Population dynamics
We investigate the conditions promoting coexistence using a stochastic
Lotka-Volterra competition model with two competing species regulated by
the population density of its own and the opposing species. Although
this design is similar to the classic form of this model (May &
MacArthur 1972; May 1974), instead of integrating all fitness responses
of environmental conditions, we leave them as continuous responses to
the environment (i.e. temperature) and explicitly model environmental
conditions through time. That is, the population dynamics follow
\(\frac{\text{dN}_{i}}{\text{dt}}={r_{i}N_{i}w}_{i}\left(T\right)\left(1-\frac{N_{i}}{K_{i}w_{i}\left(T\right)}-\frac{\alpha N_{j}}{K_{i}w_{i}\left(T\right)}\right)-d_{i}N_{i}(1-w_{i}\left(T\right))\),
(1)
where \(i,j\) identifies the species, \(T\) is temperature, and \(N\)stands for the size of the population. Similarly, \(K\),\(r\), \(\alpha\), and \(d\) denote carrying capacities,
intrinsic growth rates, levels of interspecific interaction, and
temperature-dependent mortality, respectively.
Although most parameters remain constant, the population growth rate
changes with temperature and depends on the thermal performance function
(\(w_{i}\left(T\right)\)), which is adapted from previously published
estimates of thermal performance curves (Deutsch et al. 2008;
Vasseur et al. 2014),
\(w_{i}\left(T\right)=\left\{\par
\begin{matrix}\exp\left(-{(\frac{(T-T_{opt,\ i})}{2\sigma_{i}})}^{2}\right)+w_{\text{base}},\ \ \&T<T_{opt,i}\\
1-\left[\frac{\left(T-T_{opt,i}\right)}{\left(T_{opt,i}-T_{max,i}\right)}\right]^{2}+w_{\text{base}},\ \ \&T\geq T_{opt,i}\\
\end{matrix}\right.\ \). (2)
This function has a maximum value of 1 at \(T_{\text{opt}}\). When \(T\)is below \(T_{\text{opt}}\), thermal performance decreases exponentially
with decreasing temperature, whereas when \(T\) is above\(T_{\text{opt}}\), thermal performance decreases quadratically and
eventually reaches the minimum value, \(w_{\text{base}}\), at\(T_{\max}\). We assume that there are high temperature-adapted species
(orange) and a low temperature-adapted species (blue) (Fig. 1a).