## Abstract

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_{t}dt + σZ_{t}dW_{t}, t ≥ 0, where the conditional law of Z_{t+Δt}- Z_{t} given Z_{t} = z has mean and variance approximately zμΔt and z^{2}σ^{2}Δt 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 - σ^{2}/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}=(X^{1}_{t},..., X^{n}_{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 X^{i}_{t+Δt}-X^{i}_{t} is approximately [X^{i}μ_{i}+Σ_{j}(x^{j}D_{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 X^{i}_{t+Δt}-X^{i}_{t} and X^{j}_{t+Δt}-X^{j}_{t} is approximately x^{i}x^{j}σ_{ij}Δt for some covariance matrix Σ = (σ_{ij}). We show for such a spatially extended population that if S_{t}=X^{1}_{t}+...+ X^{n}_{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_{i}E[Y^{i}_{∞}] experienced by the population minus half of the space-time average temporal variation E[Σ_{i,j} σ_{ij}; Y^{i}_{∞}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.

Original language | English (US) |
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Pages (from-to) | 423-476 |

Number of pages | 54 |

Journal | Journal of Mathematical Biology |

Volume | 66 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2013 |

## Keywords

- Dominant Lyapunov exponent
- Evolution of dispersal
- Habitat fragmentation
- Ideal free movement
- Single large or several small debate
- Spatial and temporal heterogeneity
- Stochastic population growth