Local bifurcation analysis of an ecosystem under the effect of toxicity and self-defense

Abstract

The local bifurcation phenomena in a nonlinear ecosystem model incorporating toxicity effects and species' self-defense strategies are explored in this work. The influence of the ecological parameters on the stability of the equilibrium point is investigated. Bifurcation analysis has significant importance because it reveals the changes that occur at the equilibrium points, as well as how specific parameters affect the system. This paper achieved sufficient conditions that ensure the appearance of local bifurcation (LB), pitchfork bifurcation (PFB), transcritical bifurcation (TB), saddle-node bifurcation (SNB), and Hopf bifurcation (HB) of the system, which consists of prey, middle predator, and top predator with toxin effects and self-defense in the ecosystems. Sotomayor's theorem and Hopf bifurcation conditions help characterize the system's sensitivity to parameter variations. We observed that near the first equilibrium point, PFB occurs. At the free top predator equilibrium points, there is TB, and for the positive equilibrium points, there is SNB.  HB close to the positive equilibrium point has also been studied. Numerical analysis is employed to validate the primary theoretical findings and to illustrate how alterations in parameters have a substantial impact on maintaining biodiversity and ecological balance.