Minimal Wave Speed in a Predator-Prey System with Distributed Time Delay

Abstract

This paper is concerned with the minimal wave speed of traveling wave solutions in a predator-prey system with distributed time delay, which does not satisfy comparison principle due to delayed intraspecific terms. By constructing upper and lower solutions, we obtain the existence of traveling wave solutions when the wave speed is the minimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.

1. Introduction

Traveling wave solutions of predator-prey systems have been widely utilized to model population invasion, and the minimal wave speed of traveling wave solutions is often regarded as an important threshold to characterize the invasion feature in many examples, see Owen and Lewis [1] and Shigesada and Kawasaki [2, Chapter 8]. Moreover, Lin [3] and Pan [4] confirmed that, in a Lotka-Volterra type system, the minimal wave speed of invasion traveling wave solutions is equal to the invasion speed of the predator. Here, the invasion speed is estimated by the corresponding initial value problem when the initial value of the predator admits a nonempty compact support.

When the wave system of predator-prey system is of finite dimension, there are many important results, for example, the earlier results by Dunbar [5–7]. But when the corresponding wave system is of infinite dimension, there are some open problems on the minimal wave speed, for example, in the following system [8]:where , , , , , , , , and in which , , , , and are constants such that

For (1), a traveling wave solution is a special solution with the formwhere is the wave profile and is the wave speed. Therefore, and satisfywith

In Pan [8], the author defined a threshold given byand showed the existence (nonexistence) of desired traveling wave solutions if the wave speed When the wave speed , the author presented the existence of traveling wave solutions under special conditions. Besides [8], there are also some results on the existence of traveling wave solutions of predator-prey models similar to (1) when the wave speed is large; see Huang and Zou [9], K. Li and X. Li [10], and Lin et al. [11].

The purpose of this paper is to confirm the existence of nontrivial traveling wave solutions of (1) without other conditions when the wave speed Since Pan [8, Theorem 3.5] also holds when , we shall not investigate the limit behavior as and focus on the existence of positive solution of (5) satisfyingMotivated by Lin and Ruan [12] on an abstract result of traveling wave solutions of delayed reaction-diffusion systems, we shall construct proper upper and lower solutions similar to those in Fu [13] and Lin [14] to study the existence of traveling wave solutions.

2. Main Results

When , we defineBy these constants, we first present our main conclusion as follows.

Theorem 1. Assume that holds. Then (5) admits a bounded positive solution satisfying (1), if ;(2), if ;(3), if

We shall prove the result by three lemmas, which will study three cases , , and For this purpose, we first show the following result in Lin and Ruan [12].

Lemma 2. Suppose that , , , and are continuous functions and (A1), , ;(A2)they are twice differentiable except a set containing finite points of and are continuous and bounded if ;(A3)when , they satisfy(A4)they satisfy the following inequalities: for Then (5) has a positive solution such that

Remark 3. In the above lemma, and are a pair of (generalized) upper and lower solutions of (5). That is, the existence of positive solutions of (5) can be obtained by the existence of (generalized) upper and lower solutions of (5).

Lemma 4. Assume that Then (1) of Theorem 1 holds.

Proof. For simplicity, we shall denote by and defineLet be a constant such that (K1) is monotone if ;(K2) or Moreover, select with and such that (M1) implies ;(M2);(M3) implies , and such that (N1) implies ;(N2) Select such that (L1) implies , where such that (L2) implies ;(L3)The admissibility of , , , and is clear by the limit behavior of these functions as Mathematically, we first fix , then select , and finally define , Here, and are independent of each other.
We now define where such that is continuous by (K1)-(K2) and If these functions satisfy (12), then our result holds by Lemma 2. Now, we are in a position of verifying these inequalities. For , we shall prove the first inequality of (12) when If and , then and the first inequality of (12) is clear. When , then and the verification on the first inequality of (12) is finished.
When the second inequality on is concerned, it is also clear if When , then (M1) leads to Note that then the definition of implies that the desired inequality is true ifor On the one hand, (M2) leads to At the same time, we have by (M3). Therefore, (23) is true, as is the case for the second inequality of (12).
On the third inequality of (12), it is clear if Otherwise, With these results, we obtain by (L1)-(L2). Therefore, it suffices to prove that or which is true by (L3).
We now consider , that is, the forth inequality of (12). When , the definition implies by (N1) as well as Thus, the desired inequality is true if since such that , which holds by (N2). The proof is complete.

Lemma 5. Assume that Then (2) of Theorem 1 is true.

Proof. Similar to the proof of the previous lemma, it suffices to construct proper upper and lower solutions. When , let Fix such that Select such that Let be the smaller root of . Clearly, if , then
By these constants, we define where , , and are positive constants satisfying that (N1) is large such that , ;(Q1) such that implies ;(Q2), and (R0) is a constant such that implies ,(R1) such that implies , where satisfies(R2) such that , On the first inequality of (12), if , then and the result is clear. If , then which completes the verification on On the second inequality, it is evident if When , we have Therefore, by (N1).
On the third inequality, it is clear if When , (Q1) implies Since then the third is true by (Q2).
We now consider the fourth inequality, which is clear if When , we have which implies Moreover, (R0) and (R1) imply that and so by (R2), which completes the verification and proof.

Lemma 6. Assume that Then (3) of Theorem 1 is true.

Proof. Utilizing the parameters similar to those in Lemmas 4 and 5, we define where and such that and are continuous. Similar to the proof of Lemmas 4 and 5, we can complete the proof.

Before ending this paper, we make the following remarks on the minimal wave speed.

Remark 7. In Lin [15] and Pan [16], the authors studied the asymptotic spreading of (1) if , in which one species spreads in the minimal wave speed of traveling wave solutions. However, (1) does not satisfy the comparison principle of classical predator-prey systems in [15, 16]; there are also some technique problems in estimating the asymptotic spreading of (1), which will be further investigated in our future research.

Remark 8. From Pan [8], we see that a traveling wave solution with large wave speed decays exponentially as . However, when the minimal wave speed is concerned, it does not decay exponentially as .

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.

Acknowledgments

Dongfeng Li was supported by the National Key Research and Development Program of China (no. 2016YFC0402502).