Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ_t = μZ_tdt + σZ_tdW_t, t ≥ 0, where the conditional law of Z_t+Δt- Z_t given Z_t = z has mean and variance approximately zμΔt and z²σ²Δt when the time increment Δt is small. The long-term stochastic growth rate lam_t→∞t^-1 log Z_t for such a population equals μ - σ²/2. Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model X_t=(X¹_t,..., Xⁿ_t), t ≥ 0, for the population abundances in n patches: the conditional law of X_t+Δt given X_t = x is such that the conditional mean of Xⁱ_t+Δt-Xⁱ_t is approximately [Xⁱμ_i+Σ_j(x^jD_ji)]Δt where μ_i is the per capita growth rate in the ith patch and D_ij is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of Xⁱ_t+Δt-Xⁱ_t and X^j_t+Δt-X^j_t is approximately xⁱx^jσ_ijΔt for some covariance matrix Σ = (σ_ij). We show for such a spatially extended population that if S_t=X¹_t+...+ Xⁿ_t denotes the total population abundance, then Y_t = X_t/S_t, the vector of patch proportions, converges in law to a random vector Y_∞ as t → ∞, and the stochastic growth rate lam_{t → ∞} t^-1 log S_t equals the space-time average per-capita growth rate Σ_i μ_iE[Yⁱ_∞] experienced by the population minus half of the space-time average temporal variation E[Σ_i,j σ_ij; Yⁱ_∞Y^j_∞] experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multi-scale landscapes (e. g. insects on plants in meadows on islands). Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of density-dependent feedbacks, ideal-free dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.