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

Qi, Hao; Zhao, Wencai

doi:https://doi.org/10.1155/2022/5313931

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5313931 | https://doi.org/10.1155/2022/5313931

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

Hao Qi¹and Wencai Zhao¹

Academic Editor: Chang Phang

Received13 Mar 2022

Accepted03 May 2022

Published15 Jun 2022

Abstract

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.

1. Introduction

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 [6]. Beddington et al. [7] proved that time delay could affect the stability of the dynamical model. Gazi et al. [8] 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. [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 diffusion and analyzed the global asymptotic stability of the positive equilibrium point of the model. Vinoth et al. [11] 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. [15] 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 [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 [21–27]. 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 [32–37]. 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 ; 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.

2. Preliminaries

For the theoretical derivation, we first give the relevant definitions and lemmas of Caputo calculus.

Definition 1 (see [21]). The fractional integral of order for a function is defined aswhere , , and is the Gamma function, .

Definition 2 (see [21]). The Caputo fractional derivative of order for a function is defined aswhere is a positive integer, , and . So, specifically, if ,

Lemma 1 (see [22]). Define . Suppose that there exist , such that and for ; then,

Lemma 2 (see [21]). Define , . Suppose is times continuous differentiable function and is piecewise continuous on ; we havewhere .

Lemma 3 (see [38]). 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 [24]). 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 [23]). 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 [26]). Assume is a continuous and differentiable function. For , , , .

Lemma 7 (see [23]). 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) thatwhereAs a result, suppose implies that the transversality criteria are true. That is the proof of Lemma 8.

With the support of Lemmas 7 and 8, the under theorem can be derived.

Theorem 7. In the case of , and , we have the following results:(1)The positive equilibrium of system (2) is asymptotically stable when .(2)System (2) exhibits a Hopf bifurcation at when , i.e., it has a branch of periodic solution bifurcating from near

4.2. The Bifurcation of System (2) Caused by Delay

In the following discussion, time delay is taken as the bifurcation parameter of system (2), and the Hopf bifurcation criterion of the system is obtained through theoretical analysis.

The characteristic equation about system (2) is available:where

The real and imaginary parts of are represented by and . Suppose is a purely imaginary root of (72), where , and we have

With the help of (74), we havewhere , , and . From (75), one has

In terms of , we obtain

Support the equation of (76) has a positive real root , and we makewhere is provided by (77).

Once eliminating from (72), thenwhere

Labels and represent the real and imaginary parts of . Suppose is a purely imaginary root of (79), where , and we can derive from (79) that

By means of (81), we havewhere , , and . It is obtained from (82) that

Because of , we obtain

Suppose (83) has a positive real root; we makewhere is provided by (84).

In order to better search for the criterion of the occurrence for bifurcation, the following hypotheses are available and essential: , where , , , and are described in the following.

Lemma 9. Let be the root of (29) near meeting and , so the following transversality criteria are true:where are the critical frequency and the bifurcation point individually.

Proof. Differentiating equation (72) with respect to , we haveSo, we can obtainwhereDefine and be the real and imaginary parts of individually. and be the real and imaginary parts of individually. After several algebraic calculations, we receive from (88) thatwhereAs a result, suppose implies the transversality criteria are true. That is the proof of Lemma 9.

With the support of Lemmas 7 and 8, the under theorem can be derived.

Theorem 8. In the case of , and , we have the following results:(1)The equilibrium of system (2) is asymptotically stable when (2)System (2) occurs a Hopf bifurcation at when , i.e., it has a branch of periodic solution bifurcating from near

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.

5. Numerical Results

Now, we verify our theoretical consequences by numerical simulations. For practical reasons, we only analyze the positive equilibrium , rather than , , or . We set , , , , , , , and ; then, system (2) can be transformed into

We can easily verify that the system only has a positive equilibrium, .

Case 1. We set , and then, we can verify that the system meets the condition of Theorem 6; Figure 1 indicates the positive equilibrium of system (92) is stable. Particularly, if , Figure 2 shows that of system (92) is globally asymptotically stable.
To discuss the bifurcation points about system (92), let us define .

Case 2. We fix and choose as the branch parameter to consider the bifurcation of system (92). If , we can figure out that . In Figure 3, it implies system (92) is asymptotically stable when and , and we can clearly find that the system is not stable and Hopf bifurcation appears when and . Then, we can calculate that when . Based on Theorem 7, system (92) is asymptotically stable if and . However, if we increase from 0.3 to , system (92) is unstable and Hopf bifurcation occurs, see Figure 4.

Case 3. We fix and choose as the branch parameter to consider the bifurcation of system (92). When , we can calculate that . In Figure 5, it implies system (92) is asymptotically stable when and , and we can clearly find that system (92) is not stable and Hopf bifurcation appears when and . Then, we can get that when . By Theorem 8, system (92) is asymptotically stable when and . However, if we increase from 1 to , system (92) is unstable and Hopf bifurcation occurs, see Figure 6.
In order to study the impact of fractional order on bifurcation points, we make the following simulation results.

Case 4. We choose . When , we can calculate . At this time, we take , and Hopf bifurcation appears in system (92). When , we can calculate ; then, we take ; system (92) is asymptotically stable. Look at Figure 7. Figure 8 shows the impact of fractional order on .

Case 5. Similarly, we take . When , we can obtain . At this time, we take , and Hopf bifurcation appears in system (92). When , we can get , and we take ; system (92) is asymptotically stable (see Figure 9). Figure 10 shows the impact of fractional order on .

6. 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by Shandong Provincial Natural Science Foundation of China (no. ZR2019MA003).

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Copyright © 2022 Hao Qi and Wencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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