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

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 theminimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.


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][6][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  ∈ R,  > 0,  1 > 0,  2 > 0,  1 > 0,  2 > 0,  1 ∈ R,  2 ∈ R, and in which  1 ≥ 0,  2 ≥ 0,  1 ≥ 0,  2 ≥ 0, and  ≥ 0 are constants such that For (1), a traveling wave solution is a special solution with the form where is the wave profile and  > 0 is the wave speed.Therefore, ( 1 ,  2 ) and  satisfy with In Pan [8], the author defined a threshold given by and 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) satisfying lim Motivated 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.

Main Results
When  =  * , we define By these constants, we first present our main conclusion as follows.
Theorem 1. Assume that  =  * holds.Then ( 5) admits a bounded positive solution ( 1 ,  2 ) satisfying We shall prove the result by three lemmas, which will study three cases For this purpose, we first show the following result in Lin and Ruan [12].

Lemma 2. Suppose that 𝜙
(A2) they are twice differentiable except a set  containing finite points of R and are continuous and bounded if  ∈ R \ ; (A3) when  ∈ , they satisfy (A4) they satisfy the following inequalities: for  ∈ R \ .
Proof.For simplicity, we shall denote  * = 2√ 1  1 by  and define Let  > 0 be a constant such that Moreover, select  > 1 with and 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  1 < 0 such that  1 () 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  1 (), we shall prove the first inequality of (12) when  ̸ =  1 .If  >  1 and  1 () = 1, then and the first inequality of ( 12) is clear.When  1 () = (− + )  1  < 1, then and the verification on the first inequality of ( 12) is finished.
When the second inequality on then the definition of  2 implies that the desired inequality is true if 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 With these results, we obtain by ( L1)-(L2).Therefore, it suffices to prove that or which is true by (L3).
On the third inequality, it is clear if  1 () = 0.When   1  −   1  > 0, (Q1) implies then the third is true by (Q2).We now consider the fourth inequality, which is clear if  (46)