Population dynamics under random switching

TL;DR

This paper studies the long-term behavior of multiple populations modeled by piecewise deterministic Markov processes, providing conditions for their persistence or extinction based on environmental switching dynamics.

Contribution

It introduces sharp criteria for population persistence and extinction in PDMP models, extending ecological modeling beyond traditional stochastic differential equations.

Findings

Derived conditions for population persistence and extinction.

Applied theory to ecological examples demonstrating practical relevance.

Analyzed invasion rates (Lyapunov exponents) for boundary measures.

Abstract

Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often, environmental stochasticity is modeled by systems of stochastic differential equations. However, this type of stochasticity is not always the best suited for ecological modeling. Instead, biological systems can be modeled using piecewise deterministic Markov processes (PDMP). For a PDMP the process follows the flow of a system of ordinary differential equations for a random time, after which the environment switches to a different state, where the dynamics is given by a different system of differential equations. Then this is repeated. The current paper is devoted to the study of the dynamics of $n$ populations described by $n$-dimensional Kolmogorov PDMP. We provide sharp conditions for persistence and extinction, based on the invasion rates (Lyapunov exponents) of the ergodic probability measures supported on the boundary of the positive orthant. In order to showcase the applicability of our results, we apply the theory in some interesting ecological examples.