Abstract
In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.
1. Introduction
Spatiotemporal pattern formations are one of the core problems arising in biology and ecology. Hence, it has attracted the extensive attentions from a huge number of investigations, including biologists, chemists, and mathematicians. It is well known that diffusion not only can induce spatial patterns, but also can drive spatiotemporal patterns, including oscillatory patterns. In this paper, we are mainly concerned with the spatiotemporal oscillatory patterns of a particular reaction-diffusion predator-prey system arising from ecology. In the model of our concern, the prey is assumed to exhibit herd behavior, so that the predator interacts with the prey along the outer corridor of the herd of prey, and the interaction terms are supposed to use the square root of the prey population rather than simply the prey population, where the use of the square root properly represents the assumption that the interactions occur along the boundary of the population (see [1–15]). The reaction-diffusion system takes in the following form:where and stand for the population densities of the prey and predator at time and the position , respectively; represents the interaction strength between the predator and the prey; and are the diffusion rates of and , respectively. The spatial domain is supposed to be one dimensional, , with ; stands for the death rate of ; is the saturation rate. The boundary conditions are supposed to be the no flux boundary conditions.
System (1) has been studied extensively in the past few years (see [16–25]). For example, in [16], by using as the bifurcation parameter, the authors showed that, as the parameter changes, the stability of the constant positive steady state varies, which induces the occurrence of Hopf bifurcating periodic solutions; however, their works tend to be numerical, rather than analytical. In [22], in the special case when , Tang and Song introduced the time delay into the system. By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the time delay is used to be the bifurcation parameter. In [25], in the special case when , Xu and Song introduced the time delay into the system. By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the parameter is chosen as the bifurcation parameter. In [17], G. Gimmelli, Bob W. Kooi, and E. Venturino considered the ecoepidemic models with prey group defense and feeding saturation in the form of ordinary differential equations. The Hopf bifurcation, zero-Hopf bifurcation, and global bifurcations are considered.
In this paper, however, we do not impose the special case when ; instead, we consider the more general case when . This is the first highlight of our paper. Secondly, in the existing literatures, the bifurcation parameter is to be chosen as either , or the diffusion coefficient, while, in our paper, use , or equiv. the first component of the positive equilibrium , as the bifurcation parameter. The different choice of the bifurcation parameter makes our theoretical analysis quite different. This is the second highlight of our paper; finally, the Hopf bifurcation analysis in this paper is mainly based on the theoretical framework of the abstract Hopf bifurcations due to Yi, Wei, and Shi in [26]. In [26], the authors used the same methods to consider a kind of homogeneous diffusive predator-prey system with Holling type-II functional response subject to the homogeneous Neumann boundary conditions. They found that, under suitable conditions, for any eigenmode , the predator-prey system in [26] has Hopf bifurcation points. In this paper, however, we will show that the Hopf bifurcation points of the system are much more complex than those in [26]. This is the third highlight of our paper.
The main analytical tools we used in the discussions of the paper include center manifold theory, normal form methods, and the basic theory in reaction diffusion equations. To better illustrate and support our theoretical bifurcation analysis, we include some delicate numerical simulations in the paper.
The remaining parts of this paper are organized as follows: in Section 2, we not only prove the existence of Hopf bifurcations, but also consider the detailed properties of Hopf bifurcations by using center manifold methods and normal form methods; in Section 3, we include numerical simulations to support our theoretical analysis; in Section 4, we end up our discussions by drawing some conclusions.
2. Hopf Bifurcation Analysis
System (1) has three nonnegative constant steady state solutions: , , and , where
Clearly, is a positive equilibrium solution if and only if . In the following, we will always assume that holds and will choose as the main bifurcation parameter (or equivalently as a parameter) by fixing and .
The linearized operator of system (1) evaluated at is given bywhere
It follows from [26] that the eigenvalues of the operator are determined by the eigenvalues of , with
The characteristic equation of is where
To consider the stability/instability of the positive constant steady state solution , as well as the Hopf bifurcations bifurcating from , we need to know the signs of and .
Lemma 1. DefineThen, has three zero points, denoted by , satisfying , such that and And in particular, for the largest zero point , it satisfies that .
Proof. By (8), we have Solving , we have At , we have We argue that . In fact, letting , we have Since and , we have . Since , we have . Then, there exists a , such that and .
On the other hand, by and , there exists a such that and . Clearly, .
Since , we can conclude that at a critical point in . Thus, we have .
At , we have We argue that . In fact, letting , we have . Noticing that , we can conclude that there exists a , such that . Since , we have . In particular, So far, we have completed the proof.
We are now identifying Hopf bifurcation values which satisfy the Hopf bifurcation condition (due to [26]), taking the following form: there exists a , such thatand for the unique pair of complex eigenvalues near the imaginary axis ,
From (7), it follows that for . Hence any potential bifurcation point must be in the interval . Let be the eigenvalues of for close to the Hopf bifurcation value . Then, and where
Lemma 2. Let be defined in (19). Then for any , and there exists a unique , such that .
Proof. By , we have , where . At , we have Again, letting , we have where we use the fact that . By , we conclude that there exists a such that . We thus complete the proof.
According to Lemmas 1 and 2, we can draw the graphs of the functions and in a coordinate system (Figure 1). Thus, the transversality condition (16) is always satisfied as long as . Moreover when , the real part of one pair of complex eigenvalues of becomes positive when decreases crossing , and when , the real part of one pair of complex eigenvalues of becomes negative when decreases crossing . That is, in , the constant steady state loses stability when decreases across a bifurcation point, but, in , it regains the stability when decreases across a bifurcation point.
From the aforementioned discussions, the determination of Hopf bifurcation points reduces to describing the setwhen a set of parameters is given. In the following we fix and , but we choose appropriately.
