Generalizing the Levins metapopulation model to time varying colonization and extinction rates

TL;DR

This work generalizes the Levins metapopulation model to deterministic time variation in per-patch colonization and extinction ($c(t)$ and $e(t)$). By formulating the nonautonomous ODE $p' = c(t) p(1-p) - e(t) p$ and introducing upper/lower averages via the Bohl exponents $\beta^-(\mathcal{D})$, $\beta^+(\mathcal{D})$ for $\mathcal{D}(t)=e(t)-c(t)$, the authors derive persistence or extinction criteria that extend the classical $e<c$ threshold to time-varying environments. They provide precise asymptotics for $p(t)$ under the persistence and extinction regimes and discuss partial results and open problems when the averages have mixed signs. The paper also connects these results to exponential dichotomy theory, presents similarities and differences with the classic model, and extends the framework to three extended forms of colonization/extinction interactions (propagule rain and rescue; rescue-only; mainland–island). Overall, the approach offers a rigorous, nonautonomous perspective on metapopulation persistence and highlights new avenues for understanding persistence under environmental variability.

Abstract

The metapopulation theory explores the population persistence in fragmented habitats by considering a balance between the extinction of local populations and recolonization of empty sites. In general, the extinction and colonization rates have been considered as constant parameters and the novelty of this paper is to assume that they are subject to deterministic variations. We noticed that an averaging approach proposed by C. Puccia and R. Levins can be adapted to construct the upper and lower averages of the difference between the extinction and colonization rates, whose sign is useful to determine either the permanence or the extinction of the metapopulation. In fact, we use these averages to revisit the classical model introduced by R. Levins. From a mathematical perspective, these averages can be seen as Bohl exponents whereas the corresponding analysis is carried out by using tools of non autonomous dynamics. Last but not least, compared with the Levins model, the resulting dynamics of the time varying model shares the persistence/extinction scenario when the above stated upper and lower averages have the same sign but also raises open questions about metapopulation persistence in the case of the averages have different sign.

Figures (1)

• Figure 1: Fraction of occupied patches by considering the colonization and extinction rates: $c(t)=\alpha+\sin(t) \quad \textnormal{and} \quad e(t)=\beta+\cos(t)$. The images of the first row are obtained by considering the parameters $(\alpha,\beta)=(3.5,2)$ and $(\alpha,\beta)=(2.5,2)$, which leads respectively to the averages $\beta^{+}(\mathcal{D})=\beta^{-}(\mathcal{D})=-1.5$ and $\beta^{+}(\mathcal{D})=\beta^{-}(\mathcal{D})=-0.5$, the persistence of the metapopulation is observed. The images of the second row are obtained by considering the parameters $(\alpha,\beta)=(1.5,2)$ and $(\alpha,\beta)=(1.1,2)$, leading to $\beta^{+}(\mathcal{D})=\beta^{-}(\mathcal{D})=0.5$ and $\beta^{+}(\mathcal{D})=\beta^{-}(\mathcal{D})=0.9$, the extinction of the metapopulation is observed.

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Definition 1
• Definition 2
• Proposition 1
• Remark 3
• Proposition 2
• proof
• Theorem 1
• proof