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. The 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. The results show that Hopf bifurcation can advance with increasing fractional order and another delay. In conclusion, numerical simulation verifies and illustrates the theoretical results.
In the ecosystem, no species exists in isolation. The different 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 significance 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 [1–5].
In nature, the phenomenon of time delay is exhibited universally in biological population. The phenomenon of time delay is mainly caused by many factors such as gestation, maturation, and food digestion of population. The phenomenon of time delay signifies 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 . Beddington et al.  proved that time delay could affect the stability of the dynamical model. Gazi et al.  researched the influence 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.  analyzed the time-delay predator-prey system including prey shelter and demonstrated the global asymptotic stability of the system. Yan et al.  considered the predator-prey model with delayed reaction diffusion and analyzed the global asymptotic stability of the positive equilibrium point of the model. Vinoth et al.  put forward a delayed prey-predator system with additive Allee effect, 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. Therefore, time delay has a profound impact on the stability behavior of biological systems.
Differential 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 [12–14]. 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 [15–19]. Deka et al.  proposed and analyzed a one-predator and two-prey 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 , 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 , 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 [21–27]. So far, fractional calculus has been applied to many domains, such as neural network , medicine , finance system , and safety communication . 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 [32–37]. Rihan et al.  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.  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 ; the biological significance of each variable and parameter of model (2) is the same as that of model (1). The initial conditions are . The model is established on the sense of Caputo derivative.
The 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. The Hopf bifurcation of system (2) is studied in Section 4. Some numerical simulations are presented in Section 5. Conclusions are drawn in the end.
For the theoretical derivation, we first give the relevant definitions and lemmas of Caputo calculus.
Definition 1 (see ). The fractional integral of order for a function is defined aswhere , , and is the Gamma function, .
Definition 2 (see ). The Caputo fractional derivative of order for a function is defined aswhere is a positive integer, , and . So, specifically, if ,
Lemma 1 (see ). Define . Suppose that there exist , such that and for ; then,
Lemma 2 (see ). Define , . Suppose is times continuous differentiable function and is piecewise continuous on ; we havewhere .
Lemma 3 (see ). Assume represents the complex plane, for and , and ; then,for , signifies the real part of the complex number , and is the following Mittag–Leffler function described by
Lemma 4 (see ). Consider the system:with initial condition , where and , , if meets the local Lipschitz criteria with respect to :so it has a unique solution of (10) on , where
Lemma 5 (see ). Define , , and ; think about the following nonlinear fractional system of the same order:
If the eigenvalues of the Jacobian matrix corresponding to the equilibrium point of the system meet the following criterion , then system (13) is asymptotically stable.
Lemma 6 (see ). Assume is a continuous and differentiable function. For , , , .
Lemma 7 (see ). Think about the under n-dimensional linear fractional-order time-delay system:
Among them, , and the initial conditions are provided for , . It is defined as
If all roots of have negative real parts, so the zero solution of system (14) is Lyapunov globally asymptotically stable.
3. Analysis of the Nondelayed Model
First, we research the delay-free system of (2):
The nonnegativity and boundedness, existence, and uniqueness of solutions about systems (2) and (16) are discussed in Sections 3.1 and 3.2. The 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.
3.1. Nonnegativity and Boundedness of Solutions
Think about the biological implications of reality, it is significant to analyze the nonnegativity of the system. To prove the following theorem, let denote the collection of entire positive real numbers containing 0, .
Theorem 1. The solutions about system (16) from are nonnegative and uniformly bounded.
Proof. When , then ; we desire to obtain the solution from is nonnegative, i.e., , for . Suppose it exhibits a constant , ; according to which is a continuous function, there exists and . Define ; then, when , from system (16), one obtains . However, according to the definition of , and ; moreover, ; by Lemma 1, we have . Hence, we derive a contradiction; therefore, . Likewise, we can demonstrate .
For boundedness, we think about the following function:According to system (16), one haswhere . Therefore,By Lemma 2, making Laplace transform of both sides of (19), we obtainwhere . From this, we can obtainMaking inverse Laplace transform of (21), thenBy Lemma 3, one hasAccording toso we haveHence,where, if , we have .
Furthermore, the set attracts all the solutions of system (16), where
3.2. Existence and Uniqueness of Solutions
Theorem 2. System (16) only exhibits a solution for any given initial value .
