On a predator-prey system with random switching that never converges to its equilibrium

TL;DR

This paper proves that in a predator-prey system with random switching between two deterministic Lotka-Volterra models, the populations do not stabilize but oscillate wildly, confirming a previous conjecture and analyzing related linear systems.

Contribution

It demonstrates that when two deterministic predator-prey models share an equilibrium, the stochastic switching causes populations to oscillate indefinitely, and proves a conjecture about the growth rate in linear switched systems.

Findings

Populations oscillate between 0 and infinity under switching.

Almost sure non-convergence to the equilibrium point.

Linear switched systems with imaginary eigenvalues grow exponentially.

Abstract

We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka-Volterra system for an exponential random time after which it switches to a different deterministic Lotka-Volterra system. This switching procedure is then repeated. The resulting process is a Piecewise Deterministic Markov Process (PDMP). In the case when the equilibrium points of the two deterministic Lotka--Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between $0$ and $\infty$. This proves a conjecture of Takeuchi et al (J. Math. Anal. Appl 2006). The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume $(Y_t, I_t)$ is a PDMP that evolves according to $\frac{dY_t}{dt}=A_{I_t} Y_t$ where $A_0,A_1$ are $2\times2$ matrices and $I_t$ is a Markov chain on $\{0,1\}$ with transition rates $k_0,k_1>0$. If the matrices $A_0$ and $A_1$ are not proportional and have purely imaginary eigenvalues, then there exists $\lambda >0$ such that $\lim_{t \to \infty} \frac{\log \| Y_t \|}{t} = \lambda.$