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

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]:

[mathematical expression not reproducible], (1)

where x [member of] R, t > 0, [d.sub.1] > 0, [d.sub.2] > 0, [r.sub.1] > 0, [r.sub.2] > 0, [u.sub.1] [member of] R, [u.sub.2] [member of] R, and

[mathematical expression not reproducible], (2)

in which [a.sub.1] [greater than or equal to] 0, [a.sub.2] [greater than or equal to] 0, [b.sub.1] > 0, [b.sub.2] [greater than or equal to] 0, and [tau] [greater than or equal to] 0 are constants such that

[mathematical expression not reproducible]. (3)

For (1), a traveling wave solution is a special solution with the form

[u.sub.i] (x, t) = [[phi].sub.i] ([xi]), [xi] = x + ct, i = 1, 2, (4)

where ([[phi].sub.1], [[phi].sub.2]) [member of] [C.sup.2](R, [R.sup.2]) is the wave profile and c > 0 is the wave speed. Therefore, ([[phi].sub.1], [[phi].sub.2]) and c satisfy

[mathematical expression not reproducible] (5)

with

[mathematical expression not reproducible]. (6)

In Pan [8], the author defined a threshold given by

[c.sup.*] := max {2[square root of ([d.sub.1][r.sub.1])], 2[square root of ([d.sub.2][r.sub.2])]} (7)

and showed the existence (nonexistence) of desired traveling wave solutions if the wave speed c > [c.sup.*] (c < [c.sup.*]). When the wave speed c = [c.sup.*], 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.sup.*]. Since Pan [8, Theorem 3.5] also holds when c = [c.sup.*], we shall not investigate the limit behavior as [xi] [right arrow] and focus on the existence of positive solution of (5) satisfying

[mathematical expression not reproducible]. (8)

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.sup.*], we define

[mathematical expression not reproducible]. (9)

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

Theorem 1. Assume that c = [c.sup.*] holds. Then (5) admits a bounded positive solution ([[phi].sub.1], [[phi].sub.2]) satisfying

(1) [mathematical expression not reproducible];

(2) [mathematical expression not reproducible];

(3) [mathematical expression not reproducible].

We shall prove the result by three lemmas, which will study three cases [d.sub.1][r.sub.1] > [d.sub.2][r.sub.2], [d.sub.1][r.sub.1] < [d.sub.2][r.sub.2], and [d.sub.1][r.sub.1] = [d.sub.2][r.sub.2]. For this purpose, we first show the following result in Lin and Ruan [12].

Lemma 2. Suppose that [[[phi].bar].sub.1]([xi), [[bar.[phi]].sub.i]([xi]), [[[phi].bar].sub.2]([xi]), and [[bar.[phi]].sub.2]([xi]) are continuous functions and

(A1) [mathematical expression not reproducible];

(A2) they are twice differentiable except a set E containing finite points of R and

[mathematical expression not reproducible] (10)

are continuous and bounded if [xi] [member of] R \ E;

(A3) when x [member of] E, they satisfy

[mathematical expression not reproducible]; (11)

(A4) they satisfy the following inequalities:

[mathematical expression not reproducible] (12)

for [xi] [member of] R \ E.

Then (5) has a positive solution ([[phi].sub.1]([xi]), [[phi].sub.2]([xi)) such that

[mathematical expression not reproducible]. (13)

Remark 3. In the above lemma, ([[bar.[phi]].sub.1]([xi]), [[bar.[phi]].sub.2]([xi])) and ([[[phi].bar].sub.1]([xi]), [[[phi].bar].sub.2]([xi])) 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 [d.sub.1][r.sub.1] > [d.sub.2][r.sup.2]. Then (1) of Theorem 1 holds.

Proof. For simplicity, we shall denote [c.sup.*] = 2 [square root of ([d.sub.1][r.sub.1])] by c and define

[[gamma].sub.3] = c + [square root of ([c.sup.2] - 4[d.sub.2][r.sub.2])]/2[d.sub.2]. (14)

Let K > 0 be a constant such that

(K1) [mathematical expression not reproducible];

(K2) [mathematical expression not reproducible].

Moreover, select [eta] > 1 with

[eta][[gamma].sub.2] < min {[[gamma].sub.2] + [[gamma].sub.1]/2, 2[[gamma].sub.2], [[gamma].sub.3]} (15)

and M > 1 + [b.sub.2] such that

(M1) [mathematical expression not reproducible];

(M2) [mathematical expression not reproducible];

(M3) [mathematical expression not reproducible],

and N > 1 such that

(N1) [mathematical expression not reproducible];

(N2) N > -4[r.sub.2](1 + [a.sub.2])/([d.sub.2][[eta].sup.2][[gamma].sup.2.sub.2] - c[eta][[gamma].sub.2] + [r.sub.2]) + 1.

