Stability and Bifurcation Analysis of a Fractional-Order Food Chain Model with Two Time Delays

In this study, the stability and bifurcation problems of a fractional food chain system with two kinds of delays are studied. Firstly, the nonnegative, bounded, and unique properties of the solutions of the system are proved. e asymptotic stability of the equilibrium points of the system is discussed. Furthermore, the global asymptotic stability of the positive equilibrium point is deduced by using Lyapunov function method. Secondly, the system takes two kinds of time delays as bifurcation parameters and calculates the critical values of Hopf bifurcation accurately. e results show that Hopf bifurcation can advance with increasing fractional order and another delay. In conclusion, numerical simulation veries and illustrates the theoretical results.


Introduction
In the ecosystem, no species exists in isolation. e di erent populations are all related to each other. Predator relationship, competition relationship, reciprocity relationship, and parasitic relationship are the main population relationships. In these major relationships, predator-prey relationship is universal in nature and is of great signi cance to complex ecosystems. It is precisely because of the important application background and practical value of the predation system that the food chain system has been researched extensively by many scholars [1][2][3][4][5].
In nature, the phenomenon of time delay is exhibited universally in biological population. e phenomenon of time delay is mainly caused by many factors such as gestation, maturation, and food digestion of population. e phenomenon of time delay signi es that the related properties of the system are related to not only the present state but also the previous period. Aiello and Freedman studied a single population system with a time delay and stage structure [6]. Beddington et al. [7] proved that time delay could a ect the stability of the dynamical model. Gazi et al. [8] researched the in uence of harvest and discrete time delay on the prey-predator populations and obtained the discrete time-delay length required to remain the stability of the system. Jana et al. [9] analyzed the time-delay predator-prey system including prey shelter and demonstrated the global asymptotic stability of the system. Yan et al. [10] considered the predator-prey model with delayed reaction di usion and analyzed the global asymptotic stability of the positive equilibrium point of the model. Vinoth et al. [11] put forward a delayed preypredator system with additive Allee e ect, and the local asymptotic stability of the model at equilibrium point was studied. Numerous studies have shown that a population system with time delay could exhibit more complex nonlinear dynamic behaviors. erefore, time delay has a profound impact on the stability behavior of biological systems.
Di erential equation theory has been widely used in automation system, aerospace technology, information engineering, and so on. In these practical applications, the system usually has some parameters. If the parameters of the system change, the topological structure of the phase diagram in phase space also changes; then, the phenomenon is called bifurcation [12][13][14]. Hopf bifurcation theory has become a classical tool to research the generation and extinction periodic solutions of small amplitude differential equations. When a parameter passes a marginal value, the equilibrium point will lose stability and a periodic solution will appear [15][16][17][18][19]. Deka et al. [15] proposed and analyzed a one-predator and twoprey system with a general Gauss type, and the stability and direction of the Hopf bifurcation were proved by regarding the mortality of the predator as the bifurcation argument. In [16], a predator-prey model with discrete time delay of habitat complexity and sanctuary for prey was proposed and the occurrence criterion of Hopf bifurcation was obtained by taking the time lag as argument. In [20], Guo et al. established a food chain system with a couple of time lags and Holling II type functions: Among them, the biological significance of each parameter of system (1) is well illustrated in Table 1.
At the same time, the existence of the positive equilibrium point was proved, and the occurrence criterion of Hopf bifurcation was obtained by taking the time lag as the parameter.
Fractional-order calculus is a method that rises recently. It is a method that extends the ordinary integral calculus to nonintegral calculus [21][22][23][24][25][26][27]. So far, fractional calculus has been applied to many domains, such as neural network [28], medicine [29], finance system [30], and safety communication [31]. A great deal of studies have proved that the fractional dynamical system is to a higher degree suitable to biological systems because the fractional differential is connected with the entire time zone, while the integer differential is only related to a particular moment. Because biological systems generally have the characteristics of heredity and memory, so more and more scholars believe that the method of fractional calculus can better characterize the behavior of biological system. At present, some scholars have spread the classical integer-order differential systems to the fractional-order differential systems [32][33][34][35][36][37]. Rihan et al. [32] studied a fractional-order food chain model with time delay as well as infection in predators; sufficient criterion for asymptotic stability of the stable condition of the model was established. Huang et al. [36] discovered that the bifurcation dynamics of the model could be resultfully controlled as long as other parameters of the system are determined, and the extended feedback delay or fractional order is carefully adjusted.
Based on the above discussion, model (1) is extended in this study to obtain the following fractional-order food chain model: where 0 < θ ≤ 1; the biological significance of each variable and parameter of model (2) is the same as that of model (1). e initial conditions are e model is established on the sense of Caputo derivative. e rest of the study is organized as follows. Several definitions as well as lemmas are addressed in Section 2. In Section 3, the corresponding nondelay system of (2) is discussed. e Hopf bifurcation of system (2) is studied in Section 4. Some numerical simulations are presented in Section 5. Conclusions are drawn in the end.

