Hybrid control of Hopf bifurcation in a Lotka-Volterra predator-prey model with two delays

Abstract

In this paper, the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays is studied by using a hybrid control strategy. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are established. In addition, the onset of an inherent bifurcation is delayed. Based on the normal form theory and the center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Numerical simulation results confirm that the hybrid controller is efficient in controlling Hopf bifurcation.

Hopf bifurcation analysis in a predator–prey model with two time delays and stage structure for the prey

“…In the remainder of this section, by using the algorithms in [18] and using a similar calculation process to that in [10], we obtain the coefficients used in determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions: …”

“…They discussed complete analysis of local and global dynamical behaviors of this model by Filippov qualitative theory. Peng et al [10] proposed a hybrid control strategy in a predator-prey model. Their numerical simulation results showed that the hybrid controller is efficient in controlling a Hopf bifurcation.…”

Hopf Bifurcation and Hybrid Control of a Delayed Ecoepidemiological Model with Nonlinear Incidence Rate and Holling Type II Functional Response

“…Therefore, the analysis method is very similar to Section 2; we will omit the local stability and Hopf bifurcation analysis of system (24). We obtain the corresponding critical value of time delay ( ) ( ) as…”

“…The aim of bifurcation control is to design a controller to modify the bifurcation properties of a given nonlinear system, thereby achieving some desirable dynamical behaviors. From the control theory point of view, many effective control methods have been proposed, such as the state feedback control [15,16] and hybrid control strategy [17][18][19][20][21][22][23][24]. Especially, the hybrid control has also been widely used recently.…”

“…Kiani et al [21] used the hybrid control method for a three-pole active magnetic bearing (AMB), and the method showed that the power usage decreased in the hybrid control method comparing to a simple linear control. Peng et al [24] studied the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays by using a hybrid control strategy.…”