Traveling Wave Solution in a Diffusive Predator-Prey System with Holling Type-IV Functional Response

Abstract

We establish the existence of traveling wave solution for a reaction-diffusion predator-prey system with Holling type-IV functional response. For simplicity, only one space dimension will be involved, the traveling solution equivalent to the heteroclinic orbits in . The methods used to prove the result are the shooting argument and the invariant manifold theory.

1. Introduction

The paper will study the traveling wave solution for a diffusive predator-prey system with Holling type-IV functional response, which is as follows: All parameters are positive constant. The functions and are the densities of the prey and predator, respectively; and are diffusive rates of the prey and predator, respectively; is the carrying capacity of the prey; is the death rate of the predator; and is the growth factor of the prey. We may refer to [1, 2] for more biological implications.

Recently, the system (1) and some related systems have been studied by many researchers for an understanding of the most basic features of a spatially distributed interaction; we can refer to [3–10]. Gardner [8] proved the existence of traveling wave solutions for a diffusive predator-prey system with Holling type-II functional response by using the connection index. Numerical simulation in Owen and Lewis [11] shows that a diffusive predator-prey system with Holling type-II functional response, when the diffusive rates of the prey and the predator are not zero, possesses traveling wave solutions. Huang et al. [12] proved theoretically that the numerical simulation in [11] is true. Huang et al. considered the system and they obtained that if , , and , then there are nonnegative solutions of system (2) satisfying , , , .

Dunbar [13] studied the following system: and obtained the following.(a)If , then there exist traveling wave front solutions of the system (3) satisfying , , , .(b)If , then there exist traveling wave front solutions of the system (3) satisfying , , , .

Dunbar [14] investigated the system and obtained that if and , then there is a bounded solution of (4) satisfying , , , and .

Li and Wu [15] studied a system with Holling type-III functional response and proved the existence of traveling wave solutions by using the shooting argument in together with a Lyapunov function [16], LaSalle’s invariance principle [17], and the Hopf bifurcation theorem [18]. We may refer to Murray [19], Mischaikow and Reineck [20], and Volpert et al. [21] for more results.

We notice that the Holling type-II and the Holling type-III functional response are monotonic in the first quadrant, while the Holling type-IV functional response considered in this paper is nonmonotonic in the first quadrant. It is an interesting problem to know whether the above results are available for the system (1). We should mention that although the techniques used here are similar to those in [12–15, 22], there are several differences. Firstly, it is a more complex system. The systems studied in [13, 22] are the ones with the Lotka-Volterra functional response. The systems studied in [12, 14, 15] are the ones with the Holling type-II or Holling type-III functional response. Secondly, we construct a different Wazewski set and a new Lyapunov function. For simplicity, we assume that can be considered to correspond to a situation in which the prey species is evenly distributed. We should mention that the assumption is not essential.

For further simplification, taking and dropping the stars on , and the primes on , for convenience, we obtain There are several reasonable parameter restrictions. We assume that or equivalently that , so that the satiation effect is great enough. We also assume that and , which ensure that the system (6) has positive equilibrium point corresponding to constant coexistence of the two species. Obviously, the system (6) has four equilibria points: (0, 0), (, 0), (, ), and (, ), which are equilibria of the corresponding ODE system without diffusion, where In this paper, we also require that , which ensures that equations (6) has only a positive equilibrium. We notice that , so the system (6) has only one positive equilibrium point. The equilibrium (0, 0), representing the absence of both species, is a saddle point. The equilibrium (, 0), representing the population of the prey at the environmental carrying capacity in the absence of predators, is unstable. The equilibrium (, ), representing the time constant coexistence of both species, is stable. We establish the traveling wave solution connecting the equilibria (, 0) and (, ), which is called the “waves of invasion”; see Chow and Tam [23].

The paper is organized as follows. In the next section, we first recall a lemma which is a variant of Wazewski's Theorem and then we state the result on the existence of traveling wave solution. Section 3 is devoted to prove the result.

2. Main Result

In order to establish the existence of traveling wave solution of the system (6), we assume that the solution has the special form , , where the wave speed parameter is positive. Substituting , , into the system (6), the responding system becomes Here denotes the differentiation with respect to the variable . We require that the traveling wave solutions and are nonnegative and satisfy the boundary conditions We write the system (6) as a first order system in In this section a variant of Wazewski's Theorem, which is a formalization and extension of the shooting method, is stated. This proposition recognizes that the flow defined by the solutions of a differential system gives a topological mapping between regions of phase space. The statement and the proof of Wazewski's Theorem are given in [24].