Preliminaries
For the theoretical derivation, we first give the relevant definitions and lemmas of Caputo calculus.
ink about the under n-dimensional linear fractional-order time-delay system: Among them, ∀θ i ∈ (0, 1), and the initial conditions

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If all roots of det(Λ(s)) � 0 have negative real parts, so the zero solution of system (14) is Lyapunov globally asymptotically stable.

Analysis of the Nondelayed Model
First, we research the delay-free system of (2): e nonnegativity and boundedness, existence, and uniqueness of solutions about systems (2) and (16) are discussed in Sections 3.1 and 3.2. e local stability of the equilibrium points of system (2) is discussed, and the global asymptotic stability of the positive equilibrium point of system (2) is demonstrated in Section 3.3.

Nonnegativity and Boundedness of Solutions.
ink about the biological implications of reality, it is significant to analyze the nonnegativity of the system. To prove the following theorem, let R + denote the collection of entire positive real numbers containing 0, η + � (x, y, z) ∈ η: x, y, z ∈ R + .

Theorem 1.
e solutions about system (16) from η + are nonnegative and uniformly bounded.
However, according to the definition of t 1 , x(0) > 0 and For boundedness, we think about the following function: According to system (16), one has where k � min r 2 , r 3 > 0. erefore, By Lemma 2, making Laplace transform of both sides of (19), we obtain where Υ(s) � L W(t) { }. From this, we can obtain Making inverse Laplace transform of (21), then By Lemma 3, one has According to so we have 4 Journal of Mathematics Hence, where, if t ⟶ ∞, we have E θ,1 (− kt θ ) ⟶ 0. Furthermore, the set D attracts all the solutions of system (16), where □

Existence and Uniqueness of Solutions
Proof. According to eorem 1, the solutions of system (16) from η + are nonnegative and uniformly bounded; then, there exists a constant P, such that max , Let X and X be any two solutions to system (16); we can derive Journal of Mathematics where δ � max M 1 , M 2 , M 3 and M 1 � r 1 + 2a 1 P + P+ 2mP 2 + a 2 P + 2a 2 mP 2 , M 2 � P + mP 2 + r 2 + a 3 P + a 2 P + a 2 mP 2 + a 4 P, and M 3 � a 3 P + a 4 P + r 3 . Hence, Q(X) meets the Lipschitz criteria about X. By Lemma 4, it has only a solution X(t) of system (16) 3. Stability Analysis of Balance Point. For analyzing the possible equilibrium of system (16), we first present the following assumptions: in which x * is the positive root of the following equation: We can find the following four biologically feasible equilibrium points: e Jacobian matrix about system (16) at arbitrary point (x, y, z) is as follows Proof.
(3) Using the given conditions, we can obtain all characteristic roots of equation (39) are As a result, by Lemma 5, the equilibrium E 2 about system (16) is locally asymptotically stable. (16) is globally asymptotically stable. Proof.

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As a result, by Lemma 5, the equilibrium E 3 about system (16) is locally asymptotically stable.

