Abstract

In this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decay exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy method and the squeezing technique with some new flavors to handle the nonlocal dispersals.

1. Introduction

In this paper, we investigate the stability of traveling wave fronts for a three species predator-prey model:with the initial conditionwhere are positive constants, and denote the densities of two cooperative preys at time t and location x, respectively, denotes the density of the predator at time t and location and are the intrinsic growth rates of and , respectively, is the death rate of , h and k are interspecific cooperative coefficients between two preys, and are the predation rates, and and are the conversion rates. are probability functions of the random dispersal of individuals and satisfy the following assumptions:   is the eigenvalue of the characteristic equation of model (1) 

Wu [1] investigated the spreading speed for a predator-prey model with one predator and two preys:with the initial conditionwhere and denote the densities of two competitive preys at time t and location x, respectively and denotes the density of the predator at time t and location x. The parameter , is the diffusion coefficients of , respectively, the remaining parameters are the same as model (1). Under certain conditions, the author characterized the asymptotic spreading speed by the parameters of model (3). The interaction of three species eological systems has been studied before (see [2–4]).

Yu and Pei [5] studied the stability of traveling wave fronts for the cooperative system with nonlocal dispersals:with the initial condition

The authors adopt the weighted energy method and the squeezing technique to prove the stability of the traveling wave fronts. The weighted energy method for treating time-delayed reaction diffusion equations was firstly introduced by Mei et al. [6]. Then, by combining the squeezing argument, it was developed for proving the global stability of wavefronts by [7–9]. For the nonlocal model using a single integrodifferential equation, the existence, uniqueness, and stability of traveling waves have been widely studied in [10–14]. For the multicomponent nonlocal systems, the existence of traveling waves was also investigated in [15, 16], while the stability of traveling waves is less investigated and can only be found in [5, 17]. Many researchers are widely focused on the complex dynamics of biological systems such as stochastic delay population system [18] and many researchers have studied the Lotka–Volterra time delay models with two competitive preys and one predator [19]. Note that the composite population systems with stochastic effects and time delays present some complex dynamics; thus, this causes widespread researchers concern [20, 21].

Inspired by works [1, 5], we will study the stability of traveling wave fronts for problems (1) and (2). We consider the case when both preys are cooperative. In this case, there is the constant state:

The rest of this paper is organized as follows. In Section 2, we present some preliminaries and summarize our main results. Section 3 is concerned with the proof of the main result by the technical weighted energy method and establishes the desired priori estimate, and then we get it by the squeezing technique. Finally, we make a simple summary in Section 4.

Before stating our main result, we introduce some notations.

Notations. Throughout this paper, denotes a generic constant, while represents a specific constant. Let I be an interval, typically . is the space of square integrable functions defined on I, and is the Sobolev space of the functions defined on the interval I whose derivatives also belong to . Further, be the weighted space with a weight function with the norm defined as be the weighted Sobolev space with the norm given byLet be a number and be a Banach space. We denote by the space of the -valued continuous functions on , and as the space of the -valued functions on . The corresponding spaces of the -valued functions on are defined similarly.

2. Preliminaries and Main Result

In this section, we consider the existence of traveling wave fronts for the Cauchy problems (1) and (2) by using the comparison principle and squeezing technique. Traveling waves solution of system (1) is a special solution with the form , , and with is the wave speed, where the wave profile and satisfies

In order to state our stability result, we need to give the following two propositions in [5].

Proposition 1. Assume that - hold. Then there exists a positive number such that, for any , (1) and (2) admits a traveling wavefront satisfying

Proposition 2. Assume that - hold. Let , where is the Banach space of all bounded and uniformly continuous functions from to with the usual supremum norm . Then, (1) and (2) admit a unique, bounded, and nonnegative solution on with the initial data , and for any and . Furthermore, letting and be the solutions of (1) with no diffusion term with the initial data and , respectively, and ifthen

Propositions 1 and 2 are based on reference [5] and are generalized from two dimension to three dimension. Their proofs are similar to the ones of Theorem 3.5 in [22] and Lemma 3.2 in [23], respectively.

In order to state our stability result, we need to add the following conditions:   

Define three functions on η as follows:

According to -, it is easily checked

Therefore, by continuity, there exists such that and .

In addition, define three functions on ξ as follows:where is a traveling wave front of (10). We can easily provewhich imply thatwhere is chosen to be large enough.

Define a weight function by

Now, we state the stability result.

Theorem 1. Assume that - hold. If any traveling wave front of (10) with the wave speed wherethe initial data satisfyand the initial perturbation satisfies

Then, the nonnegative solution of problems (1) and (2) exists uniquely and satisfieswhere is defined by (19). Moreover, converges to the traveling wave front exponentially in time t, i.e.,for all , where C and μ are some positive constants.

