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.

National Key Research and Development Program of China2016YFC04025021. 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]:(1)∂u1x,t∂t=d1Δu1x,t+r1u1x,tF1u1,u2x,t,∂u2x,t∂t=d2Δu2x,t+r2u2x,tF2u1,u2x,t,where x∈R, t>0, d1>0, d2>0, r1>0, r2>0, u1∈R, u2∈R, and (2)F1u1,u2x,t=1-u1x,t-a1∫-τ0u1x,t+sdη11s-b1∫-τ0u2x,t+sdη12s,F2u1,u2x,t=1-u2x,t-a2∫-τ0u2x,t+sdη22s+b2∫-τ0u1x,t+sdη21s,in which a1≥0, a2≥0, b1≥0, b2≥0, and τ≥0 are constants such that(3)ηijsisnondecreasingon-τ,0,ηij0-ηij-τ=1,i,j=1,2.

For (1), a traveling wave solution is a special solution with the form(4)uix,t=ϕiξ,ξ=x+ct,i=1,2,where (ϕ1,ϕ2)∈C2(R,R2) is the wave profile and c>0 is the wave speed. Therefore, (ϕ1,ϕ2) and c satisfy(5)d1ϕ1′′ξ-cϕ1′ξ+r1ϕ1ξF1ϕ1,ϕ2ξ=0,ξ∈R,d2ϕ2′′ξ-cϕ2′ξ+r2ϕ2ξF2ϕ1,ϕ2ξ=0,ξ∈Rwith (6)F1ϕ1,ϕ2ξ=1-ϕ1ξ-a1∫-τ0ϕ1ξ+csdη11s-b1∫-τ0ϕ2ξ+csdη12s,F2ϕ1,ϕ2ξ=1-ϕ2ξ-a2∫-τ0ϕ2ξ+csdη22s+b2∫-τ0ϕ1ξ+csdη21s.

In Pan [8], the author defined a threshold given by(7)c∗≔max2d1r1,2d2r2and showed the existence (nonexistence) of desired traveling wave solutions if the wave speed c>c∗(c<c∗). When the wave speed c=c∗, 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 c=c∗. Since Pan [8, Theorem 3.5] also holds when c=c∗, we shall not investigate the limit behavior as ξ→∞ and focus on the existence of positive solution of (5) satisfying(8)limξ→-∞ϕiξ=0,i=1,2.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.

2. Main Results

When c=c∗, we define(9)γ1=c-c2-4d1r12d1,γ2=c-c2-4d2r22d2.By these constants, we first present our main conclusion as follows.

Theorem 1.

Assume that c=c∗ holds. Then (5) admits a bounded positive solution (ϕ1,ϕ2) satisfying

limξ→-∞ϕ1(ξ)/-ξeγ1ξ=1, limξ→-∞ϕ2(ξ)/eγ2ξ=1 if c=c∗=2d1r1>2d2r2;

limξ→-∞ϕ1(ξ)/eγ1ξ=1, limξ→-∞ϕ2(ξ)/-ξeγ2ξ=1 if c=c∗=2d2r2>2d1r1;

limξ→-∞ϕ1(ξ)/-ξeγ1ξ=1, limξ→-∞ϕ2(ξ)/-ξeγ2ξ=1 if c=c∗=2d2r2=2d1r1.

We shall prove the result by three lemmas, which will study three cases d1r1>d2r2, d1r1<d2r2, and d1r1=d2r2. For this purpose, we first show the following result in Lin and Ruan [12].

Lemma 2.

