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

The study of intra-guild interactions 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 in an intra-guild, three species system (i.e., in which two species are predators of the third, but such that one of the predators also consumes the other). 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 explorations revealed sequences of transcritical and Hopf bifurcations underlying 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.