Periodic pulse control of Hopf bifurcation in a fractional-order delay predator–prey model incorporating a prey refuge

Abstract

This paper is concerned with periodic pulse control of Hopf bifurcation for a fractional-order delay predator-prey model incorporating a prey refuge. The existence and uniqueness of a solution for such system is studied. Taking the time delay as the bifurcation parameter, critical values of the time delay for the emergence of Hopf bifurcation are determined. A novel periodic pulse delay feedback controller is introduced into the first equation of an uncontrolled system to successfully control the delay-deduced Hopf bifurcation of such a system. Since the stability theory is not well-developed for nonlinear fractional-order non-autonomous systems with delays, we investigate the periodic pulse control problem of the original system by a semi-analytical and semi-numerical method. Specifically, the stability of the linearized averaging system of the controlled system is first investigated, and then it is shown by numerical simulations that the controlled system has the same stability characteristics as its linearized averaging system. The proposed periodic pulse delay feedback controller has more flexibility than a classical linear delay feedback controller guaranteeing the control effect, due to the fact that the pulse width in each control period can be flexibly selected.

On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system

“…Since the stability change of an equilibrium involves the appearance of Hopf bifurcation, we need to verify the reasonability of the above linearized approximation by the equivalence of stability of equilibrium for systems (3.2) and (3.3). Following a similar idea as in [36], we prove the equivalence of stability of equilibrium for systems (3.2) and (3.3) in the sense that lim t→+∞ū (t) = 0, lim…”

Hopf Bifurcation and Control of a Fractional-Order Delay Stage Structure Prey-Predator Model with Two Fear Effects and Prey Refuge

“…Without loss of generality, we assume that all positive roots are ω k (k = 1, 2, ..., K). By substituting each ω k into Equation ( 25) and the corresponding critical value of τ k can be obtained (for the exact mathematical expressions, please refer to [40]). In relation to the actual meaning of delay, we only pay attention to the value of τ when Hopf bifurcation occurs firstly, so the bifurcation critical value of delay is…”

Stability and Bifurcation Control for a Generalized Delayed Fractional Food Chain Model

“…Biological system control plays an important role in maintaining ecosystem balance and species diversity [35][36][37][38][39]. In 1992, K. Pyragas [40] designed a linear feedback controller with delay, aiming at managing bifurcation of the system.…”