Periodic pulse control of Hopf bifurcation in a fractional-order delay predator\u2013prey model incorporating a prey refuge

Xiuduo Liu,Hui Fang

doi:10.1186/s13662-019-2413-9

Xiuduo Liu, Hui Fang

Open Access

https://doi.org/10.1186/s13662-019-2413-9

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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.

Highlights

Long-range temporal memory exists in many population systems [1,2,3,4]
Since the Caputo fractional derivative of q-order for the function f ∈ Cn([t0, ∞), R) (n – 1 < q ≤ n ∈ Z+) at time t depends on the total effect of the usual nth-order integer derivative on the interval [t0, t] [7, 8], a fractional-order derivative is associated with whole domain for a biological process [9]
Yang et al proposed new and interesting fractional derivatives without singular kernel [11, 12], and analytic and computational methods for solving nonlinear fractional-order partial differential equations [13, 14], which can be effectively used in the modeling of the fractional-order heat flow [15, 16]

Summary

Introduction

Long-range temporal memory exists in many population systems [1,2,3,4]. Ecological memory was originally defined as “the capacity of past states or experiences to influence present or future responses of the community” [5], and as “the degree to which an ecological process is shaped by its past modifications of a landscape” [6]. Based on the above analysis, we will add a time delay into the model proposed by Li et al [19], and consider a fractional-order delay predator–prey model incorporating a prey refuge in this paper. In order to reduce the control time costs and to facilitate implementation, we introduce a periodic pulse controller in our proposed fractional-order delay predator–prey model incorporating a prey refuge. A periodic pulse delay feedback controller is introduced into the first equation of the original system to control the Hopf bifurcation successfully. 4, the existence and uniqueness of the solution for our proposed system was studied, Hopf bifurcation of uncontrolled delayed fractional-order predator–prey model incorporating a prey refuge is analyzed, and a periodic pulse delay feedback controller is designed to control the Hopf bifurcation for the proposed model. We consider the following fractional-order predator–prey model with time delay: m)x(t)y(t τ.

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Conclusions and discussions