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Environmental Variation, Stochastic Extinction, and Competitive Coexistence

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Environmental Variation, Stochastic Extinction, and Competitive Coexistence vol. 172, no. 5 the american naturalist november 2008 E-Article Environmental Variation, Stochastic Extinction, and Competitive Coexistence Peter B. Adler1,* and John M. Drake2,† 1. Department of Wildland Resources and the Ecology Center, The great difficulty in the opportunistic hypothesis is that Utah State University, Logan, Utah 84322; since, if many species are present in a really variable environ- 2. Odum School of Ecology, University of Georgia, Athens, ment which is controlling their competition, chance extinction Georgia 30602 is likely to be an important aspect of the process. (Hutchinson Submitted November 25, 2007; Accepted May 16, 2008; 1961, pp. 141–142) Electronically published September 24, 2008 Temporal variation is a ubiquitous feature of ecosystems, and ecologists have long been interested in how the mag- nitude of variation structures natural populations and abstract: Understanding how environmental fluctuations affect communities (Andrewartha and Birch 1954; Lewontin and population persistence is essential for predicting the ecological im- Cohen 1969). Exogenous environmental variation has pacts of expected future increases in climate variability. However, been shown to affect all levels of biological organization, two bodies of theory make opposite predictions about the effect of from the evolution of individual traits (Cohen 1966; Ven- environmental variation on persistence. Single-species theory, com- mon in conservation biology and population viability analyses, sug- able and Brown 1988) to the stability of ecosystem pro- gests that environmental variation increases the risk of stochastic cesses (Tilman and Downing 1994; Bai et al. 2004). This extinction. By contrast, coexistence theory has shown that environ- basic research must now provide the foundation for un- mental variation can buffer inferior competitors against competitive derstanding the ecological impacts of expected increases exclusion through a storage effect. We reconcile these two perspec- in climate variability. Global circulation models consis- tives by showing that in the presence of demographic stochasticity, tently forecast increases in the frequency of extreme events environmental variation can increase the chance of extinction while such as severe storms and droughts (Karl and Trenberth simultaneously stabilizing coexistence. Our stochastic simulations of 2003; Salinger 2005). As precipitation and temperature a two-species storage effect model reveal a unimodal relationship depart from their historical ranges of variability, theory between environmental variation and coexistence time, implying maximum coexistence at intermediate levels of environmental var- will be crucial for predicting how natural populations will iation. The unimodal pattern reflects the fact that the stabilizing respond. influence of the storage effect accumulates rapidly at low levels of Unfortunately, such predictions are complicated by dis- environmental variation, whereas the risk of extinction due to the agreement between two bodies of ecological theory. The combined effects of environmental variation and demographic sto- single-species models used in population viability analysis, chasticity increases most rapidly at higher levels of variation. Future an important tool in conservation biology, suggest that increases in environmental variation could either increase or decrease environmental variation typically increases the risk of sto- an inferior competitor’s expected persistence time, depending on the chastic extinction due to geometric averaging of popula- distance between the present level of environmental variation and the optimal level anticipated by this theory. tion growth rates over time and high spatial synchrony of species with restricted ranges (Boyce 1992; Menges 2000; Keywords: climate variability, competition, demographic stochasticity, but see Drake 2005; Boyce et al. 2006). By contrast, co- plant communities, population dynamics, simulation. existence theory has shown that environmental variation can buffer rare species against extinction: temporary con- ditions resulting in high individual fitness can allow in- * Corresponding author; e-mail: [email protected]. ferior competitors to escape competitive exclusion (Ches- † E-mail: [email protected]. son and Warner 1981; Chesson and Huntly 1989). Am. Nat. 2008. Vol. 172, pp. E186–E195. ᭧ 2008 by The University of Empirical tests in natural systems indicate that this co- Chicago. 