(p)/2 0 . When this occurs, there is a unique positive stationary distribution for this strategy and the population persists (Braumann 1999). Whether such a p exists reduces to maximizing the function p ͸ Ϫ 2 ≥ ͸ p g(p) i pfii(0) j(p)/2subject to the linear constraintspii0 for all i andi p 1 . Under the assumption that **͹ 1 the optimal p, call itp , satisﬁesi pi 0 and the covariance matrix S is invertible, Evans et al. (2012) showed that

1TSϪ1r Ϫ 1 p* p SϪ1 r Ϫ 1 ,(B1) ()1TSϪ11

p p * where1 (1, … , 1) andr (f1(0), … , fk(0)) . Equation (1) deﬁnes the optimal vectorp provided that it belongs to p ≥ ͸ p the interior of the probability simplexD {p : pii0, i p 1} . Otherwise, one can perform similar analyses on the faces of the probability simplex D. Hence, persistence requires thatg(p*) 1 0 . p 22p ( p * p For the special case wherejiijj and ijrj fori jf and i(0) rp for all i, symmetry implies that i 1/k, in which case

11 * p Ϫ ͸ g(p ) r 2 jij, 2 i, j k

11 1 p Ϫ ͸͸22ϩ r 22j jr, 2 ()iikk(j

j 221 k Ϫ 1 j p r ϪϪ r p r Ϫ [1 Ϫ (k Ϫ 1)r], 2 ()kk 2k as claimed in the main text.

APPENDIX C

The Evolutionarily Stable Strategy (ESS)

Assume the covariance matrix S is positive deﬁnite and p is an ESS. Then p is a local maximum for the function . I p ͸ p ≥ q (p, q) subject to the constraintsh(q) i qii1 andq 0 for all i. By the method of Lagrange multipliers there exists a scalar l such that

Ѩh ѨI(p, q) l p l (p) ppF ޅ[f (pNˆ (p))] Ϫ ͸ p j (C1) Ѩ Ѩ ii jij qiiqq pp j 1 for all i such thatpi 0 . Multiplying both sides of equation (C1) bypi and summing with respect to i yields

p ͸͸p ޅ ˆ Ϫ 2 l piiiil p [f (pN(p))] j(p). (C2) ii

On the other hand, at the stochastic equilibrium, the stochastic growth rate of the population equals zero:

2 p ͸ ޅ ˆ Ϫ j(p) 0 piii[f (pN(p))] . (C3) i 2 Combining the last two equations yields j(p)2 l p Ϫ . 2 This expression for l combined with equation (C1) yields the ESS characterization (10) provided in the main text. Sinceq . I(p, q) is a strictly concave down function, any p satisfying the Lagrange multiplier condition is a global maximum for this function and, consequently, an ESS. Unlike the deterministic case for which an ESS p for patch selection only satisﬁes the weak noninvasibilty condition I(p, q) ≤ 0 for allq ( p , any ESS p for patch selection in a stochastic environment satisﬁes the strong noninvasibility condition:I(p, q) ! 0 for allq ( p . This observation implies that the stochastic growth rate is negative in all occupied p patches whenever multiple patches are occupied. To understand why, consider the mutant strategy q whereqi 1 for 1 I p ޅ ˆ Ϫ 2 some patch i selected by the ESS (i.e.,piiii0 ). Since the invasion rate(p, q) [f (pN(p))] j /2 equals the stochastic growth rate in patch i andI(p, q) ! 0 by the ESS noninvasibility condition, the stochastic growth rate in patch i is negative. p Ϫ Ϫ In the special case of linear per capita growth ratesfii(N ) b i(1 N ii/K ) d i , the ESS characterization becomes

p ޅ[Nˆ (p)] j p ϪϪijkϩ ͸ Ϫ 0 biijkji1 d pp j .(C4) ()K jk ()2 i To solve forޅ[Nˆ (p)] , one can use the fact that the stochastic growth rate at the stochastic equilibrium is zero:

