Bifurcations and multi-stability in an eco-epidemic model with additional food

Abstract

In this work, we propose a Leslie–Gower prey–predator model where prey is afflicted with an incurable illness, and the predator may choose to eat the provided extra food. Our study aims to control the existing disease in the system with the provision of alternative food. To achieve the goal, we investigate the suggested model and its disease-free subsystem theoretically and numerically. The scope of our analysis is broadened to encompass both local and global bifurcations. Hopf-bifurcation, transcritical bifurcation, saddle-node bifurcation, homoclinic bifurcation, heteroclinic bifurcation, all occur due to stability transitioning between steady states or cycles. Numerical results indicate that the additional food parameter αA contributes to the complex dynamics of the system. A slight modification in αA can significantly change the characteristics of the entire system. In a specific range of αA, all of these unanticipated changes render the system bi-stable and multi-stable. In such cases, we plot their basins of attraction. Consequently, a set of starting values for which the system is disease-free is obtained. We also illustrate the phenomenon of global stability toward the positive equilibrium. Furthermore, the infection rate is capable of altering the dynamics of the system. Through a subcritical Hopf-bifurcation, it can control the oscillations in species around their positive steady state. However, ample energy from the alternative food may lead to disease eradication even for higher infection rates.