Stability analysis of an eco-epidemic predator–prey model with Holling type-I and type-III functional responses

Abstract

A predator–prey model with Holling type-I and type-III functional responses, where the disease spreads between the prey, is considered in this paper. In consideration of the ecological balance, a harvest term is added to the predator. The positivity and boundedness of the solutions are discussed. Then, the conditions of the equilibrium points are analysed. According to the Routh–Hurwitz criterion, the local stability of equilibrium points can be analysed. For the disease-free equilibrium point, harvest rate h is selected as the bifurcation parameter. For the positive equilibrium point of the system, we choose infection rate b as the bifurcation parameter. By calculating and analysing the corresponding characteristic equations, the existence of Hopf bifurcation at equilibrium points is investigated. On the basis of high-dimensional bifurcation theory, we can obtain formulas which can decide the direction, period and stability of Hopf bifurcation of the system. To substantiate the theory, time history, bifurcation diagram and phase diagrams at different equilibrium points are drawn. In a disease-free environment, it may occur that the predator will prey on the prey in large numbers and eventually leads to the death of the prey. According to the numerical results, it can be seen that proper harvesting of predators is conducive to the stable development of the population. In a diseased ecology, when the infection rate experiences \(b^{*}\), the stability of the system changes and the prey population can adapt to such changes better. It helps to eliminate some old and weak species to reduce the consumption of resources.