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Traveling wave solutions in a Lotka-Volterra type competition recursion
Advances in Difference Equations volume 2014, Article number: 173 (2014)
This paper is concerned with the traveling wave solutions of a competitive integrodifference system with Lotka-Volterra type nonlinearity. The existence of traveling wave solutions is proved by constructing generalized upper and lower solutions. The asymptotic behavior of traveling wave solutions is established by combining the theory of asymptotic spreading with the idea of contracting rectangles. The nonexistence of monotone traveling wave solutions is also confirmed by the theory of asymptotic spreading.
In this paper, we study the traveling wave solutions of the following integrodifference system with Lotka-Volterra type nonlinearity:
in which , and , , , are constants. For , is the probability function describing the random walk of individuals under consideration, and it is the so-called kernel function. In particular, we take the following conditions in this paper:
(A1) is Lebesgue measurable and integrable on ℝ and ;
(A2) , , and for each , .
For the parameters in (1.1), we also give the following assumptions:
(A4) , , where
(A5) , ;
(A6) , ;
(A7) , .
We now give some illustration of (A3)-(A7) from the viewpoint of both the literature and the population dynamics of the corresponding difference system
If the inter-specific vanishes () and (A3) holds in (1.2), then or and is persistent in population dynamics (see Murray [, Section 2.3] for the dynamics). Condition (A4) provides a positive invariant region of the difference system (1.2), namely, . (A5) indicates that the inter-specific is weak, namely, even if the competitor () takes the maximal value (), the species () still persists. The coexistence steady state of (1.2) exists if (A6) holds. (A7) leads to a comparison principle appealing to the difference system (1.2) in . Finally, if (A3)-(A7) hold, then (1.2) has a stable steady state. In particular, there are different parameters such that part or all of (A3)-(A7) hold. For example, if , , , then (1.2) satisfies (A3)-(A7); if , , , then (1.2) satisfies (A3)-(A6), if , , , then (A3) and (A4) hold.
Hereafter, a traveling wave solution of (1.1) is a special solution having the form
in which is the wave speed and describes the wave profile that propagates in the media ℝ. Then and c must satisfy
for . To describe determinative transition processes between different states, proper asymptotic behavior satisfied by the traveling wave solutions is often necessary. In this paper, we require the following asymptotic boundary conditions:
Furthermore, we also consider the following stronger ones:
provided that (A6) is true. In population dynamics, (1.3) with (1.4) or (1.3) with (1.5) could formulate the successful invasion of two competitors.
To study the existence of traveling wave solutions of competitive recursions of two competitors, Lin et al.  established an abstract scheme and the existence of traveling wave solutions was reduced to the existence of upper and lower solutions. Since the competitive system in  does not generate monotone semiflows when the synchronous invasion of two competitors is concerned, the asymptotic behavior of traveling wave solutions cannot be confirmed by the monotonicity of them (see [3–15] for monostable traveling wave solutions of (local) monotone recursions). In , the asymptotic behavior of traveling wave solutions was obtained by that of upper and lower solutions. The method was also applied to several competitive systems; see [16–18] and the references cited therein. Without the requirements of upper and lower solutions, it is difficult to obtain the asymptotic behavior of traveling wave solutions . In fact, for coupled systems with general kernels, it is not an easy job to construct proper upper and lower solutions satisfying the asymptotic behavior in , and [16–18] just obtained the existence of traveling wave solutions of some systems with special kernels. Although the kernel functions in [2, 16–18] satisfy some special conditions, the verification of upper and lower solutions is still very complex, and the nonexistence of nontrivial traveling wave solutions of models in [2, 16–18] remains open.
To simplify the construction of upper and lower solutions and provide a more general result of the existence of traveling wave solutions of recursions, Lin  further considered the traveling wave solutions of recursions and gave some simpler conditions. By the theory in , the existence of traveling wave solutions can be obtained by the existence of upper and lower solutions which are easy to construct. Moreover, by the properties of the corresponding difference systems, the asymptotic behavior of traveling wave solutions was also studied. Moreover, for the model investigated by [2, 17], Lin  also obtained the nonexistence of nontrivial traveling wave solutions by the theory of asymptotic spreading.
