Abstract
The issue of bifurcation control for a novel fractional-order two-prey and one-predator system with time delay is dealt with in this paper. Firstly, the characteristic equation is investigated by picking time delay as the bifurcation parameter, and some conditions for the appearance of Hopf bifurcation are obtained. It is shown that time delay can give rise to periodic oscillations and each order has an important impact on the occurrence of Hopf bifurcation for the controlled system. Then, it is illustrated that the control result is obviously influenced by the feedback gain. It is also noted that the inception of the bifurcation can be postponed if the feedback gain decreases. Finally, two simulation examples are carried out to verify the chief theoretical results.
1. Introduction
The dynamic relationship between prey and predator has currently aroused widespread concern from many researchers due to its ubiquitous existence and great importance [1, 2]. Parrish and Saila first constructed the mathematical model of two-prey and one-predator model [3]. Subsequently, numerous academicians researched the dynamical behaviors of the system and took distinct functional responses into consideration [4–7]. Unfortunately, the impact of time delay had not been considered. In fact, predators do not immediately increase their number after capturing the prey, but it takes time to digest the prey [8, 9]. So it is indispensable to take time delay into account in the prey-predator systems.
Constructing dynamical models with fractional-order differential equations has more merits in comparison with integer-order ones due to the memory and hereditary properties of fractional calculus [10–13]. Without doubt, the fractional-order differential equations are more in accordance with the real world than the integer-order ones since the fractional-order derivative is relevant to the entire time scale for a biological process; howbeit the integer-order ones only stress a certain factor or variation at specific time [14, 15]. It is well known that fractional-order differential equations relating to memory exist widely in biological systems [16, 17]. Since the ecosystem is a heterogeneous, interactive involute system and various species have otherness, the pattern creation of the integer differential equation can barely depict the ecosystem as a whole. Applying the fractional-order differential equations, we can analyze the nature of the ecosystem by understanding the individual state. Besides, the fractional-order ones can reduce errors caused by the parameters neglected in simulating the real life phenomenon [18, 19]. Therefore, it is indubitable that fractional-order differential equations can better precisely describe and model the relevant phenomena in the ecological environments.
Bifurcation theory has been greatly investigated in prey and predator systems [20, 21]. It is proverbial that Hopf bifurcations in inter-order ordinary differential systems have been profoundly studied [22, 23]. In the wake of development of the fractional-order calculus, Hopf bifurcations of delayed fractional-order models have recently aroused gigantic attention [24–27]. It is acknowledged that the occurrence of Hopf bifurcation may cause unforeseen harmful damage on biomedicine or biological systems [28, 29]. Luckily, bifurcation control is an efficient method to improve stability. By the aid of it, we can devise a controller to reduce some bad bifurcation dynamics and get some dynamical behaviors that we want. Bifurcation control of prey and predator model with time delay is mainly concentrated on state feedback control, which is also commonly used in biological control. Biological control method mainly by changing biome structure is a very efficacious technique in the practical ecosystem control [30, 31]. Nevertheless, the problem of bifurcation control for a delayed fractional-order two-prey and one-predator system has not been investigated before.
Inspired by the above discussion, we are focused on the topic of a delayed fractional two-prey and one-predator system with disparate orders via a linear state feedback control tool in this paper. The highlights of this paper are the following: (i) a linear time-delayed feedback controller is exploited to complete the bifurcation control in a delayed two-prey and one-predator model and give its biological means. (ii) The stability behaviors of uncontrolled model can be greatly exalted because of the designed controller. (iii) Comparative study is executed. It is displayed that gestation delay has a significant impact on the stability of the proposed model, the bifurcation value of the controlled system is negatively related to the order, and the effects of bifurcation control in the integer-order system are not as obvious as the corresponding fractional-order one when choosing the same feedback gain.
The framework of this paper reads as follows: Some basic tools are given in Section 2. In Section 3, the mathematical model is introduced. Main results are presented in Section 4. Some simulations are carried out in Section 5. Finally, a conclusion is given to summarize our work.
2. Basic Tools
In this section, we present the Caputo definition and the equilibrium of fractional-order system. Our works are based on the Caputo derivative.
Definition 1 (see [10]). The fractional-order integral of non-integer order for a function is defined aswhere , is the Gamma function,
Definition 2 (see [10]). The Caputo fractional-order derivative is defined bywhere
Particularly, when , .