On one hand, it is clear that is always an element of for any since for any , and for any . This corresponds to the Hopf bifurcation of spatially homogeneous periodic solution which has been known from the studies of ODE model. Clearly, is also the unique value for the Hopf bifurcation of spatially homogeneous periodic solution for any .
In the following, we will look for the spatially nonhomogeneous Hopf bifurcation for . By the properties of (stated in Lemma 1), it follows that , and in particular, has a unique critical point , with , at which achieves a local maximum . In particular, by Lemma 2, .
Theorem 3. Let be the critical point of , with , at which achieves a local maximum . (1)Suppose that and that, for sufficiently small , holds. Define For and , there exist points , with , satisfying such that the system (1) undergoes a Hopf bifurcation at or .(2)Suppose that and that, for sufficiently small , holds. Define (a)Then for and , with for some , there exist points , with satisfying such that the system (1) undergoes a Hopf bifurcation at or .(b)For and , there exist points , with , satisfying Then for any , there exist points , satisfying , such that the system (1) undergoes a Hopf bifurcation at or . Moreover:
The bifurcating periodic solutions from are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system.
The bifurcating periodic solutions from (or ) are spatially nonhomogeneous.
Proof. (1)Suppose that holds. Then, and for any . In this case, define For and , we define to be the unique root of satisfying . These points satisfy Clearly and for . In this case, the transversality condition always holds true. Since and , we can assume that , where is an arbitrary positive number. Indeed if , then we have The quadratic function is positive for all if (2)Suppose that holds. Then, for any and for any . Define Then for and , with for some , we define and to be the roots of satisfying . These points satisfy Clearly and for . In this case, the transversality condition always holds true. For and , we define to be the unique root of satisfying . These points satisfy Clearly and for . In this case, the transversality condition always holds true. Now we only need to verify whether for all for the aforementioned three cases, and, in particular, . Here we derive a condition on the parameters so that for all so that . Since and , we can assume that , where is an arbitrary positive number. Indeed if , then we have The quadratic function is positive for all if Then, applying Theorem 2.1 in [27], we have completed the proof.
In the following, we will consider the bifurcation direction and the stability of the bifurcating periodic solutions. We have the following results.
Theorem 4. For the Hopf bifurcation points obtained in Theorem 3, the following conclusions hold: (1)The Hopf bifurcation at is supercritical if . It is subcritical if .(2)The spatially homogeneous periodic solutions bifurcating from the constant positive steady state at are locally orbitally stable if , and they are unstable if , where is precisely defined in (63).
Proof. We rely on the abstract results obtained in [26] to determine the stability and bifurcation direction of the bifurcating periodic solution. To that end, we calculate in the context of [26]. We use to stand for . When , we let be such thatwhere ( and ), andAs in [26], we define We rewrite the original system in the following abstract equation:wherewith , where and are shown in (4). At , the system (43) reduces towhere .
Let and decompose , with , . For any , there exist and such that Thus the system (45) is reduced to the following system in coordinates:whereAs in [26], we write in the following form:where are symmetric multilinear forms. For simplicity, we write and .
Then, the reaction-diffusion system restricted to the center manifold is given bywhere , , , and The dynamics of (48) can be determined by the dynamics of (51).
We write the Poincaré normal form of (43) (for in a neighborhood of ) in the following form: where is a complex variable, , and are complex-valued coefficients.
Then, we have Thus,Then, it remains to calculate , , , and .
After direct calculations, we observe that all of the , , , , , and are all zeros. And in particular, Then, from [26], we have and Again, by [26], we have and Then, from [26], we have which implies that . Therefore, So far, based on the aforementioned computations, we haveThen, from [26], we complete the proof.
3. Numerical Simulations
In this section, we include some numerical simulations to support our theoretical analysis. We use the symbolic mathematical software Matlab (Version 7.0) to plot numerical graphs.
The model (1) has five different system parameters, which are: , , , , and . On one hand, we choose the system parameters in the following way:
Under (64) and after direct calculations, we have =. In this case, the positive equilibrium solution is locally asymptotically stable, and no Hopf bifurcations can be observed. This is shown in Figure 2, where the initial condition is taken by , (Figure 2).
On the other hand, we choose the system parameters in the following way:
Under (65) and after direct calculations, we have =. In this case, is unstable and a family of periodic solutions bifurcating from can be observed. This is shown in Figure 3, where the initial condition is taken at , (Figure 3).
4. Conclusions
In this paper, we considered a new homogeneous diffusive predator-prey system with herd behavior in terms of the square root of the prey population. By using the classical Hopf bifurcation theorem applicable to the general reaction-diffusion equations due to [26], we are capable of showing the existence of both spatially homogeneous/inhomogeneous periodic solutions bifurcating from the positive constant steady state solutions. For the homogeneous periodic solutions, we derive the formulae to determine their stability and instability. In this paper, there are three highlights: we consider the more general case when , instead of . We use , or equiv. the first component of the positive equilibrium , as the bifurcation parameter, instead of and the diffusion coefficient; we show that the Hopf bifurcation points of the system are much more complex than those in [26]. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.
Data Availability
Data sharing is not applicable to this paper as no datasets were generated or analysed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
Prof. Meng Lin, Prof. Yanyou Chai, and Prof. Xuguang Yang contributed to the Hopf bifurcation analysis. Dr. Yufeng Wang contributed to the numerical simulations in the revision stage of the manuscript. All authors read and approved the final manuscript.
Acknowledgments
The final version of the paper was completed when the corresponding author Yanyou Chai visited University of Central Florida (UCF), and he would like to thank the hospitality of Prof. Z. Shuai in UCF. This work is supported by the National Natural Science Foundation of China (No. 61801141) and the Fundamental Research Funds for the Central Universities.