In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by systems of differential equations (ODE) that switch at random times. We follow one ODE for an exponential time, after which we switch to a different ODE and the process gets repeated. This is one way of adding environmental noise to the system.
Results/Conclusions
We are able to give sharp conditions for coexistence and for extinction. As applications, we look at a 2d Lotka-Volterra predator-prey model and a 2d competitive Lotka-Volterra model. We show that if the switching is between two deterministic environments E and F the behavior of the switched system can be non-intuitive. In particular, one can see in the competitive model that even if species 1 persists and species 2 goes extinct in environments E and F once we introduce the switching we can get: 1) Reversal: species 1 goes extinct and species 2 persists. 2) Coexistence: both species persist. Our results show how environmental noise can facilitate coexistence for multiple species competing for the same resource.