3. Stability

In this section, the existence of the solution to problems (1) and (2) can be proved via the upper and lower solutions method (see [24]). We first examine the case when two species cooperate while remaining in the same area without diffusive movement:where all parameters are positive and have four constant equilibria and coexistence equilibrium provided that .

Li and Lin [22] investigated the existence of traveling wave fronts connecting to the constant equilibrium for system (5) by using the known theory [25]. Integrodifferential system (5) is related to the classic Laplacian diffusion system, for example,where δ is the Dirac delta (see, Medlock et al. [26]), and then (5) reduces to

Because time delay often affects the evolutionary process, the system was incorporated into discrete and nonlocal delays while the existence of traveling wave fronts was obtained (see, [27–29]). The existence of traveling wave solutions of system (1) is similar to them. That is, the initial data satisfiesand the initial perturbation satisfiesand then the nonnegative solution of problems (1) and (2) exists uniquely and satisfies

Here, we omit the details.

Defineand then we obtain

Therefore, it follows from the comparison principle that

Letting where , and it follows from (32) and (33) that

3.1. A Prior Estimate

This section is devoted to establishing a prior estimate, which is the core of this paper. The approach is the weighted energy method.

Definewhere

In order to get the basic estimate, we must prove that and , for some constant C. We need the following key lemma.

Lemma 1. Assume that - hold. For any , there exists some positive constant C such that

Proof. We prove for some positive constant . Since , we have and , that is,

Case 1. . Since, and is nonincreasing, it follows from (37) that

Case 2 . In this case, and . Thus, it follows from (37) thatNext, setting , we have . We can prove in the same way.

Lemma 2. Assume that - hold. For any , there exists some positive constant C such that

Proof. We prove for some positive constant .

Case 3. . It follows from and the monotonicity of that

Case 4. . According to and , we knowIn view of the above argument and letting , we have . We can prove in the same way. This completes the proof.
We easily check the following result.

Lemma 3. Assume that - hold. For any , there exists some positive constant C such thathold for all , and .
Now we are going to derive the energy estimates for , and in the weighted Sobolev space .

Lemma 4. Assume that - hold. For any , it holdsfor some positive constant C.

Proof. From (1) and (10), satisfieswith the initial data . Multiplying the three equations of (52) by , and , respectively, where is defined like that, we haveIntegrating (53)–(55) over with respect to ξ and t and noting the vanishing term at far fieldwe obtainBy using the Cauchy–Schwarz inequality , we getSimilarly, we obtainSubstituting (60)–(63) into (57)–(59), respectively, we haveAdding the three inequalities (64), (65), and (66), we obtainwhere , and are defined in (36). According to (36), we can verifyfor some positive constant C. This completes the proof.

Lemma 5. Assume that - hold. For any , it holdsfor some positive constant C.

Proof. Differentiating (52) with respect to ξ, we haveMultiplying the three equations of (70) by , and , respectively, it holdsAccording to , and , we haveIntegrating (71)–(73) over with respect to ξ and t and according to the Cauchy–Schwarz inequality, we obtainAdding the three inequalities (75), (76), and (77), we obtainwhere , and are given by (36), andNote that , and are bounded on . There exists a positive constant such thatAccording to Lemma 4, we obtain the following inequalityBy using the Young inequality and (51) we haveChoosing such that and combining with (81), we havefor some positive constant C. This completes the proof.
Combining Lemma 4 and Lemma 5 and noting that on R, we obtain the following result.

Lemma 6. Assume that - hold. For any , it holdsfor some positive constant C and all .

3.2. Proof of Theorem 1

Using Sobolev embedding inequality and , we have

In view of Lemma 6, we obtain the following resultsfor . Similarly, for we can also verify

Hence, by the squeezing technique, we obtain

This completes the proof of Theorem 1.

4. Conclusions

Inspired by the works of [1, 5], in this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. Since the predation models of the three species add to the computational difficulty, it takes a lot of effort to the priority estimation, and the key point in proving the stability of the traveling front is devoted to establishing a prior estimate by using the weighted energy method. By the standard Sobolev embedding inequality and the squeezing technique, we prove the stability of the traveling wave solution.

In recent years, there has been great progress in modeling and analysis dynamical behavior of predator-prey population involving both time delay and spatial diffusion. In a pioneer work, some researchers have studied a scalar reaction diffusion equation with a single discrete delay by using the phase-plan technique; we can take more attention and initiate the study of traveling wave solutions to delayed reaction diffusion systems on the basis of this paper. Futhermore, we will consider the existence and stability of traveling wave fronts for three-dimensional diffusion systems with convolution delay by the comparison principle and squeezing technique in the future. Another interesting and difficulty problem is the stability of the traveling wave solution under quasi-monotone or nonquasi-monotone assumptions. We leave these issues for future research.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 11671227) and the Natural Science Foundation of Shandong Province of China (no. ZR2019MA067).