Select L > 1 such that

(L1) [mathematical expression not reproducible] such that

2[[gamma]'.sub.1] - [[gamma].sub.1] > 0, [[gamma]'.sub.1] + [[gamma].sub.2] - [[gamma].sub.1] > 0; (16)

(L2) [mathematical expression not reproducible];

(L3) [mathematical expression not reproducible].

The admissibility of L, M, N, and K is clear by the limit behavior of these functions as [xi] [member of] -[infinity]. 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

[mathematical expression not reproducible], (17)

where [[xi].sub.1] < 0 such that [[bar.[phi]].sub.1] ([xi) is continuous by (K1)-(K2) and

[mathematical expression not reproducible]. (18)

If these functions satisfy (12), then our result holds by Lemma 2. Now, we are in a position of verifying these inequalities. For [[bar[phi]].sub.1]([xi]), we shall prove the first inequality of (12) when [xi] [not equal to] [[xi].sub.1]. If [xi] > [[xi].sub.1] and [[bar.[phi]].sub.1] = 1, then

[mathematical expression not reproducible] (19)

and the first inequality of (12) is clear. When [mathematical expression not reproducible], then

[mathematical expression not reproducible] (20)

and the verification on the first inequality of (12) is finished.

When the second inequality on [[bar.[phi]].sub.2]([xi]) is concerned, it is also clear if [mathematical expression not reproducible]. When [mathematical expression not reproducible], then (M1) leads to

[mathematical expression not reproducible]. (21)

Note that

[mathematical expression not reproducible] (22)

then the definition of [[gamma].sub.2] implies that the desired inequality is true if

[mathematical expression not reproducible] (23)

or

[mathematical expression not reproducible]. (24)

On the one hand, (M2) leads to

M ([d.sub.2][[eta].sup.2][y.sup.2.sub.2] - c[eta][[gamma].sub.2] + [r.sub.2]) + 2[r.sub.2][b.sub.2] < 0. (25)

At the same time, we have

[mathematical expression not reproducible] (26)

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 [[bar.[phi]].sub.1]([xi]) = 0. Otherwise,

[mathematical expression not reproducible]. (27)

With these results, we obtain

[mathematical expression not reproducible] (28)

by (L1)-(L2). Therefore, it suffices to prove that

[mathematical expression not reproducible] (29)

or

[mathematical expression not reproducible], (30)

which is true by (L3).

We now consider [[bar.[phi]].sub.2]([xi]), that is, the forth inequality of (12). When [[bar.[phi]].sub.2]([xi]) > 0, the definition implies

[mathematical expression not reproducible] (31)

by (N1) as well as

[mathematical expression not reproducible] (32)

Thus, the desired inequality is true if

[mathematical expression not reproducible] (33)

since [xi] < 0 such that [mathematical expression not reproducible], which holds by (N2). The proof is complete.

Lemma 5. Assume that [d.sub.1][r.sub.1] < [d.sub.2][r.sub.2]. 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 = 2[square root of ([d.sub.2][r.sub.2])], let

[[gamma].sub.3] = c + [square root of ([c.sup.2] - 4[d.sub.1][r.sub.1])]/2[d.sub.1]. (34)

Fix [eta] > 1 such that

[mathematical expression not reproducible]. (35)

Select [N.sub.1] > 1 such that

[mathematical expression not reproducible]. (36)

Let [[xi].sub.2] < -1 be the smaller root of [mathematical expression not reproducible]. Clearly, if N [right arrow] [infinity], then [[xi].sub.2] [right arrow] -[infinity].

By these constants, we define

[mathematical expression not reproducible], (37)

where N, Q, and R are positive constants satisfying that

(N1) [mathematical expression not reproducible];

(Q1) [mathematical expression not reproducible];

(Q2) [mathematical expression not reproducible],

and

(R0) [R.sub.0] > 1 is a constant such that [xi] < -[R.sup.2.sub.0] < -1 implies [[[phi].bar].sub.2]([xi]) [less than or equal to] [[bar.[phi]].sub.2]([xi]),

(R1) R [greater than or equal to] [R.sub.1] > [R.sub.0] such that [xi] < -[R.sup.2.sub.1] implies [mathematical expression not reproducible], where [y'.sub.2] satisfies

[mathematical expression not reproducible], (38)

(R2) R > [R.sup.2] > [R.sub.1] such that [mathematical expression not reproducible].

On the first inequality of (12), if [mathematical expression not reproducible], then

[mathematical expression not reproducible] (39)

and the result is clear. If [mathematical expression not reproducible], then

[mathematical expression not reproducible], (40)

which completes the verification on [[bar.[phi]].sub.1]([xi]). On the second inequality, it is evident if [[bar.[phi]].sub.2]([xi]) = 1 + [b.sub.2]. When [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (41)

Therefore,

[mathematical expression not reproducible]. (42)

by (N1).