Proof. According to Theorem 1, the solutions of system (16) from are nonnegative and uniformly bounded; then, there exists a constant , such that . Define a mapping , in whichLet be any two solutions to system (16); we can derivewhere and , , and . Hence, meets the Lipschitz criteria about . By Lemma 4, it has only a solution of system (16) with initial value .
3.3. Stability Analysis of Balance Point
For analyzing the possible equilibrium of system (16), we first present the following assumptions:(i): .(ii): and , in which is the positive root of the following equation:
We can find the following four biologically feasible equilibrium points:(1)(2)(3); it exhibits if the condition is true, where and (4); it exhibits if the condition is true, where and
The Jacobian matrix about system (16) at arbitrary point is as follows
Theorem 3. The trivial equilibrium of system (16) is unstable.
Proof. The Jacobian matrix at is as follows:Let ; the characteristic equation of (32) isSo, the roots of the characteristic equation are . Therefore,Consequently, by Lemma 5, the trivial equilibrium of system (16) is unstable.
Theorem 4. If is met, the boundary equilibrium of system (16) is locally asymptotically stable.
Proof. The corresponding Jacobian matrix at is shown below:At this point, the characteristic equation corresponding to (35) isSo, the roots of the characteristic equation are . Owing to the assumptions,Consequently, of system (16) is locally asymptotically stable by Lemma 5.
Theorem 5. In the case of , if one of the following conditions is met,(1)(2)(3) and , where and are given in the following proof; then, the boundary equilibrium of system (16) is locally asymptotically stable
Proof. In the case of , the boundary equilibrium of system exhibits. The Jacobian matrix at is as follows:The characteristic equation of (38) iswhere , , and .(1)If , we can find that and ; all characteristic roots of equation (39) are ; therefore, Hence, by Lemma 5, the equilibrium about system (16) is locally asymptotically stable.(2)If , we can find that and . Obviously, Accordingly, by Lemma 5, the equilibrium about system (16) is locally asymptotically stable.(3)Using the given conditions, we can obtain all characteristic roots of equation (39) are . Owing to , therefore,As a result, by Lemma 5, the equilibrium about system (16) is locally asymptotically stable.
Theorem 6. If is satisfied, then the positive equilibrium about system (16) is globally asymptotically stable.
Proof. The Jacobian matrix at is as follows:The characteristic equation of (43) iswhere , , and . Based on our assumptions, we haveBy the Routh–Hurwitz criterion, all roots of (44) are negative real parts; therefore,As a result, by Lemma 5, the equilibrium about system (16) is locally asymptotically stable.
Let us consider the Lyapunov function:Obviously, for any , except for the positive equilibrium .
By Lemma 6, we haveSince , then we have . Thus, is globally asymptotically stable.
4. Analysis of the Delayed Model
The 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 and , the stability of system (2) needs to be rediscussed. Next, the stability and branch of system (2) are studied by selecting and as key parameters, and the critical bifurcation value is discussed precisely.
4.1. The Bifurcation of System (2) Caused by Delay
In the following analysis, we focus on time delay as the bifurcation parameter of system (2) and obtain the critical value of Hopf bifurcation of the system.
Making transformation, , , and . In consequence, system (2) is able to be transformed into
The linearized scheme from system (49) results inwhere
The characteristic equation of system (50) is as shown below:where
The real and imaginary parts of are represented by and . Suppose is a purely imaginary root of (52), where ; it follows from (52) that
In view of (54), we derive thatwhere , , and . It is apparent from (55) that
In terms of , we obtain
Suppose the equation of (56) has a positive real root ; we makewhere is provided by (57).
If , then (52) becomeswhere
Suppose that and represent the real and imaginary parts of , is a purely imaginary root of (59), and ; we can get that
Based on (61), we havewhere , , and . It is apparent from (62) that
In the light of , we obtain
Suppose the equation of (63) has a positive real root, we makewhere is provided by (64).
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, . Similarly, 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 , , , and are described in the following.
Lemma 8. Let be the root of (17) near meeting and , so the following transversality criteria are true:where are the critical frequency and the bifurcation point individually.
Proof. After differentiating equation (52) about , we haveSo, we can obtainwhereLet and be the real and imaginary parts of individually. and be the real and imaginary parts of severally. After several algebraic calculation, we get from (68) thatwhere