Mathematical Analysis of a Fractional-Order Predator-Prey Network with Feedback Control Strategy

This paper examines the bifurcation control problem of a class of delayed fractional-order predator-prey models in accordance with an enhancing feedback controller. Firstly, the bifurcation points of the devised model are precisely figured out via theoretical derivation taking time delay as a bifurcation parameter. Secondly, a set comparative analysis on the influence of bifurcation control is numerically studied containing enhancing feedback, dislocated feedback, and eliminating feedback approaches. It can be seen that the stability performance of the proposed model can be immensely heightened by the enhancing feedback approach. At the end, a numerical example is given to illustrate the feasibility of the theoretical results.


Introduction
e predator-prey model is a class of considerable models in the sphere of ecological models. e prey-predator model is one of the basic topics in ecology as a result of the pervasive importance and existence, which constitutes the convoluted food chains and food networks. e original predator-prey system was formulated in [1,2]. It is generally accepted that time delays have been merged into biological models to elaborately delineate the authentic dynamical predator-prey use taking into account the time required for resource regeneration time, maturation period, reaction time, feeding time, and gestation period [3,4]. Many excellent results have been attained on the study of predator-prey models [5][6][7].
Fractional calculus is merged into complicated, dynamical systems which extremely renovate the theory of the design and control performance for complex systems. Scholars discovered that physical phenomena in nature can be depicted more accurately by fractional-order models in comparison with classical integer-order ones [8]. Recently, quite a few researchers introduced fractional calculus into the predator-prey model and constructed fractional predator-prey models, for example, design and control of various ecological models [9][10][11], secure communication [12,13], system control [14,15], and so on. Furthermore, modelling and control based on the theory of the fractional calculus of complex systems can greatly enhance the capability of discrimination, design, and control for dynamic models since fractional calculus possesses infinite memory and more degrees of freedom [16]. Consequently, a number of available theories and principles have been further renovated on the basis of fractional calculus. Modelling and control for fractional order dynamical models have recently grown a hot research issue [17][18][19][20][21][22]. In fact, lots of biological models exhibit fractional dynamics thanks to possessing memory effects. Lately, fractional calculus has been successfully introduced into predator-prey models, and some interesting phenomena have been studied. In [23], the authors consider a fractional order delayed predator-prey system with harvesting terms. In [24], Mondal et al. found that the solutions of the fractional-order predator-prey system converge to the respective equilibrium more slowly as fractional order decreases. In [25], Chinnathambi and Rihan proposed that fractional order can strengthen the stability of the prey-predator system and hamper the occurrence of oscillation behaviors. Fractional dynamics of the predator-prey model without delays have been captured [26,27].
e Hopf bifurcation has been widely researched for nonlinear systems, and a number of significant results have been obtained [28][29][30]. It should be pointed out that bifurcation problems of conventional delayed integer-order models have been fully studied, due to the maturity of theoretical methods. Nevertheless, the bifurcations of fractional-order dynamical systems have been analyzed in many papers [31][32][33][34]. In 2018, the authors gave attention to the bifurcation problems of the delayed generalized fractionalorder prey-predator model with interspecific competition, and exact bifurcation results were obtained [32]. In [34], the bifurcation of a class of quaternion-valued fractional-order neural networks with time-varying delay was given, and it indicated that an increase in orders of fractional-order models will result in advancing the onset of bifurcation. In recent years, the problem of bifurcation control of fractional models with time-varying delay has attracted people's attention [35][36][37][38]. In [37], research has found that the bifurcation phenomena can be controlled by adjusting extended feedback delay or fractional order. In [38], a newly invented fractional-order PD scheme involving variable order was proposed to control the generation of bifurcation for integer-order small-world network, and innate bifurcation can be efficiently unperturbed if dedicated control gains are set. e performance of nonlinear systems can be elevated utilizing bifurcation control methods. Recently, bifurcation control strategies have been used to delay the onset of bifurcation in fractional-order systems with time delay [39][40][41][42], such as dislocated feedback control, speed feedback control, and enhancing feedback control [43,44]. In [43], the author considered that the feedback coefficients were smaller than the ones of ordinary feedback control in controlling the hyperchaotic Lorenz system. In [44], it was revealed that the hyperchaotic Lorenz system can be efficiently controlled based on the enhancing feedback control approach in comparison with the addressed feedback ones by selecting relatively simple external inputs and small necessary feedback coefficient. Actually, it is difficult to completely control the dynamical properties of a complex system relying on a unique feedback variable. In this instance, a larger feedback gain needs to be selected for procuring the anticipated dynamical behaviors of a nonlinear system. Hence, to acquire high-quality performance of the devised fractionalorder dynamical systems, it is essential to adopt the enhancing feedback approach to control the onset of bifurcation. Up to now, few results with respect to the bifurcation control of fractional order predator-prey systems with delays by adopting the enhancing feedback control method have been obtained.
Based on the above motivations, so far, to the best of the authors' knowledge, no work has concerned the Hopf bifurcation control for a delayed fractional-order predatorprey model with feedback control. e main contributions can be summarized in three aspects: (1) An improved fractional-order predator-prey model is constructed by introducing an enhancing feedback control strategy. (2) A larger feedback gain is selected for controlling the onset of bifurcation of the created system through dislocated feedback scheme. (3) It detects that the control effects of the proposed system can be largely hoisted by using enhancing feedback approach than dislocated feedback one. is implies that the devised enhancing feedback method can attenuate the control amount compared with dislocated feedback ones. It is pointed out that the devised enhancing feedback method can attenuate the control amount compared with dislocated feedback ones. e organization of this paper is as follows. In Section 2, several useful definitions and lemmas of fractional-order calculus are recalled. In Section 3, the discussed system is proposed. In Section 4, by analyzing the associated characteristic equation, the existence of the Hopf bifurcation of the delayed fractional-order predator-prey model with feedback control is established. In Section 5, one illustrative example is provided to demonstrate the theoretical results. Section 6 is the conclusion of the paper.