0003-0147/2008/17205-50077$15.00. All rights reserved. existence mechanism may be important in communities DOI: 10.1086/591678 of zooplankton (Ca´ceres 1997), annual plants (Pake and Environmental Variation and Coexistence E187 Venable 1995; Adondakis and Venable 2004; Facelli et al. Extinction in a Single-Species System 2005), and prairie grasses (Adler et al. 2006). As previous authors have noted (Higgins et al. 2000; Changes in the abundance of seeds over time are set de- Levine and Rees 2004), an important difference between terministically (following Chesson and Huntly 1989) by these two perspectives is the focus on competitive inter- the nonlinear difference equation actions in coexistence models. But these two disciplines also have addressed subtly different questions. Single- ˆ lg(Ett)N N ϩ p s[1 Ϫ g(E )]N ϩ ,(1) species conservation studies typically use stochastic sim- t 1 tt ϩ 1 ag(Ett)N ulations to evaluate mean time to extinction in models that include a quasiextinction threshold or formally in- where N is the abundance of seeds at time t, s is the corporate demographic stochasticity (reviewed in Boyce t survival of seeds in the seed bank, l is the per capita 1992; Menges 2000), whereas coexistence theory uses de- fecundity, a is the intraspecific competition coefficient, terministic, analytical models without extinction thresh- and g is the germination rate. This model allows popu- olds or demographic stochasticity to evaluate whether co- lations to decline arbitrarily close to, but not reach, ex- existence is stable in the long term (e.g., Chesson 1990). tinction. To incorporate demographic stochasticity, we Here we aim to reconcile these perspectives by studying ϩ draw the realizedNttϩ1 density of seeds at time1 from the combined effects of environmental variation and de- ˆ a Poisson distribution with expectationN ϩ , that is, mographic stochasticity on mean coexistence times in a t 1 modified version of Chesson and Huntly’s (1989) two- ∼ ˆ species annual plant model. Our approach recognizes that Ntϩ1 Pois(Ntϩ1). (2) environmental variation can increase the probability of extinction while simultaneously stabilizing coexistence by Adding demographic stochasticity means that actual seed buffering inferior competitors against competitive exclu- densities take integer values and that extinction (N p 0 ) sion. To understand the combined effect of these two forces is possible. on the duration of coexistence, we first study them in- We simulated annual germination rates by drawing a dividually. We show that the probability of stochastic ex- sequence of independent and identically distributed ran- tinction in this model is low at low levels of environmental dom variates, E, from a continuous normal distribution variation but then increases quickly at higher levels of 2 with mean 0 and variancejE , and then we used the inverse variation. In contrast, the stabilizing effects of environ- logit transformation to convert E to germination rates in mental variation accumulate most rapidly at low levels of the 0–1 range: environmental variation. As a result, in simulations of the two-species model with demographic stochasticity, we find e E a unimodal relationship between environmental variation g(E) p .(3) ϩ E and coexistence times, implying that coexistence is max- 1 e imized at an intermediate level of environmental variation. 2 We explore how the location of this optimum depends on As the variance of E,jE , increases, extreme values for the degree of competitive asymmetry and the correlation germination occur more frequently (fig. 1). We conducted 2 in species’ responses to the environment. our simulations at values ofjE of 0, 0.4, 1, 2.5, 5, 7.5, and 10. We used simulations to estimate the probability that the population will reach extinction within short time periods, Methods given a specified variance for E. Simulations were initial- p ized at low population density (N0 2 , where N0 is the While the effect of environmental variation on extinction realized density and not the mean of the Poisson process), in single-species systems has been studied for many pop- iterated for 100 time steps, and replicated 10,000 times. ulation models (Alvarez 2001), research on coexistence We did not follow each run to extinction because the mediated by temporal variability is based on a limited set computing time would be prohibitive. The probability of of competition models (Chesson and Warner 1981; Ches- extinction was estimated as the fraction of runs in which son and Huntly 1989; Chesson 2000). To facilitate our density fell to 0. As a baseline for comparison, we set comparison of these distinct perspectives, we first dem- s p 0.5 anda p 1 . We repeated the simulations for values onstrate the single-species approach by using a one-species of l of 10, 15, 25, and 100 to represent a range from small, special case of Chesson and Huntly’s (1989) competition slow-growing populations to larger, fast-growing popu- model. lations. E188 The American Naturalist 2 Figure 1: Distribution of germination rates as a function of variation in the
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