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.