Impacts of Tempo and Mode of Environmental Fluctuations on Population Growth: Slow- and Fast-Limit Approximations of Lyapunov Exponents for Periodic and Random Environments.

Populations consist of individuals living in different states and experiencing temporally varying environmental conditions. Individuals may differ in their geographic location, stage of development (e.g., juvenile versus adult), or physiological state (infected or susceptible). Environmental conditions may vary due to abiotic (e.g. temperature) or biotic (e.g. resource availability) factors. As the survival, growth, and reproduction of individuals depend on their state and environmental conditions, environmental fluctuations often impact population growth. Here, we examine to what extent the tempo and mode of these fluctuations matter for population growth. We model population growth for a population with d individual states and experiencing N different environmental states. The models are switching, linear ordinary differential equations $x^{\text{'}}\left(t\right)=A\left(\sigma \left(\omega t\right)\right)x\left(t\right)$ where $x\left(t\right)=(x_1\left(t\right),\dots ,x_d\left(t\right))$ corresponds to the population densities in the d individual states, $\sigma \left(t\right)$ is a piece-wise constant function representing the fluctuations in the environmental states $1,\dots ,N$ , $\omega$ is the frequency of the environmental fluctuations, and $A\left(1\right),\dots ,A\left(n\right)$ are Metzler matrices representing the population dynamics in the environmental states $1,\dots ,N$ . $\sigma \left(t\right)$ can either be a periodic function or correspond to a continuous-time Markov chain. Under suitable conditions, there exists a Lyapunov exponent $\Lambda \left(\omega \right)$ such that ${lim}_{t\to \infty }\frac{1}{t}log{\sum }_i{x}_i\left(t\right)=\Lambda \left(\omega \right)$ for all non-negative, non-zero initial conditions x(0) (with probability one in the random case). For both random and periodic switching, we derive analytical first-order and second-order approximations of $\Lambda \left(\omega \right)$ in the limits of slow ( $\omega \to 0$ ) and fast ( $\omega \to \infty$ ) environmental fluctuations. When the order of switching and the average switching times are equal, we show that the first-order approximations of $\Lambda \left(\omega \right)$ are equivalent in the slow-switching limit, but not in the fast-switching limit. Hence, the mode (random versus periodic) of switching matters for population growth. We illustrate our results with applications to a simple stage-structured model and a general spatially structured model. When dispersal rates are symmetric, the first-order approximations suggest that population growth rates decrease with the frequency of switching, which is consistent with earlier work on periodic switching. In the absence of dispersal symmetry, we demonstrate that $\Lambda \left(\omega \right)$ can be non-monotonic in $\omega$ . In conclusion, our results show that population growth rates often depend both on the tempo ( $\omega$ ) and the mode (random versus deterministic) of environmental fluctuations.