Preliminaries
In this section, let us recall some definitions and lemmas of fractional derivatives, which can be used in the proofs of main result of Section 4.
Lemma 2 (see [45]). e following n-dimensional linear fractional-order model with delays is examined: Its fractional order is defined as q � (l 1 , l 2 , . . . , l n ). It is defined as

Mathematical Model Description
A delayed ratio-dependent fractional-order predator-prey system is proposed in this paper. e mathematical model is depicted by where the relative variables and parameters of systems (7) are interpreted in Table 1.
In this paper, we consider the delayed fractional-order version of (7) which is given as follows: where ϕ 1 , ϕ 2 represent the feedback control gains.
Remark 1. Hence, we have the observation that model (8) degenerates into the integer-order model in [46] when (8) develops into the enhancing time-delayed feedback control model.

Symbols
Interpretation Denotes the population densities of prey at time t y 2 (t) Denotes the population densities of predator at time t u u ∈ (0, 1] is fractional order Time delay for both the densities of the predator and the prey Computational Intelligence and Neuroscience Remark 2. E * of system (7) is consistent with system (8), which does not rely on the values of control parameters ϕ 1 and ϕ 2 � 0. is implies that E * is immutable in the devised controllers.
To capture the brilliant control effects, the following essential assumption is presented in this paper: According to [45], this paper is devoted to finding out the conditions of bifurcation for model (8) using time delay as a bifurcation parameter. en, quite a few comparative studies on bifurcation control are carried out. It can be seen that the stability performance of the controlled model can be excessively ameliorated on the basis of enhancing feedback control in comparison with the dislocated feedback control.

Main Results
In this section, time delay shall be selected as a bifurcation parameter to investigate the problem of bifurcation control for the predator-prey model (8) by utilizing enhancing feedback approach. e existence bifurcation and bifurcation point for the proposed model shall be determined.
It follows from the first equation of equation (19) that Define the bifurcation point where τ (k) is defined by equation (23). In the following, we will consider the stability of model (10) when τ � 0. If τ is removed, the characteristic equation (16) develops into where It is obvious from φ 1 > 0, φ 2 > 0 that the two roots of equation (25) have negative parts satisfying Lemma 1. us, the positive equilibrium of fractional-order model (8) is asymptotically stable.
Remark 3. Due to the higher order of equation (22), it is not always easy to deal with all the positive real roots of it theoretically. However, it is simple to procure the concrete of these positive real roots of equation (22) with the help of Maple numerical software. Hence, the values of τ 0 can be accurately concluded. predator-prey model based on dislocated feedback strategy in [47]. Based on the dislocation feedback method, an extended delayed feedback method was designed to control bifurcation of a delayed fractional predator-prey model despite selecting a small feedback gain in [37]. It should be pointed out that the extended feedback delay plays an essential role in postponing Hopf bifurcation for such model. In this paper, by choosing enhancing feedback method, the bifurcation of the proposed model can be easily controlled provided that a set of smaller feedback gains is selected. Contrarily, the bifurcation of the devised model can be controlled by using the dislocated feedback approach only if a larger feedback gain is requested. It exhibits that the onset of the bifurcation for the fractional delayed predator-prey model can be delayed and satisfactory bifurcation control effects are realized compared with the dislocated feedback approaches in this paper. is shows that the devised enhancing feedback method can reduce the control cost compared with dislocated feedback ones.