Suppose that ϕ_1(ξ), ϕ¯1(ξ), ϕ_2(ξ), and ϕ¯2(ξ) are continuous functions and

0≤ϕ_1(ξ)≤ϕ¯1(ξ)≤1, 0≤ϕ_2(ξ)≤ϕ¯2(ξ)≤1+b2, ξ∈R;

they are twice differentiable except a set E containing finite points of R and(10)ϕ_1′ξ,ϕ¯1′ξ,ϕ_2′ξ,ϕ¯2′ξ,ϕ_1′′ξ,ϕ¯1′′ξ,ϕ_2′′ξ,ϕ¯2′′ξ

are continuous and bounded if ξ∈R∖E;

when x∈E, they satisfy(11)ϕ_1′ξ-≤ϕ_1′ξ+,ϕ_2′ξ-≤ϕ_2′ξ+,ϕ¯1′ξ+≤ϕ¯1′ξ-,ϕ¯2′ξ+≤ϕ¯2′ξ-;

they satisfy the following inequalities:(12)d1ϕ¯1′′ξ-cϕ¯1′ξ+r1ϕ¯1ξ1-ϕ¯1ξ-a1∫-τ0ϕ_1ξ+csdη11s-b1∫-τ0ϕ_2ξ+csdη12s≤0,d2ϕ¯2′′ξ-cϕ¯2′ξ+r2ϕ¯2ξ1-ϕ¯2ξ-a2∫-τ0ϕ_2ξ+csdη22s+b2∫-τ0ϕ¯1ξ+csdη21s≤0,d1ϕ_1′′ξ-cϕ_1′ξ+r1ϕ_1ξ1-ϕ_1ξ-a1∫-τ0ϕ¯1ξ+csdη11s-b1∫-τ0ϕ¯2ξ+csdη12s≥0,d2ϕ_2′′ξ-cϕ_2′ξ+r2ϕ_2ξ1-ϕ_2ξ-a2∫-τ0ϕ¯2ξ+csdη22s+b2∫-τ0ϕ_1ξ+csdη21s≥0

for ξ∈R∖E.

Then (5) has a positive solution (ϕ1(ξ),ϕ2(ξ)) such that(13)ϕ_1ξ≤ϕ1ξ≤ϕ¯1ξ,ϕ_2ξ≤ϕ2ξ≤ϕ¯2ξ,ξ∈R.

Remark 3.

In the above lemma, (ϕ¯1(ξ),ϕ¯2(ξ)) and (ϕ_1(ξ),ϕ_2(ξ)) 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 d1r1>d2r2. Then (1) of Theorem 1 holds.

Proof.

For simplicity, we shall denote c∗=2d1r1 by c and define(14)γ3=c+c2-4d2r22d2.Let K>0 be a constant such that

(-ξ+K)eγ1ξ is monotone if ξ≤0;

K>1 or supξ≤0(-ξ+K)eγ1ξ>1.

Moreover, select η>1 with (15)ηγ2<minγ2+γ12,2γ2,γ3

The admissibility of L, M, N, and K is clear by the limit behavior of these functions as ξ→-∞. Mathematically, we first fix K, then select M, and finally define N, L. Here, N and L are independent of each other.

We now define (17)ϕ¯1ξ=-ξ+Keγ1ξ,ξ≤ξ1,1,ξ≥ξ1,ϕ_1ξ=-ξ-L-ξeγ1ξ,ξ≤-L2,0,ξ>-L2,where ξ1<0 such that ϕ¯1(ξ) is continuous by (K1)-(K2) and (18)ϕ¯2ξ=mineγ2ξ+Meηγ2ξ,1+b2,ϕ_2ξ=max0,eγ2ξ-Neηγ2ξ.

If these functions satisfy (12), then our result holds by Lemma 2. Now, we are in a position of verifying these inequalities. For ϕ¯1(ξ), we shall prove the first inequality of (12) when ξ≠ξ1. If ξ>ξ1 and ϕ¯1(ξ)=1, then (19)1-ϕ¯1ξ-a1∫-τ0ϕ_1ξ+csdη11s-b1∫-τ0ϕ_2ξ+csdη12s≤0and the first inequality of (12) is clear. When ϕ¯1(ξ)=(-ξ+K)eγ1ξ<1, then (20)d1ϕ¯1′′ξ-cϕ¯1′ξ+r1ϕ¯1ξ=-ξ+Kd1γ12-cγ1+r1eγ1ξ-2d1γ1-ceγ1ξ=0and the verification on the first inequality of (12) is finished.

