Bifurcation for a fractional-order Lotka-Volterra predator–prey model with delay feedback control

Abstract: This paper addresses the bifurcation control of a fractional-order Lokta-Volterra predator– prey model by using delay feedback control. By employing time delay as a bifurcation parameter, the conditions of bifurcation are gained for controlled systems. Then, it indications that the onset of bifurcation can be postponed as feedback gain decreases. An example numerical results are ultimately exploited to validate the correctness of the the proposed scheme.


Introduction
Dynamical relationships between predator and prey exist widely in real world, which play a key role in linking complex food chains and food networks [1,2]. Previously, to unravel these dynamics and their biological functions, several predator-prey models have been proposed. Lotka-Volterra predatorprey system, one of the most celebrated predator-prey models, is being paid more and more attention in recent years [3][4][5][6][7][8]. Given the importance of the Lotka-Volterra model in the study of ecosystem, many efforts have been undertaken over the years to investigate its dynamical properties, including, dynamical behavior, stability, persistent property, anti-periodic solution, periodic solution and almost periodic solution [9][10][11][12][13][14][15]. In 2008, Yan and Zhang [16] considered the following form of predator-prey model to investigate the effects of time delay on stability and bifurcation: 11 x(t − τ) − a 12 y(t − τ)], y(t) = y(t)[−r 2 + a 21 x(t − τ) − a 22 y(t − τ)], (1.1) where x(t) and y(t) denote the population densities of prey and predator at time t, respectively; τ is the feedback time delay of the prey to the growth of the species itself; r 1 > 0 denotes intrinsic growth rate of the prey and r 2 > 0 denotes the death rate of the predator; a i j (i, j = 1, 2) are all positive constants. As a matter of fact, fractional calculus is merged into complicated, dynamical systems which extremely renovate the theory of the design and control performance for complex systems. The scholars discovered that some real world problems in nature can be depicted more accurately by fractional-order systems in comparison with classical integer-order ones [17,18]. Furthermore, the biological process is in relation to the entire time information of the model in the light of the traits of the fractional derivative, whereas the classic integer-order derivative places a high value on the information at a given time [19,20]. Recently years, many scholars have done a lot of research on the basic theory of fractional differential equations and the dynamics analysis of fractional order predator-prey or eco-epidemiological models (see [21][22][23][24][25][26][27][28][29]). As in [30][31][32], the authors considered the fractional-order delayed predator-prey systems.
Normally, quite a few bifurcation control schemes can be adopted to handle bifurcation dynamics, such as dislocated feedback control, speed feedback control and enhancing feedback control. Actually, it is challenging to exhaustively control the dynamical properties of an involute system relying on a unique feedback variable. In [33], Xiao et al. found that the onset of Hopf bifurcations can be lagged or advanced by the proposed fractional-order PD controller by selecting proper control parameters. Paper [34], an extended delayed feedback controller is subtly designed to control Hopf bifurcation for a delayed fractional predator-prey model, and it is detected that both extended feedback delay and fractional order can delay the onset of bifurcation for the proposed system. It is point out that the performance of nonlinear fractional dynamic systems can be elevated by utilising bifurcation control methods [35,36]. In addition, several control design analysis methods is a valid tool for the amelioration of the stabilization/synchronization of nonlinear systems [37][38][39][40][41][42][43]. However, to the best of our knowledge, there are few papers to investigate the existence of Hopf bifurcation to fractional-order delay Lotka-Volterra predator-prey with feedback control by using time delay as a bifurcation parameter.
Inspired by the above discussions, in this paper, we consider the following fractional-order delayed Lokta-Volterra predator-prey with feedback control: where q ∈ (0, 1] is fractional order, and k is negative feedback gains, x * is equilibrium point of system (1.1), other paraments are same as system (1.1). Obviously, system (1.2) degenerates into the model in [16] when k = 0 and q = 1.
The main contributions can be sum up in three key points: 1) One new fractional-order Lotka-Volterra predator-prey control model with feedback control and feedback gain is considered.
The joint effects of feedback gain and feedback delay on the controlled system are investigated. 2) Two primary dynamical properties-stability and oscillation-of the delayed fractional-order Lotka-Volterra predator-prey model with feedback control are investigated.
3) The influences of the order on the Hopf bifurcation are obtained. 4) One numerical simulation is given to illustrate the effectiveness of the proposed controllers. Throughout of this paper, we address the following assumption: (H1) r 1 a 21 − r 2 a 11 > 0. Suppose (H1) holds, the positive equilibrium point E * = (x * , y * ) of system (1.2) is unique, described by Our main purpose of this work is by applying time delay as a bifurcation parameter, some conditions of bifurcation are gained for controlled system (1.2).
The rest of this paper is structured as follows. In Section 2, we state some basic necessary definitions and lemmas. In Section 3, we study the existence of Hopf bifurcation of system (1.2). In Section 4, simulation is illustrated to verify the theoretical results. To the end, a brief conclusion is given.

Preliminaries
In this section, we introduce some definitions and lemmas of fractional derivatives, which serve as a basis for the proofs of main result of Section 3.
Generally speaking, there are three extensively used fractional operators, that is to say, the Riemann-Liouville definition, the Grünwald-Letnikov definition, and the Caputo definition. Since the Caputo derivative only requires the initial conditions which are based on integer-order derivative and represents well-understood features of physical state, it is more benefiting to real world questions. With this concept in mind, we shall apply the Caputo fractional-order derivative to model and analyze the stability of the proposed fractional-order algorithms in this paper.
Consider the n-dimensional linear fractional-order system with multiple time delays: det Next section, we will establish some sufficient conditions for the existence Hopf bifurcation of system (1.2).
Proof. By exploit implicit function theorem and differentiating Eq (3.3) with respect to k, we obtain It can be noted that C 3 (s) = 0, then we have Let C r l , C i l be the real and imaginary parts of C l (s), respectively. C r l , C i l are the real and imaginary parts of C l (s), respectively. Let χ 1 , χ 2 be the real and imaginary parts of χ(s), respectively. Let ϕ 1 , ϕ 2 be the real and imaginary parts of ϕ(s), respectively.
According to Lemma 3.1 it is not difficult to arrive at the following Theorem.

Conclusion
The bifurcation control of a fractional-order Lotka-Volterra predator-prey model has been carefully studied by delay feedback control. The criteria of bifurcation have been derived for controlled systems by choosing delay as a bifurcation parameter. It detects that the emergence of bifurcation can be delayed with the decrement of feedback gain. A simulation example is finally used to verify the efficiency of the devised strategy. It is worth noting that there will be several future directions to apply the methods from employing time delay as a bifurcation parameter to more complex ones like models with different delays, or to study the Hopf bifurcation of fractional-order systems with higher dimension.