For convenience, in what follows, we use the notion to denote the Caputo fractional-order derivative operator and assume . In virtue of the definition of the equilibrium point for two-dimensional fractional-order system in [14], the definition of equilibrium points for the n-dimension fractional-order system is given here.
Definition 3. Consider the following high-dimension fractional-order system:where , Therefore, the equilibrium solutions are defined by , and we can get the equilibrium points
3. The Mathematical Model
The dynamics of a two-prey and one-predator delayed fractional-order system with inconsistent orders by means of bifurcation control is discussed in this paper, and the mathematical system is as follows:Variables and parameters are demonstrated in Table 1.
Remark 4. In system (4), is a delay due to gestation of the predator. is the feedback control delay. The linear time-delayed feedback controller can alter the domain of stability and the inception of the bifurcation value [32]. In the field of biological control, the linear time-delayed feedback tactic indicates that we capture or release predators according to past data (the time scale is ). For example, in the field of fisheries management, when the density of predator in the past is higher than the present density, we may reduce the growth rate of predator; conversely, we increase the growth rate of predator.
For the sake of handling the problem easily, we assume that ; then system (4) is converted into
4. Dynamical Behaviors of Controlled System
On account of the biological significance, we concentrate on solutions that are nonnegative and bounded. The following result can be obtained to ensure the non-negativity and boundedness of the solutions of system (5). Let
Theorem 5. All the solutions of system (5) which start in are non-negative and uniformly bounded.
Proof. In order to prove the solutions of system (5) are non-negative, we first prove for all . Suppose that it is not true; then there exists a constant such that . There must be a constant such that . DenoteIt is obvious that . On the other hand, it is easy to get for Based on the results in [33], one can deduce , which leads to a contradiction. In the same manner, we have that for all . It is apparent that for all in the fact that
Next we explore that the solution is uniformly bounded. It follows from the first equation of system (5) that According to the results in [34] and the fractional comparison theorem, we can get . Similarly, we have . If is not uniformly bounded, it will result in for , which contradicts the fact of the non-negativity of
It is easy to obtain the three species equilibriums , whereand .
To get our chief results, the following hypothesis is essential: (H1) , , and
To acquire the main outcome, the linear transformation of system (5) is performed. Using the transformation , , , then the system can be obtained asThe linearization system of the system (8) can be shown aswhereThen the characteristic equation for system (9) can be expressed asObviously, (11) can be rewritten equivalently aswhere
Suppose that is a root of (12), . By bringing it into (12) and separating the imaginary parts and the real parts, it leads towhere , are the real and imaginary part of , which are displayed in Appendix A.
From (14), an easy calculation getsUsing the fact , it is obvious thatHence, it deduces from thatWe assume that (16) has at least one positive real root. Define the bifurcation valuewhere is defined by (17).
To find the condition of the appearance for Hopf bifurcation, we give the following hypothesis: (H2) ,
where , , , are defined by Appendix B.
Lemma 6. Let be the root of (12) near meeting , ; then the following transversality condition meets
Proof. Differentiating both sides of (12) with respect to , it can be deduced thatwhere are the derivatives of HencewhereBy straightforward computation, it can be derived from (21) thatwhere , are the real and imaginary parts of and , are the real and imaginary parts of .
Apparently, the hypothesis hints that transversality condition is satisfied.
To obtain the stability of system (5) when , we give the following assumptions:
(H3) ,
(H4) ,
(H5) and
Lemma 7. If (H3)–(H5) hold, then the positive equilibrium point of the fractional-order system (5) is asymptotically stable when
Proof. If there exists no delay, (12) turns intoWhen the assumptions (H3)–(H5) are met, it is not difficult to check from Routh-Hurwitz criterion that the three eigenvalues of characteristic equation (24) all are negative real parts. Therefore, the positive equilibrium point of the fractional-order system (5) is asymptotically stable when .
Remark 8. Based on the theory of stability for fractional-order system, it is obvious that the conditions in Lemma 7 are only sufficient conditions. If the conditions (H3)–(H5) take place by other conditions that can make sure that all the roots of (24) satisfy , , then Lemma 7 may be still true which indicates that the fraction order can improve the stability compared with inter-order ones.
On account of Lemmas 6–7 we can achieve the following theorem.