When the second inequality on ϕ¯2(ξ) is concerned, it is also clear if ϕ¯2(ξ)=1+b2<eγ2ξ+Meηγ2ξ. When ϕ¯2(ξ)=eγ2ξ+Meηγ2ξ<1+b2, then (M1) leads to (21)r2ϕ¯2ξ1-ϕ¯2ξ-a2∫-τ0ϕ_2ξ+csdη22s+b2∫-τ0ϕ¯1ξ+csdη21s≤r2ϕ¯2ξ1+b2ϕ¯1ξ≤r2eγ2ξ+Meηγ2ξ1+b2eγ1ξ/2=r2eγ2ξ+Meηγ2ξ+r2b2eγ2+γ1/2ξ+Meηγ2+γ1/2ξ.Note that (22)d2ϕ¯2′′ξ-cϕ¯2′ξ=d2γ22-cγ2eγ2ξ+Md2η2γ22-cηγ2eηγ2ξthen the definition of γ2 implies that the desired inequality is true if(23)Md2η2γ22-cηγ2+r2eηγ2ξ+r2b2eγ2+γ1/2ξ+Meηγ2+γ1/2ξ≤0or (24)2Md2η2γ22-cηγ2+r2+2r2b2eγ2+γ1/2-ηγ2ξ+Meγ1ξ/2≤0.On the one hand, (M2) leads to (25)Md2η2γ22-cηγ2+r2+2r2b2<0.At the same time, we have (26)d2η2γ22-cηγ2+r2+2r2b2eγ1ξ/2<0by (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 ϕ_1(ξ)=0. Otherwise, (27)ϕ_1′ξ=γ1ϕ_1ξ-eγ1ξ+L2-ξeγ1ξ,ϕ_1′′ξ=γ12ϕ_1ξ-2γ1eγ1ξ+Lγ1-ξeγ1ξ+L4-ξ3eγ1ξ.With these results, we obtain (28)d1ϕ_1′′ξ-cϕ_1′ξ+r1ϕ_1ξ=d1γ12-cγ1+r1ϕ_1ξ-2d1γ1-ceγ1ξ+Leγ1ξd1γ1-ξ-c2-ξ+d14-ξ3=Ld1eγ1ξ4-ξ3,r1ϕ_1ξ-ϕ_1ξ-a1∫-τ0ϕ¯1ξ+csdη11s-b1∫-τ0ϕ¯2ξ+csdη12s≥-r1ϕ_12ξ-r1a1ϕ¯1ξ∫-τ0ϕ¯1ξ+csdη11s-r1b1ϕ¯1ξ∫-τ0ϕ¯2ξ+csdη12s≥-r11+a1ϕ¯12ξ-2r1b1ϕ¯1ξϕ¯2ξ≥-r11+a1e2γ1′ξ-2r1b1eγ1′+γ2ξby (L1)-(L2). Therefore, it suffices to prove that (29)Ld1eγ1ξ4-ξ3≥r11+a1e2γ1′ξ+2r1b1eγ1′+γ2ξor (30)Ld14-ξ3≥r11+a1e2γ1′-γ1ξ+2r1b1eγ1′+γ2-γ1ξ,which is true by (L3).

We now consider ϕ_2(ξ), that is, the forth inequality of (12). When ϕ_2(ξ)>0, the definition implies (31)r2-ϕ_22ξ-a2ϕ_2ξ∫-τ0ϕ¯2ξ+csdη22s+b2ϕ_2ξ∫-τ0ϕ_1ξ+csdη21s≥r2-ϕ_22ξ-a2ϕ_2ξ∫-τ0ϕ¯2ξ+csdη22s≥r2-ϕ¯22ξ-a2ϕ¯2ξ∫-τ0ϕ¯2ξ+csdη22s≥-4r21+a2e2γ2ξby (N1) as well as (32)d2ϕ_2′′ξ-cϕ_2′ξ+r2ϕ_2ξ=eγ2ξd2γ22-cγ2+r2-Neηγ2ξd2η2γ22-cηγ2+r2=-Neηγ2ξd2η2γ22-cηγ2+r2.Thus, the desired inequality is true if (33)N>-4r21+a2d2η2γ22-cηγ2+r2+1>1since ξ<0 such that e2γ2ξ<eηγ2ξ, which holds by (N2). The proof is complete.

