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).