Abstract
Hopf bifurcation analysis of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response is investigated. By analyzing the corresponding characteristic equations, the conditions for the stability and existence of Hopf bifurcation for the system are obtained. In addition, a hybrid control strategy is proposed to postpone the onset of an inherent bifurcation of the system. By utilizing normal form method and center manifold theorem, the explicit formulas that determine the direction of Hopf bifurcation and the stability of bifurcating period solutions of the controlled system are derived. Finally, some numerical simulation examples confirm that the hybrid controller is efficient in controlling Hopf bifurcation.
Highlights
In the natural world, transmissible diseases in ecological environment cannot be ignored
Referring to diseases that are transmissible in different populations, Guo et al [12] studied an ecoepidemiological model with disease spreading within the predator population as follows: ẋ (t) a12x (t) S (t) 1 + mx (t) a21x (t) S (t) 1 + mx (t) βS (t) I (t), (1)
The parameters r, a11, a12, a21, m, r1, r2, and β in model (1) are all positive constants in which r is the intrinsic growth rate of prey and r/a11 represents the carrying capacity of prey; only the susceptible predators have the ability to capture the prey with capturing rate a12; a21/a12 is the conversion rate of the susceptible predators; m is the half-capturing saturation constant, r1 is the natural death rate of the susceptible predator, r2 is the natural and disease-related mortality rate of the infected predator, and β > 0 is called the disease transmission coefficient
Summary
Transmissible diseases in ecological environment cannot be ignored. Motivated by the works of Guo et al [12] and Capasso and Serio [13] and based on the influence about the time delay, in this paper, we consider the following ecoepidemiological model with nonlinear incidence rate, time delay, and Holling type II functional response:. It even causes the system to explode, which may be harmful to the ecological balance Based on this point, a hybrid control strategy by combining the state feedback control and perturbation parameter is used in order to postpone the onset of an inherent bifurcation and enlarge the stable range in model (2). A brief conclusion is given in the last section to conclude this work
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