Lemma 5.

Assume that d1r1<d2r2. 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 c=2d2r2, let (34)γ3=c+c2-4d1r12d1.Fix η>1 such that (35)η∈1,min2,γ3γ1,γ1+γ2/2γ1.Select N1>1 such that (36)supξ<-1-ξ+N-ξeγ2ξ≥1+b2,N>N1.Let ξ2<-1 be the smaller root of -ξ+N-ξeγ2ξ=1+b2. Clearly, if N→∞, then ξ2→-∞.

By these constants, we define (37)ϕ¯1ξ=min1,eγ1ξ,ϕ_1ξ=max0,eγ1ξ-Qeηγ1ξ,ϕ¯2ξ=1+b2,ξ≥ξ2,-ξ+N-ξeγ2ξ,ξ<ξ2,ϕ_2ξ=0,ξ≥-R2,-ξ-R-ξeγ2ξ,ξ<-R2,where N, Q, and R are positive constants satisfying that

N>N1 is large such that -d2/4+r2b2-ξ5/2eγ1ξ/N+r2b2-ξ2eγ1ξ<0, ξ<ξ2;

Q≥Q1>1 such that eγ1ξ-Q1eηγ1ξ>0 implies -ξ+N-ξeγ2ξ<eγ2ξ/2;

Q≥r1(1+a1)+r1b1/-(d1η2γ12-cηγ1+r1)+Q1,

and

R0>1 is a constant such that ξ<-R02<-1 implies ϕ_2(ξ)≤ϕ¯2(ξ),

R≥R1>R0 such that ξ<-R12 implies ϕ¯2(ξ)<eγ2′ξ, where γ2′ satisfies(38)2γ2′-γ2>0,γ2′+γ1-γ2>0,γ2′∈0,γ2,

R>R2>R1 such that d2/4(-ξ)3-r2(1+a2)e(2γ2′-γ2)ξ/R2>0, ξ≤-R22.

On the first inequality of (12), if ϕ¯1(ξ)=1<eγ1ξ, then (39)1-ϕ¯1ξ-a1∫-τ0ϕ_1ξ+csdη11s-b1∫-τ0ϕ_2ξ+csdη12s≤0and the result is clear. If ϕ¯1(ξ)=eγ1ξ<1, then (40)1-ϕ¯1ξ-a1∫-τ0ϕ_1ξ+csdη11s-b1∫-τ0ϕ_2ξ+csdη12s≤1,d1ϕ¯1′′ξ-cϕ¯1′ξ+r1ϕ¯1ξ=d1γ12-cγ1+r1eγ1ξ=0,which completes the verification on ϕ¯1(ξ). On the second inequality, it is evident if ϕ¯2(ξ)=1+b2. When ϕ¯2(ξ)=-ξ+N-ξeγ2ξ<1+b2, we have (41)r2ϕ¯2ξ1-ϕ¯2ξ-a2∫-τ0ϕ_2ξ+csdη22s+b2∫-τ0ϕ¯1ξ+csdη21s≤r2ϕ¯2ξ1+b2ϕ¯1ξ=r2ϕ¯2ξ+r2b2ϕ¯2ξeγ1ξ,ϕ¯2′ξ=γ2ϕ¯2ξ-eγ2ξ-Neγ2ξ2-ξϕ¯2′′ξ=γ22ϕ¯2ξ-2γ2eγ2ξ-Nγ2eγ2ξ-ξ-Neγ2ξ4-ξ3/2.Therefore, (42)d2ϕ¯2′′ξ-cϕ¯2′ξ+r2ϕ¯2ξ×1-ϕ¯2ξ-a2∫-τ0ϕ_2ξ+csdη22s+b2∫-τ0ϕ¯1ξ+csdη21s≤d2γ22ϕ¯2ξ-2γ2eγ2ξ-Nγ2eγ2ξ-ξ-Neγ2ξ4-ξ3/2-cγ2ϕ¯2ξ-eγ2ξ-Neγ2ξ2-ξ+r2ϕ¯2ξ+r2b2ϕ¯2ξeγ1ξ=-d2Neγ2ξ4-ξ3/2+r2b2-ξ+N-ξeγ2ξeγ1ξ=eγ2ξ-d2N4-ξ3/2+r2b2-ξ+N-ξeγ1ξ=eγ2ξ-ξ3/2-d2N4+r2b2-ξ5/2+N-ξ2eγ1ξ=Neγ2ξ-ξ3/2-d24+r2b2-ξ5/2eγ1ξN+r2b2-ξ2eγ1ξ≤0by (N1).

