A model of predation and survival in a system of three interacting species

TL;DR

This paper develops a mathematical model of a three-species ecosystem with predator-prey interactions, revealing how small changes can lead to instability and potential extinction, emphasizing the need for predictive population management.

Contribution

It introduces a Lotka-Volterra based model for a three-species system with complex interactions, analyzing bifurcations that indicate system vulnerability.

Findings

Sequences of bifurcations lead to system instability.

External noise can trigger transitions to vulnerable states.

Population control strategies are essential to prevent extinction.

Abstract

The study of interactions between multiple species in an ecosystem is an active and impactful direction of inquiry. This is true in particular for fragile systems in which even small perturbations of their functional parameters can produce dramatic effects like species endangerment or extinction, leading the system to enter an unsustainable regime and eventually collapse. In this context, it is important to understand which factors can lead to such effects and for which systems, so that one can act proactively and timely to prevent them. We built and studied a mathematical model that captures the natural interactions between three species, in which two species are predators of the third, but such that one of the predators also consumes the other (to which we refer as Owls, Snakes and Mice). The nonlinear components of the model were documented on existing literature and assembled as a system of Lotka-Volterra ordinary differential equations. Our analytical computations and numerical exploration explorations revealed sequences of transcritical and Hopf bifurcations that underlie counterintuitive transitions of the system into regions of vulnerability to external noise. We conclude that, in order to avoid extinction,one needs to rigorously prescribe a well-documented, prediction-based approach to population control.