Abstract
On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.
1. Introduction
For a long time, the dynamic characteristics of interaction between predator population and prey population has been a central issue in ecology and biomathematics due to its general appearance and potential importance [1]. General speaking, the interaction between predator population and prey population includes four cases: competition, predation, mutualism, and parasitism [2]. During the past few decades, a great deal of valuable research fruits on dynamical behavior of the above four-type predator-prey models has been covered. For example, Alidousti and Ghafari [3] investigated the Hopf bifurcation and limit cycle of the fractional-order predator-prey model; Sasmal and Takeuchi [4] studied the stability behavior of all equilibria, bifurcation nature, global features, and multistability of a predator-prey model; Ryu and Ko [5] discussed the asymptotic peculiarity for positive solutions for a prey-predator model; Guo et al. [6] proved the appearance of traveling waves in a prey-predator system. Zhang et al. [7] revealed the effect of the fear factor on the periodic solution of a prey-predator model. As for concrete works, we refer the readers to [8–25].
In real life, any biological or environmental coefficients will change with time. So, the parameters in predator-prey models are usually not fixed constants. They are often functions with respect to time. In particular, the influence of a periodically varying environment on the dynamics of predator-prey models plays a vital role in maintaining population balance. Furthermore, the capture of the prey from the predator throughout its past time has an important effect on the present birth rate of the predator [26, 27]. Thus, it is of importance to establish the various type predator-prey models with periodic coefficients. Based on this idea, in 2010, Lv et al. [27] established the predator-prey model involving periodic coefficients as follows:where and stand for the densities of competing species at time , stands for the density of cooperating species at time , And and denote -periodic function . The parameter is the delay. Applying the fixed point theory and constructing Lyapunov functions, Lv et al. [27] set up the condition to ensure the global stability of periodic solutions for model (1). For details, one can see [27].
To reveal the Hopf bifurcation nature of model (1), Xu [26] assumes that all biological and environmental coefficients remain constants and only the feedback time delays of all species to the growth of the species themselves exist and are same [26]. Then, model (1) is rewritten as the following form:
With the Hopf bifurcation theory, normal form theory, and center manifold principle, Xu [26] obtained a sufficient condition to guarantee the stability behavior and the appearance for bifurcation phenomenon of model (2). Also, they have derived the concrete expression to find the nature of bifurcation periodic solution.
However, the mentioned works above are only restricted to the integer-order differential systems. Recently, the dynamics of fractional-order dynamical models has attracted great attention of many authors due to their extensive application in numerous areas, such as electromagnetic waves, medicine, mechanics, network science, biology, and finance [28–30]. Many scholars argue that fractional calculus is a powerful tool to depict real phenomena of the object world due to its owned hereditary and memory properties of various practical dynamical models [31]. In recent several decades, fractional calculus has attracted more and more attention from a large number of researchers in various fields. In particular, a great deal of interesting fruits on various dynamical natures of prey-predator systems has sprung up. For instance, Mondal et al. [32] derived the condition to ensure the stability behavior for a class of fractional-order prey-predator system; El-Saka et al. [33] analyzed the local stability and bifurcation of fractional-order predator-prey models; Li et al. [34] dealt with the dynamical property of the solutions and global asymptotic stability for a class of fractional-order prey-predator system. For more relational publications, one can see [35–37]. Here, we particularly emphasize that the study on Hopf bifurcation for fractional-order dynamical models starts relatively late. Up to now, only a few literatures have been published. For example, Alidousti [38] investigated the bifurcation phenomenon of a fractional-order predator-prey model; Huang et al. [30] proposed a novel control technique of bifurcation for a fractional-order prey-predator model with delays; Xu et al. [39] revealed the effect of two delays on bifurcation for fractional-order neural networks. Xiao et al. [40] put up PD control way for Hopf bifurcations of fractional-order networks. As for more related literatures, one can see [41–44].
In terms of the above analysis, we think that it is meaningful for us to study the dynamics (especially Hopf bifurcation) of fractional-order prey-predator models. Based on the previous predator-prey model (2) and assuming that the feedback time delays of all species to the growth of the species themselves and other species exists are same, then we propose the following fractional-order competitor-competitor-mutualist Lotka–Volterra system:where and stand for the densities of competing population and stands for the density of cooperating population, , the parameter is the feedback time delay of different species, and is a constant. For more concrete meaning of coefficients for system (3), see [26, 27].
The initial condition of system (3) takes the formwhere is a constant. The key object of this work focuses on existence, uniqueness, nonnegativity, stability, and bifurcation phenomenon of model (3). Different from the methodology in [26, 27], in this paper, we will mainly discuss the various dynamics by applying fractional-order differential equation theory. Due to the introduction of fractional order, various dynamical behaviors of the predator-prey model (3) are different from the integer-order case. We think that it is necessary to reveal the effect of time delay, parameters of system, and fractional order on dynamics such as the stability and Hopt bifurcation. Thus, this study has significance in theory and practice.
We plan the structure of this manuscript as follows. Section 2 gives some related knowledge about fractional-order dynamical systems. Section 3 discusses the existence, uniqueness, nonnegativity, local stability, and Hopf bifurcation of model (3). Section 4 gives an example to support the effectiveness of the obtained key conclusions. Section 6 ends our work.
2. Basic Knowledge
In this part, we present several related definitions and lemmas on fractional-order dynamical systems that will be applied in the later proof. Let represent the set of all nonnegative real numbers.
Definition 1. (see [45]). Define Caputo fractional-order derivative as follows:where , , and .
The Laplace transform of Caputo fractional-order derivative is defined as follows:where . Especially, if , then .
