Saddle-node--transcritical interactions in a stressed predator-prey-nutrient system

TL;DR

This paper analyzes complex bifurcation interactions in a predator-prey-nutrient model affected by toxicants, revealing bistability and periodic behaviors, and introduces a new method for detecting bifurcation points.

Contribution

It provides a detailed analysis of saddle-node and transcritical bifurcation interactions in a predator-prey system and proposes a novel test function for bifurcation detection.

Findings

Identification of bistable and periodic dynamics due to bifurcation interactions

Numerical detection of codimension-two points as cusp and Bogdanov-Takens points

Proposal of a new test function for systems with transcritical curves

Abstract

We examine the interaction of transcritical and saddle-node bifurcations in a predator-prey-nutrient system that is stressed by the presence of a toxicant affecting the prey. This model, formulated by Kooi et al. ({\sl Ecol. Model.} {\bf 212}(2008), 304--318), has a two-dimensional invariant sub system with zero predator density. In the sub system, a pair of prey-nutrient equilibria is created in a saddle-node bifurcation, while predator invasion in modelled by a transcritical bifurcation of one of this pair. Interactions of these bifurcations at codimension-two points give rise to bistable, periodic and heteroclinic predator-prey-nutrient dynamics. We explain why the the codimension-two points are numerically detected as cusp and Bogdanov-Takens points when using standard test functions and propose a new test function for systems with codimension one trancritical curves.