Introduction
Environmental fluctuation not only influences an organism’s physiology and reproduction directly, it can also impact an organism’s fitness indirectly by driving species interactions (Davis et al. 1998; Tylianakis et al. 2008; Gilman et al. 2010). Although it is often assumed that niche differences need to be greater for competing species to coexist in fluctuating environments because stochastic environmental fluctuation can favor one species and exclude others by chance (May & MacArthur 1972; May 1973, 1974), depending on the intensity of disturbance, environmental fluctuation can either promote or prevent species coexistence (Hutchinson 1953; Hutchinson 1961). For example, one of the key arguments behind the intermediate disturbance hypothesis is that species can reach an equilibrium state and exclude other competing species under reduced environmental fluctuation, whereas increased fluctuation make species more vulnerable to extinction and few species can coexist (Hutchinson 1953; Hutchinson 1961; Grime 1973; Connell 1978; Roxburgh et al. 2004). Consequently, intermediate levels of disturbance are predicted to promote coexistence.
Modern coexistence theory proposes that species can only coexist when the fitness differences between them—defined as relative population growth rates in response to environmental condition and intra- or interspecific competition—are smaller than their niche differences in a shared environment (i.e. differences in resource utilization in space and time) (Carroll et al. 2011; Ke & Letten 2018). Accordingly, environmental fluctuation potentially promotes species coexistence either by equalizing the effects that minimize average fitness differences between species or by creating different temporal niches. Most theoretical models of species coexistence have focused on systems in equilibrium by assuming stationary environments. In other words, fluctuating environments are represented by the mean environmental condition because the environmental state at any given time recurs with a predictable long-run frequency (i.e. environmental states are at equilibrium) (Chesson 2017). Consequently, these studies argue that mean environmental conditions, instead of environmental fluctuation, are crucial for determining patterns of coexistence (Chesson & Huntly 1997; Fox 2013). How environmental fluctuation influences species coexistence in nonequilibrium systems remains poorly understood. Since natural systems are largely considered to be in a nonequilibrium state (Rohde 2005; Shimadzu et al. 2013; Donohue et al. 2016), considering species coexistence in nonequilibrium systems will be crucial for understanding real-world scenarios that might influence species coexistence, particularly in a period of increased global climate change where environmental fluctuation is increasing across the world.
The relationship between environmental variation and species coexistence is also likely to be context dependent. Importantly, the degree of environmental fluctuation can vary in intensity, frequency, and duration (Vasseur et al. 2014; Lawson et al. 2015), meaning it occurs at multiple temporal scales (Chan et al. 2016; Dillonet al. 2016). For example, variation in temperature lasting days or months may have different effects on adaptation such that higher long-term environmental variation tends to favor niche generalists, whereas higher short-term environmental variation tends to favor niche specialists (Gilchrist 1995; Chan et al. 2016). Likewise, fluctuation in temperature occurring near a species’ optimum may have different impacts from fluctuation occurring at unfavorable temperatures (Liu et al. 2019). Yet, few theoretical models have explicitly addressed the impacts of these different forms of environmental variation on species coexistence, especially in nonequilibrium systems.
Here, we employ the newly developed standardized approach for characterizing temperature variation across temporal scales (Dillonet al. 2016) within a stochastic Lotka-Volterra competition model framework to explore patterns of species coexistence in stochastic environments. We use thermal performance curves to explicitly describe temperature-dependent fitness (Huey & Kingsolver 1989; Angilletta Jr & Angilletta 2009). Although we focus on temperature, our approach can be applied to other climatic measures like precipitation. In addition, we limit our study to nonequilibrium (or unstable) species coexistence (Hutchinson 1961; Chesson 2000; Loreau 2010) because (1) many competing species in fluctuating environments are unlikely to exist in a state of stable coexistence (Edmunds et al. 2003; Cothran et al.2015; Donohue et al. 2016) and (2) numerous empirical studies have shown that environmental fluctuation is critical for influencing patterns of species coexistence (Shimadzu et al. 2013; Chisholmet al. 2014). Our model can thus explore environmental fluctuations of large magnitude in nonequilibrium systems, which supplements previous models focusing on stable coexistence at equilibrium states (Chesson 1994) or fixed population sizes (Ellneret al. 2016). Ultimately, our model provides a basic framework for understanding patterns of species coexistence in fluctuating and unpredictable environments, a topic that will have critical implications for studying and conserving biodiversity in an era of anthropogenic climate change.