Consider a system Here is a continuous function and satisfies the Lipschitz condition. Let be the unique solution of satisfying . For convenience, we set ; let be the set of points , where and .

Given , the notation cl() is used for the closure of . Define is called the immediate exit set of . Given , let For , define is called an exit time. Note that and if and only if .

Lemma 1. Suppose that(i)if and , then ;(ii)if , , , then there is an open set about disjoint from ;(iii), is a compact set and intersects a trajectory of only once.Then the mapping is a homeomorphism from to its image on . The proof is given in [22]. A set satisfying the conditions (i) and (ii) is called a Wazewski set.

Theorem 2. (i) If , and , then there are no nonnegative solutions of the system (10) satisfying the boundary conditions (9).
(ii) If , , , and , then there exists nonnegative solution of the system (10) satisfying the boundary conditions (9), which correspond to traveling wave solution of the system (6).

3. Proofs of the Result

The eigenvalues of the linearization of the system (10) at (, 0, 0) are If , then and are a pair of complex conjugate eigenvalues with positive real part. By Theorems 6.1 and 6.2 in [16], there is a two-dimensional unstable manifold base at ; the critical point is a spiral point on this unstable manifold, so the trajectory approaching as must have for some . It violates the requirement that the solution of the system (10) must be nonnegative. It proves the first part of Theorem 2.

We only need to discuss the case . In fact we require the stronger condition for mathematical simplicity. With the requirement there are three distinct real eigenvalues . Let the eigenvectors , , associated with , , , respectively, be Here , .

Applying Theorems 6.1 and 6.2 of [16], there exists a one-dimension strongly unstable manifold tangent to at (, 0, 0). A parametric representation for the strongly unstable manifold in a small neighborhood of (, 0, 0) is There exists a two-dimension unstable manifold tangent to the span of and at (, 0, 0). A parametric representation for the two-dimensional unstable manifold in a small neighborhood of (, 0, 0) is The idea of constructing the Wazewski set is similar to that in Dunbar [22]: it will be the complement of three blocks in , two of which are chosen so that has the same sign as so solutions entering these blocks would not have as . Thus we define the Wazewski set as follows: Here Note that is a closed set. Let By checking the vector field on , we obtain Details of proof that is the set described above are tedious. We only examine the part of as an example, which shows why the set must be excluded from to obtain . The other proofs are similar. The boundary of is , , or .(1), , and . Since , , , , and , thus the trajectory enters .(2), , and . Since , , , and , thus and we obtain the following.(i); then , , and the trajectory enters .(ii); then , , and the trajectory does not enter .(iii), . Consider the system We come to the conclusion that the -axis is an invariant manifold and the trajectory does not enter .(3), , and . Since and , we obtain the following.(i); then , , , , and the trajectory enters .(ii); then , , , , and the trajectory does not enter .(iii); then , , and . That is, , , and the trajectory enters .(4), , and . From the proof of (2), we come to the conclusion that , , and the trajectory enters .(5), , and . Since , the trajectory enters .(6), , and . Since , we obtain the following.(i); then and , which implies that the trajectory enters .(ii); similar to the proof of (2iii), the trajectory does not enter .(iii); then and ; that is, the trajectory does not enter .(7), , and . Since , , and , we obtain the following.(i); then , , , and the trajectory enters .(ii); similar to the proof of (2iii), the trajectory does not enter .(iii); then and ; that is, , and the trajectory does not enter .(iv). It is a singular point (, , 0) and is not in the immediate exit set.(v); then , , and , which implies that the trajectory enters .(8), , and . Since , then we obtain the following.(i); then , , and the trajectory does not enter .(ii); the trajectory does not enter .(iii); then , , and the trajectory enters .

In order to use Lemma 1, we construct the set on a sphere surrounding (, 0, 0) in the two-dimensional unstable manifold by Lemma 3 to Lemma 7. The specification of the arc requires the identification of the endpoints on the circle. One endpoint is the intersection of the circle with the strongly unstable manifold and the other is the intersection of the circle with the plane defined by . Lemmas 3–6 are simple comparison arguments showing that the first endpoint on the strongly unstable manifold is carried by the flow into and the other is carried into . We use the notation .

Lemma 3. Let . A solution of the system (10) having a point, corresponding to without loss of generality, such that , , and , will have and for all . In particular, it is true for trajectories on the branch of strongly unstable manifold in the octant .

Proof. Suppose, to the contrary, that there exists an such that , but . Let . Since and for , , we have and . Using that , we obtain . Since , it follows that ; that is, ; it is a contradiction with . It completes the proof.