Theorem 9. Assume that (H1)–(H5) hold; then
(i) The internal equilibrium point of system (5) is asymptotically stable for ;
(ii) System (5) undergoes a Hopf bifurcation at internal equilibrium point when ; i.e., it has a branch of periodic bifurcating from the positive equilibrium near .
5. Simulation
In this section, two numerical cases are presented to verify the main results of our work.
5.1. Example 1
Time delay is picked as the bifurcation parameter to investigate the stability and bifurcation of (5) when . For the simplicity of comparison, all the parameters are chosen from the article [9]; then the system (5) can be rewritten as
When the orders are picked as , , , it is easy to get the positive equilibrium point We can calculate the critical and the bifurcation value . From Theorem 9, we can obtain that the positive point is asymptotically stable when , which is shown in Figure 1, while the positive equilibrium is unstable; Hopf bifurcation occurs when , as is displayed in Figure 2.
(a) Time series of
(b) Time series of
(c) Time series of
(d) Phase diagrams
(a) Time series of
(b) Time series of
(c) Time series of
(d) Phase diagram
5.2. Example 2
To postpone the onset of Hopf bifurcation of uncontrolled system (25), we introduce a linear delayed feedback controller here; then the system is given by
When the orders are chosen as , , , we can get that Hopf bifurcation occurs when . To control bifurcation of system (26) and achieve dynamical behaviors that we want, we choose the feedback gain . The proposed system is controlled, but the corresponding integer-order one’s positive equilibrium is unstable, which are depicted in Figure 3. Moreover, it can obviously be seen from Figure 4 that the effects of bifurcation control are negatively correlated with feedback gain .
(a) Time series of fractional-order system
(b) Time series of integer-order system
(a) versus
(b) Time series of
(c) Time series of
(d) Time series of
Next, we will explore the impact of each order on the bifurcation value for the controlled system (26). For the reason of simplicity, is chosen as and is picked as ; meanwhile the two orders are firstly fixed, and the impact of the third one is changed, which is shown in Figures 5–7. To sum up, when the order decreases, the stability of the controlled system (26) is exalted, and has much more influence on the dynamical behaviors.
(a) versus
(b) Time series of
(c) Time series of
(d) Time series of
(a) versus
(b) Time series of
(c) Time series of
(d) Time series of
(a) versus
(b) Time series of
(c) Time series of
(d) Time series of
Remark 10. From the article [4], the positive equilibrium point is globally stable when , , , , , and . However, when we guarantee , we discover that if is big enough, Hopf bifurcation occurs. Hence gestation delay can usually lead to instability and oscillation dynamical behaviors.
Remark 11. When , and , the author found that and in [9]. Our results indicate that the fractional orders can expand the stability zone.
Remark 12. The effects of fractional order on the controlled system dynamical behaviors are considered in this paper rather than the uncontrolled one’s in [31]. It is evident that there is a negative relation between fractional order and bifurcation value. Compared with state feedback control without time delay in [30], it is highlighted that time-delayed feedback controller is more adaptable in bifurcation control since the positive equilibrium cannot be obtained by analytic expression for some complex systems [32].
Remark 13. Many scholars have researched prey-predator systems with different harvest strategies [19, 20, 35]. However, when the density of predators is low, we may release predators to keep the balance of nature [36]. In the real world, the linear delayed feedback control tactic may have more practical means since we usually capture or release predators according to past date (the time scale is ). And, by introducing such time-delayed force, we can get the desirable dynamics.
6. Conclusion
The problem of the bifurcation control for an incommensurate fractional-order two-prey and one-predator system with time delay has been investigated at length. The results state clearly that the delay of gestation can give rise to periodic oscillations and the occurrence of Hopf bifurcation can be affected by fractional orders for the controlled system. By introducing a delayed feedback controller and picking appropriate feedback gain, we can get the desired dynamical behaviors. It purports that the beginning of the bifurcation can be postponed as feedback gain decreases. Moreover, it is interesting that the proposed system is stable, but the corresponding integer-order ordinary differential one periodically fluctuates around the positive equilibrium point with the same system parameters and feedback gain. Finally, two simulation examples have been carried out to explain practicability and validity of our principal results compared with the previous ones.
Appendix
A.
The expressions of , , , and in (12)
B.
The expressions of , , , and of (21)
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
No potential conflicts of interest were reported by the authors.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No.11671346) and Nanhu Scholars Program of XYNU .