Definition 2. (see [46]). is called an equilibrium point of model (3) provided that the following systemholds.
Lemma 1. (see [47]). Given the following systemwhere . If satisfies the local Lipschitz condition with respect to , then model (3) possesses a unique solution defined on .
Lemma 2. (see [48]). Suppose that and , where . If , then is a nondecreasing function . If , then is a nonincreasing function .
Lemma 3. (see [49, 50]). For a given fractional-order model,where and , the equilibrium point of model (9) is locally asymptotically stable if all eigenvalues of evaluated near the equilibrium point satisfy .
Lemma 4. (see [51]). For given $m$-dimensional fractional-order system,where , the initial values , and , . Denoteand then, the zero solution of equation (10) is Lyapunov asymptotically stable if all roots of possess negative real parts.
3. Main Results
3.1. Existence and Uniqueness
Theorem 1. Let , where is a positive constant. Then, for every and for every , there exists a unique solution of system (3) with initial value .
Proof. We discuss this issue for model (3) in , where . Let and , and define the following mapping: , whereFor arbitrary , one haswhereIt follows from (13) thatwhereIn view of Lemma 1, one can conclude that Theorem 1 is true. We finish the proof.
3.2. Nonnegativity
Denote .
Theorem 2. Each solution of model (3) that begins with remains nonnegative.
Proof. Our object is to prove that . Let be the initial value of system (3). Assume that is a constant and such thatwhere . In view of system (3), one haswhere . By Lemma 2, one knows that , which contradicts in (17). Thus, one can conclude that . The proof is complete.
3.3. Stability and Hopf Bifurcation
In the section, we shall focus on the local stability and the appearance of Hopf bifurcation of system (3). Consider the biological implication of system (3); we only seek the sufficient condition to guarantee the local stability of the positive equilibrium and the emergence of Hopf bifurcations of system (3).
Clearly, system (3) has a unique positive equilibrium provided that the following conditionis satisfied, where
The linear system of equation (3) near takeswhere
The characteristic equation of equation (21) is computed as follows:which leads towhere
In terms of (24), we have
Let be the root of (26); then, one haswhere
In (28), letand then, (28) takes the form:
For (27), we consider two cases.
Case 1. If , then the first equation of (27) becomesHence,which leads towhereSet , and letHence,DenoteSet and then, (37) takes the formwhereLetIt follows from (38) thatAccording to the discussion above, one easily obtains the expression of . Then, one easily gets the expression of . Here, we suppose thatThen,With the aid of Matlab 7.0, one can get the root (say ) of equation (43). Then, one obtains
Case 2. If , applying the same way to this case, we can obtainThen,By Matlab 7.0, one can obtain the root (say ) of equation (46). So,SetIn the sequel, we check the transversality condition of the appearance of Hopf bifurcation. We give the hypothesis as follows: , where
Lemma 5. Suppose that is a root of equation (26) near such that ; then, one has .
Proof. In terms of equation (26), one knows thatwhich leads toThen,whereSo,By , one obtainsThis finishes the proof of Lemma 5.
Suppose that
Lemma 6. Provided that and is fulfilled, then system (3) is locally asymptotically stable.
Proof. If , then (24) takes the formBy , one knows that all roots of (57) satisfy . Thus, one knows that Lemma 6 is right. The proof finishes.
Based on the investigation above, we have the following conclusion.
Theorem 3. Provided that – hold true, then of system (3) is locally asymptotically stable if , and a Hopf bifurcation exists near if .
4. Computer Simulations
In this paper, we apply implicit Euler’s scheme which is introduced in [52] to carry our numerical simulations. Given the following fractional-order predator-prey modelwe can easily know that model (58) possesses the equilibrium point . Let . Then, . Hence, , . Then, . Thus, the hypothesis – of Theorem 3 hold true. If , the positive equilibrium point of system (58) is locally asymptotically stable. For this case, we choose . The simulation plots are displayed in Figures 1–3. Figures 1–3 reveal that if is less than , then the densities of competing species and of system (58) will be tardily close to , respectively, and the density of cooperating species of system (58) will be tardily close to 2.2559. When , then system (58) loses its stability and Hopf bifurcation behavior emerges. For this case, we choose . The numerical simulation diagrams are presented in Figures 4–6. Figures 4–6 confirm that when is greater than , then the densities of competing species and and the density of cooperating species will remain periodic motion around the positive equilibrium point , namely, Hopf bifurcation phenomenon takes place around . In order to illustrate the bifurcation phenomenon intuitively, we plot the bifurcation diagrams (see Figures 7–9). From Figures 7–9, we can easily see that the bifurcation value is 0.192. In addition, we give the relationship table between and in Table 1.
5. Conclusions
Based on the previous works on predator-prey models and noticing the effect of feedback delay between the predators and preys, we have established a new fractional-order competitor-competitor-mutualist Lotka–Volterra model. We have found the conditions to ensure the existence, uniqueness, and nonnegativity of the involved fractional-order prey-predator model. Applying the Laplace transform, stability theorem, and Hopf bifurcation theory of fractional-order dynamical systems, we set up a novel sufficient criterion to guarantee the local stability and the emergence of Hopf bifurcation of the involved fractional-order predator-prey system. The study shows that the time delay has a vital effect on controlling the stability. To check the rationality of theoretical predictions, we design the program to implement simulation experiments. The established analytical conclusions have crucial guiding significance in preserving the coexistence of biological populations. In this paper, we assume that the feedback time delay of different species is same. Of course, we can also deal with the different delay cases. We will focus on this topic in near future.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no competing interest.