Lemma 4. A trajectory on the portion of strongly unstable manifold in the octant must satisfy for all .

Proof. The solution approaches (, 0, 0) tangent to and the eigenvector at (, 0, 0) has such that . Suppose to the contrary that there exists an such that . Let ; then . Since , , it follows that ; it is a contradiction. It completes the proof.

Lemma 5. Let be a fixed number. A solution of the system (10) having a point, corresponding to without loss of generality, such that and , will have for all such that . In particular, this is true for trajectories on the branch of strongly unstable manifold in the octant .

Proof. Suppose, to the contrary, that there exists an such that , but . Let ; then and for . Substituting and , we obtain ; that is, . However, the choice of implies that ; thus . This contradiction shows that for such that .

Lemma 6. Suppose that a solution of the system (10) has a point such that Then for all , as long as , , the trajectory must satisfy that In particular, it is true for trajectories on the branch of strongly unstable manifold in the octant .

Proof. We first show that for all such that . If it is not true, then there is a first such that , , and . However, we have It is a contradiction; then for all such that .
Let ; suppose that there exists a first time such that , , and ; then . By Lemma 5, we obtain From and , we have It is a contradiction, which completes the proof.

Combining the results of these lemmas, we follow the trajectory of a solution of the system (10) on the strongly unstable manifold . Define Lemmas 3–6 show that the trajectory of a solution of the system (10) on the strongly unstable manifold is contained in . Recall the assumption that ; then which implies that in the region . Thus, for a solution of the system (10) on the strongly unstable manifold , decreases until for some finite ; the trajectory of this solution hits on the face , , and . The vector field on this face shows that a solution of the system (10) on enters the region at some finite time.

Lemma 7. In a sufficiently small neighborhood of , the two-dimensional unstable manifold intersects the plane defined by in a curve , given by and .

Proof. The proof, which is similar to that of Lemma 5 in [13], is therefore omitted.

We are interested in the portion of the curve in the region . The function can be approximated to the first order by Thus the -coordinate of points along the curve will satisfy . From the direction of the vector field on the plane defined by , , and , any trajectory passing through a point of near (, 0, 0) will immediately enter the region .

Now, we place a sufficiently small circle surrounding (, 0, 0) on the two-dimensional unstable manifold such that the circle is contained in the neighborhood of (, 0, 0) given in Lemma 7 and the conditions of Lemmas 3–6 are satisfied. The circle intersects the curve . Define to be the arc of this circle contained in the octant , whose endpoints are the intersections of the circle with and the curve .

We now prove part (ii) of Theorem 2, which requires two steps. Firstly, we use Lemma 1 to produce a trajectory which remains in the region . Secondly, we construct a Lyapunov function to prove the trajectory approaches (, , 0).

Lemma 8. There exists a point such that the solution of the system (10) remains in the region for all .

Proof. The proof, which is similar to the proof of Lemma 3.7 in [15], is therefore omitted.

Lemma 9. The solution must be in the bounded region for all , where

Proof. Since the plane defined by is an invariant manifold, the first coordinate of is strictly positive, and thus for all . Suppose enters the region ; let ; then we have , , and . We know that the -axis is an invariant manifold, . Since does not enter then . From the system (10), we obtain , which must enter the region In the region , and are decreasing, and thus is bounded below by the positive Then increases to in a finite time and enters . It is a contradiction, so for all .
From Lemma 6, we obtain that for . Since , we have for all . Suppose that there is an such that for , where . Let , so , , and . Then either or immediately enters , which is a contradiction.
Suppose that there exists an such that ; then for all . If it is not true, there exists an such that , and thus . From the system (10), we have which is impossible. So if , then continues to hold for . Thus, and for all and is strictly negative and bounded away from zero by . Then for some finite ; it is a contradiction. Notice that a trajectory starting on tangent to or has or . Since and , we have for all , which completes the proof of this lemma.

Lemma 10. The trajectory as .

Proof. In order to show the trajectory will approach the point (, , 0), we construct a Lyapunov function as follows: We obtain that is continuous and bounded below on , Here . Recall the assumption that ; then Therefore, is always nonpositive in . Moreover, if and only if , , and the largest invariant subset of this segment is the single point (, , 0). By LaSalle’s invariance principle, it follows that as , which completes the proof of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Zhengyi Lu for discussing the research and his useful remarks. The research was partially supported by the Program of Chongqing Innovation Team Project in University under Grant no. KJTD201308 and the Program of Chongqing Normal University under Grant no. 13XLB015. This work is supported by Grant from the Scientific Research Project of the University of South China (no. 2011XQD36) and supported by the construct program in USC.