Remark 5.
e influence of fractional order on the bifurcation point is adequately discussed. is means that better effects in delaying the onset of bifurcation can be achieved as fractional order decreases if feedback gain is established. At the same time, it can be seen that better control effects can be gained in delaying bifurcation of the proposed model by  Computational Intelligence and Neuroscience applying the enhancing feedback method instead of dislocated feedback one.

Numerical Example
In this section, we give one example to show the feasibility and effectiveness of the results obtained in this paper. All of the simulation results are based on Adama-Bashforth-Moulton predictor-corrector scheme. In this section, numerical simulations are presented. For the purpose of comparison, the parameters were identically derived from [46]. E * can be obtained as (y * 1 , y * 2 ) � (0.5775, 0.3465). e initial values are all designated as (y 1 (0), y 2 (0)) � (0.5, 0.3). Investigate the controlled model where α � 0.7, β � 0.9, δ � 0.6, y * 1 � 0.5775, and y * 2 � 0.3465.
Some results are available in Table 2. Figure 1 describes the asymptotic stability of system (34) when τ � 2.1 < τ 0 � 2.2072. Hopf bifurcation occurs, and the instability of system (34) with τ � 2.3 > τ 0 � 2.2072 is depicted in Figure 2. It can be observed that system (34) turns unstable when u � 1 is selected and other parameters are established (see Figure 3). Moreover, it can be seen from Computational Intelligence and Neuroscience when choosing larger ϕ 2 � −0.4. is means that the onset of bifurcation of system (34) can be controlled by taking larger feedback gain in terms of dislocated control. Similar phenomena can be observed in Figures 6 and 7. When abolishing the controllers, that is, ϕ 1 � 0, ϕ 2 � 0, it is clear that system (34) becomes unstable, which is illustrated in Figure 8. In fact, by changing the values of u, the corresponding τ 0 can be obtained by taking the enhancing feedback control and dislocated feedback control and without control approaches, respectively, which is illustrated in Figure 9. is implies that enhancing feedback control transcends others, which is verified in simulation results in Figures 10-12. When establishing u, ϕ 2 � −0.18, or ϕ 2 � 0, the values of τ 0 can be determined as ϕ 1 varies, which is depicted in Figure 13. Figure 13 also means that enhancing feedback control exceeds dislocated feedback control, as shown in Figures 14-16. e order is selected as u � 0.95 and the parameter is designated as ϕ 1 � −0.25, taking ϕ 1 � 0 and varying ϕ 2 in model (34), and the values of τ 0 can be computed, which is described in Figure 17. It can also be seen from Figure 17 that enhancing feedback control overmatches dislocated feedback control, which is very consistent with numerical simulations in

Conclusion
In this paper, the issue of bifurcation for a delayed of fractional predator-prey model with feedback control has been given. By utilizing time delay as the bifurcation parameter, a number of criteria to ensure the existence of the Hopf bifurcation for the fractional-order predatorprey model with feedback control were studied. Mathematical analysis and simulation results further reveal that better efficiency of bifurcation control has been obtained in terms of enhancing feedback approach compared to dislocated and uncontrolled ones with partially or completely removing the branch for feedback gains. It can be seen that the onset of bifurcation can be controlled for the dislocated feedback schemes, yet greater feedback gains must be taken, which increases the cost of control model. Contrarily, the stability performance of the controlled model can be extremely ameliorated on account of the designed enhancing feedback methodology by choosing smaller feedback gains. In addition, numerical results have been provided to confirm the efficiency of the derived theoretical results.