On the third inequality, it is clear if ϕ_1(ξ)=0. When eγ1ξ-Qeηγ1ξ>0, (Q1) implies (43)r1ϕ_1ξ-ϕ_1ξ-a1∫-τ0ϕ¯1ξ+csdη11s-b1∫-τ0ϕ¯2ξ+csdη12s≥-r1ϕ_12ξ-r1a1ϕ¯1ξ∫-τ0ϕ¯1ξ+csdη11s-r1b1ϕ¯1ξ∫-τ0ϕ¯2ξ+csdη12s≥-r11+a1ϕ¯12ξ-r1b1ϕ¯1ξϕ¯2ξ≥-r11+a1e2γ1ξ-r1b1eγ1+γ2/2ξ.Since (44)d1ϕ_1′′ξ-cϕ_1′ξ+r1ϕ_1ξ=eγ1ξd1γ12-cγ1+r1-Qeηγ1ξd1η2γ12-cηγ1+r1=-Qeηγ1ξd1η2γ12-cηγ1+r1then the third is true by (Q2).

We now consider the fourth inequality, which is clear if ϕ_2(ξ)=0. When ϕ_2(ξ)>0, we have (45)ϕ_2′ξ=γ2ϕ_2ξ-eγ2ξ+R2-ξeγ2ξ,ϕ_2′′ξ=γ22ϕ_2ξ-2γ2eγ2ξ+Rγ2-ξeγ2ξ+R4-ξ3eγ2ξ,which implies (46)d2ϕ_2′′ξ-cϕ_2′ξ+r2ϕ_2ξ=d2γ22-cγ2+r2ϕ_2ξ-2d2γ2-ceγ2ξ+Reγ2ξd2γ2-ξ-c2-ξ+d24-ξ3=Rd2eγ2ξ4-ξ3.Moreover, (R0) and (R1) imply that (47)r2ϕ_2ξ-ϕ_2ξ-a2∫-τ0ϕ¯2ξ+csdη22s+b2∫-τ0ϕ_1ξ+csdη21s≥-r21+a2ϕ¯22ξ≥-r21+a2e2γ2′ξ,and so (48)Rd2eγ2ξ4-ξ3-r21+a2e2γ2′ξ=Reγ2ξd24-ξ3-r21+a2e2γ2′-γ2ξR>0by (R2), which completes the verification and proof.

Lemma 6.

Assume that d1r1=d2r2. Then (3) of Theorem 1 is true.

Proof.

Utilizing the parameters similar to those in Lemmas 4 and 5, we define (49)ϕ¯1ξ=-ξ+Keγ1ξ,ξ≤ξ1,1,ξ≥ξ1,ϕ_1ξ=-ξ-L-ξeγ1ξ,ξ≤-L2,0,ξ>-L2,ϕ¯2ξ=-ξ+N-ξeγ2ξ,ξ≤ξ2,1+b2,ξ≥ξ2,ϕ_2ξ=-ξ-R-ξeγ2ξ,ξ≤-R2,0,ξ>-R2,where ξ1<0 and ξ2<0 such that ϕ¯1(ξ) and ϕ¯2(ξ) 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 τ=0, 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 ξ→-∞. However, when the minimal wave speed is concerned, it does not decay exponentially as ξ→-∞.

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).

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