# Stochastic Lotka-Volterra food chains

## TL;DR

This paper analyzes how environmental stochasticity affects species persistence and extinction in a simple predator-prey food chain modeled by stochastic Lotka-Volterra equations, extending deterministic results to stochastic settings.

## Contribution

It provides explicit criteria for species extinction or persistence in stochastic Lotka-Volterra food chains, including cases with degenerate noise and correlations.

## Key findings

- Environmental stochasticity increases extinction risk.
- Small environmental fluctuations preserve deterministic persistence.
- Adding an apex predator increases extinction likelihood.

## Abstract

We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $n-1$ predators. The $j$th predator eats the $j-1$th species and is eaten by the $j+1$th predator; this way each species only interacts with at most two other species - the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting. As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction. Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.