Analysis of the Delayed Model
e conditions for nonnegativity boundedness, existence, and uniqueness derived for system (16) also apply to system (2). Systems (2) and (16) have identical equilibrium points. Due to the impact of time lags τ 1 and τ 2 , the stability of system (2) needs to be rediscussed. Next, the stability and branch of system (2) are studied by selecting τ 1 and τ 2 as key parameters, and the critical bifurcation value is discussed precisely. (2) Caused by Delay τ 1 . In the following analysis, we focus on time delay τ 1 as the bifurcation parameter of system (2) and obtain the critical value of Hopf bifurcation of the system.

e Bifurcation of System
Making transformation, P 1 (t) � x(t) − x * , P 2 (t) � y(t) − y * , and P 3 (t) � z(t) − z * . In consequence, system (2) is able to be transformed into 8 Journal of Mathematics e linearized scheme from system (49) results in (51) e characteristic equation of system (50) is as shown below: where (53) e real and imaginary parts of U k (s) (k � 1, 2) are represented by U r k and U i k . Suppose s is a purely imaginary root of (52), where s � ω 1 (cos(π/2) + i sin(π/2)) (ω 1 > 0); it follows from (52) that In view of (54), we derive that It is apparent from (55) that

Remark 1.
If equation (56) has no positive roots, then the system does not have bifurcation points. On the contrary, if equation (56) has more than one positive root, we take the minimum of all the roots. As mentioned above, τ 10 � min τ k 10 , k � 0, 1, 2 . . . . Similarly, τ k 20 is obtained this way.
In order to better search for the criterion of the occurrence for bifurcation, the following hypotheses are helpful and essential: where E 1 , E 2 , F 1 , and F 2 are described in the following.
Proof. After differentiating equation (52) about τ 1 , we have So, we can obtain where Let E 1 and E 2 be the real and imaginary parts of E(s) individually. F 1 and F 2 be the real and imaginary parts of F(s) severally. After several algebraic calculation, we get from (68) that where E 1 � ω 10 U r 2 sin ω 10 τ 10 − U i 2 cos ω 10 τ 10 , E 2 � ω 10 U r 2 cos ω 10 τ 10 + U i 2 sin ω 10 τ 10 , As a result, suppose [H 4 ] implies that the transversality criteria are true. at is the proof of Lemma 8.
(2) System (2) exhibits a Hopf bifurcation at E 3 when τ 1 � τ 10 , i.e., it has a branch of periodic solution bifurcating from E 3 near τ 1 � τ 10 (2) Caused by Delay τ 2 . In the following discussion, time delay τ 2 is taken as the bifurcation parameter of system (2), and the Hopf bifurcation criterion of the system is obtained through theoretical analysis. e characteristic equation about system (2) is available:
Proof. Differentiating equation (72) with respect to τ 2 , we have So, we can obtain where ℵ(s) � sV 2 (s)e − sτ 2 , Define ℵ 1 and ℵ 2 be the real and imaginary parts of ℵ(s) individually. I 1 and I 2 be the real and imaginary parts of I(s) individually. After several algebraic calculations, we receive from (88) that Re ds dτ 2 | ω 2 �ω 20 ,τ 2 �τ 20 ( ) � where As a result, suppose [H 5 ] implies the transversality criteria are true. at is the proof of Lemma 9.

□
With the support of Lemmas 7 and 8, the under theorem can be derived. (2) System (2) occurs a Hopf bifurcation at E 3 when τ 2 � τ 20 , i.e., it has a branch of periodic solution bifurcating from E 3 near τ 2 � τ 20 Remark 2. In the previous work, many authors discussed Hopf bifurcations for fractional-order systems with single delay [39,40], but in this study, we study Hopf bifurcations for fractional-order systems with two delays, which is of great significance for the discussion of Hopf bifurcations for systems with multiple delays.
Remark 3. In fact, the fractional-order system has a wider stability region than the integer order system. In other words, the fractional-order number will affect the stability of the system, taking the fractional-order number as the bifurcation parameter will also cause Hopf bifurcation.
To discuss the bifurcation points about system (92), let us define θ � 0.9.

8, system
In order to study the impact of fractional order on bifurcation points, we make the following simulation results.

Conclusion
In this study, a fractional-order food chain system involving two time delays has been presented. Nonnegative, bounded, existence, and uniqueness about the solution of the system have been proved. For nondelay system, we have discussed the local stability of the system equilibrium point and proved the globally asymptotically stability of the positive equilibrium point by constructing Lyapunov functions. By using time delays as parameters to discuss the Hopf bifurcation, which has showed that when the delay exceeds the critical value, the Hopf bifurcation will appear in the system, that is to say, the system will change from stable to unstable and a periodic solution will appear. In particular, the periodic oscillation behavior of the system could be suppressed by fractional order, which has indicated that the fractional-order system has a larger range of stability region than the integer-order system.

Data Availability
No data were used to support this study. Journal of Mathematics 17