p ޅ[Nˆ (p)] j(p)2 p ͸͸ϪϪi Ϫ 0 pbii1 pd i i . ii()K 2 i Hence, ͸ Ϫ Ϫ 2 i pii(b d i) (j(p)/2) ޅ[Nˆ (p)] p . (C5) ͸ 2 i pbii/K i Combined, equations (C4)and (C5) provide an implicit algebraic characterization of the ESS for this special case of linear per capita growth rates. 1 In general, all source patches must be occupied at an ESS. Indeed, assume that patch k is a source patch (fk(0) p p I p Ϫ 1 jkk/2) and p is a strategy such thatp k0 . Letq (0,…,0,1) . Then(p, q) f k(0) j kk/2 0 and p is not an ESS. Hence, a population playing the ESS must occupy all of the source patches. To see that all the stochastic sinks are occupied in spatially uncorrelated environments, consider a strategy p such p p ␧ p thatpi 0 for some i. Without loss of generality, assume thatpk 0 . Consider the mutant strategyq( ) q with p Ϫ ␧ ! p ␧ ␧ p I ␧ qii(1 )piforkq and k . Letg( ) (p, q( )) , in which case 1 ␧2 ␧ p ͸͸͸Ϫ ␧ ޅ ˆ ϩ ␧ Ϫ Ϫ ␧ 2 Ϫ ␧ Ϫ ␧ Ϫ g( ) (1 )piii[f (pN(p))] f k(0) (1 ) pp ijijj (1 ) p iikj j kk. ii22, ji Strategy p can be an ESS only ifg (0) ≤ 0 , where p Ϫ ͸͸͸ޅ ˆ ϩ Ϫ g (0) fkiiiijijiik(0) p [f (pN(p))] ppj p j , ii, ji

p ϩ 1 ͸͸Ϫ fkijijiik(0) ppj p j . 2 i, ji ͸ This is only possible ifi piikj is sufﬁciently positive orf k(0) is sufﬁciently negative. If the correlations between patch ϩ 2 1 1 1 k and the other patches is too weak andfk(0) j(p)/2 0 , theng (0) 0 and one needspk 0 at the ESS. In particular, 1 this means that iffk(0) 0 , then the patch is occupied even if it is a stochastic sink. It also means that indeterministic sinks are occupied if the reduction of the population growth rate due to noise is greater than the reduction in of the population growth rate for individuals living in the sink patch.

To understand the effects of spatial correlations, in and of themselves, on the ESS, I assume there are n regions ͸ p 2 withk 1,…,k niii patches in each region such thati k k . Letjr be the variance for patches in region i and be the p correlation between a pair of patches in region i. Assume thatfi f for all patches i and that patches in different regions are uncorrelated. Then

n 222p ͸ ϩ Ϫ j(p) piiij k [1 r i(k i 1)], ip1 wherepi is fraction of individuals going to a patch within region i. If there are multiple patches occupied at the ESS, then characterization (10) implies 1 n Ϫ ͸ 22ϩ Ϫ p ޅ ˆ Ϫ 2 ϩ Ϫ piiij k [1 r i(1 k i)] [f(pN j(p))] p jij [1 r i(k i 1)] (C6) 2 ip1 for all patches j within region i. Since the right-hand side of equation (C6) is a decreasing function ofpj , occupied 2 ϩ Ϫ p patches in regions with larger values ofjiii[1 r(k 1)] must have smaller values ofp j . Whenj ij for all regions ޅ ˆ i, patches with a lower value ofpjj have a larger value of[f(pN(p))] and, consequently, a larger stochastic growth rate ޅ ˆ Ϫ 2 [f(pNj (p))] j /2.

APPENDIX D Supplementary Figure

Figure D1: Inﬂuence of patch number (6 vs. 30) of stochastic growth rates for populations playing the evolutionarily stable strategy (ESS). p Ϫ Ϫ p p p 22p Ϫ Ϫ p ( Parameters:fii(N ) b i(1 N ii/K ) db i with i5 ,d i0 ,K i200 , andj i8 8exp[ 3(i/k)/2] andj ij 0 fori j . Literature Cited ics restrict local adaptation in Elliott’s blueberry (Vaccinium el- liottii). Evolution 64:370–384. Barson, N., J. Cable, and C. Van Oosterhout. 2009. Population genetic Abrahams, M. 1986. Patch choice under perceptual constraints: a analysis of microsatellite variation of guppies Poecilia reticulata in cause for departures from an ideal free distributions. Behavioral Trinidad and Tobago: evidence for a dynamic source-sink meta- Ecology and Sociobiology 19:409–415. Allen, C., and D. Breshears. 1998. Drought-induced shift of a forest- population structure, founder events and population bottlenecks. woodland ecotone: rapid landscape response to climate variations. Journal of Evolutionary Biology 22:485–497. Proceedings of the National Academy of Sciences of the USA 95: Bascompte, J., H. Possingham, and J. Roughgarden. 2002. Patchy 14839–14842. populations in stochastic environments: critical number of patches Anderson, J., and M. Geber. 2010. Demographic source-sink dynam- for persistence. American Naturalist 159:128–137.

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