On the third inequality, it is clear if [[[phi].bar].sub.1]([xi]) = 0. When [mathematical expression not reproducible] implies

[mathematical expression not reproducible]. (43)

Since

[mathematical expression not reproducible] (44)

then the third is true by (Q2).

We now consider the fourth inequality, which is clear if [[[phi].bar].sub.2]([xi]) = 0. When [[[phi].bar].sub.2]([xi]) > 0, we have

[mathematical expression not reproducible] (45)

which implies

[mathematical expression not reproducible]. (46)

Moreover, (R0) and (R1) imply that

[mathematical expression not reproducible], (47)

and so

[mathematical expression not reproducible] (48)

by (R2), which completes the verification and proof.

Lemma 6. Assume that [d.sub.1][r.sub.1] = [d.sub.2][r.sub.2]. Then (3) of Theorem 1 is true.

Proof. Utilizing the parameters similar to those in Lemmas 4 and 5, we define

[mathematical expression not reproducible], (49)

where [[xi].sub.1] < 0 and [[xi].sub.2] < 0 such that [[bar.[phi]].sub.1]([xi]) and [[bar.[phi]].sub.2]([xi]) 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 [tau] = 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 [xi] [right arrow] -[infinity].

However, when the minimal wave speed is concerned, it does not decay exponentially as [xi] [right arrow] -[infinity].

https://doi.org/10.1155/2018/4873803

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).

References

[1] M. R. Owen and M. A. Lewis, "How predation can slow, stop or reverse a prey invasion," Bulletin of Mathematical Biology, vol. 63, no. 4, pp. 655-684, 2001.

[2] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, UK, 1997.

[3] G. Lin, "Invasion traveling wave solutions of a predator-prey system," Nonlinear Analysis. Theory, Methods & Applications, vol. 96, pp. 47-58, 2014.

[4] S. Pan, "Invasion speed of a predator-prey system," Applied Mathematics Letters, vol. 74, pp. 46-51, 2017.

[5] S. R. Dunbar, "Travelling wave solutions of diffusive Lotka-Volterra equations," Journal of Mathematical Biology, vol. 17, no. 1, pp. 11-32, 1983.

[6] S. R. Dunbar, "Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4," Transactions of the American Mathematical Society, vol. 286, no. 2, pp. 557-594, 1984.

[7] S. R. Dunbar, "Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits," SIAM Journal on Applied Mathematics, vol. 46, no. 6, pp. 1057-1078, 1986.

[8] S. Pan, "Convergence and traveling wave solutions for a predator-prey system with distributed delays," Mediterranean Journal of Mathematics, vol. 14, no. 3, 15 pages, 2017.

[9] J.-H. Huang and X.-F. Zou, "Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity," Acta Mathematicae Applicatae Sinica, vol. 22, no. 2, pp. 243-256, 2006.

[10] K. Li and X. Li, "Travelling wave solutions in diffusive and competition-cooperation systems with delays," IMA Journal of Applied Mathematics, vol. 74, no. 4, pp. 604-621, 2009.

[11] G. Lin, W.-T. Li, and M. Ma, "Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models," Discrete and Continuous Dynamical Systems--Series B, vol. 19, no. 2, pp. 393-414, 2010.

[12] G. Lin and S. Ruan, "Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays," Journal of Dynamics and Differential Equations, vol. 26, no. 3, pp. 583-605, 2014.

[13] S.-C. Fu, "Traveling waves for a diffusive SIR model with delay," Journal of Mathematical Analysis and Applications, vol. 435, no. 1, pp. 20-37, 2016.

[14] G. Lin, "Minimal wave speed of competitive diffusive systems with time delays," Applied Mathematics Letters, vol. 76, pp. 164-169, 2018.

[15] G. Lin, "Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator," Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 7, pp. 2448-2461, 2011.

[16] S. Pan, "Asymptotic spreading in a Lotka-Volterra predator-prey system," Journal of Mathematical Analysis and Applications, vol. 407, no. 2, pp. 230-236, 2013.

Fuzhen Wu (iD) and Dongfeng Li

Department of Basic, Zhejiang University of Water Resources and Electric Power, Hangzhou, Zhejiang 310018, China

Correspondence should be addressed to Fuzhen Wu; fuzhwu@yeah.net

Received 14 March 2018; Accepted 6 May 2018; Published 31 May 2018

Academic Editor: Nikos I. Karachalios
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Title Annotation:Research Article
Author:Wu, Fuzhen; Li, Dongfeng
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
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