In this paper, we shall establish the existence and nonexistence of (1.3) with (1.4) or (1.3) with (1.5), and we present the corresponding mathematical results by the idea in . In particular, we shall not take special general kernels , , and just add conditions (A1)-(A2) in what follows.
The rest of this paper is organized as follows. In Section 2, we investigate the existence of traveling wave solutions by constructing upper and lower solutions and applying Schauder’s fixed point theorem in a functional space equipped with the decay norm. In Section 3, the asymptotic boundary conditions (1.4) and (1.5) will be considered by combining the theory of asymptotic spreading with the idea of contracting rectangles in . Finally, the nonexistence of monotone traveling wave solutions is proved, which indicates that the threshold in the paper is the minimal wave speed of monotone traveling wave solutions of (1.1).
2 Existence of traveling wave solutions
In this paper, we shall utilize standard partial ordering in . Also let X be the set of uniformly continuous and bounded functions from ℝ to . Moreover, we denote
Let be the supremum norm in and be a constant. We define
and the decay norm
Then is a Banach space.
Before discussing (1.1), we first consider the corresponding difference equation
Lemma 2.1 For (2.1), we have the following conclusions:
(D1) if (A3)-(A4) hold, then is invariant, namely, if , , then , , ;
(D2) if (A3), (A4), and (A7) are true and , , then for each , () is monotone increasing in () and monotone decreasing in ().
We now introduce some constants. Denote
for and . Then the following result holds.
Lemma 2.2 There exists a positive constant such that implies that for any or for any . If , then has at least one positive root such that and for , . Moreover, there exists such that for all , .
By Liang and Zhao [, Lemma 3.8], Lemma 2.2 is clear and is defined by
In what follows, we suppose that is fixed. If , then we define by
In particular, P also admits the following properties.
Lemma 2.3 .
Lemma 2.3 is clear by Lemma 2.1 and (A1)-(A2), and we omit the proof here. Clearly, a fixed point of P in is a solution to (1.3). Therefore, it suffices to prove the existence of the fixed points of P by Schauder’s fixed point theorem, and we first construct a potential set of wave profiles.
For the purpose, we define continuous functions
where is a positive constant clarified later and such that
With these notations, we define the following set of potential wave profiles:
which satisfies the following nice properties.
Lemma 2.4 If is large such that
then Γ is convex and nonempty. Moreover, it is closed and bounded with respect to the decay norm .
Lemma 2.5 Assume that (A3)-(A4) hold. If is large, then .
Proof By Lemma 2.1, it suffices to verify that
If , then
If , then Lemma 2.1 implies that
and we have verified that
In a similar way, we can prove that
If , then
by (A4) and
is clear. Otherwise, leads to
In a similar way, when
is true, we have
The proof is complete. □
Lemma 2.6 Assume that (A3)-(A4) hold. Then is complete continuous in the sense of the decay norm .
The proof is provided by Lin [, Lemma 3.4] and we omit it here.
Theorem 2.7 Assume that (A3)-(A4) hold. Then for each , (1.3) has a positive solution such that
The result is evident by Schauder’s fixed point theorem and Lemmas 2.4-2.6, and we omit the proof here.
3 Asymptotic behavior of traveling wave solutions
In this section, we investigate the asymptotic behavior of positive solutions of (1.3). We first consider the initial value problem
(U1) for some , is Lipschitz continuous and monotone increasing;
(U2) there exists such that , , ; if , then , ;
(U4) and , .
In literature, (U1)-(U4) imply the comparison principle, monostability, and persistence in (3.1). More precisely, by Hsu and Zhao , we have the following conclusion.
Lemma 3.1 Assume that g satisfies (A1)-(A2).
If is bounded and uniformly continuous such that
then . Moreover, let satisfy
then for each , we have
If is bounded and uniformly continuous for each n such that
then , , .
Theorem 3.2 Assume that (A3)-(A5) hold. If is given by Theorem 2.7, then (1.4) is true.
Proof Note that a traveling wave solution is a special invariant solution; then satisfies the following inequality:
for all . Define
Clearly, admits the following properties:
(b1) is Lipschitz continuous and monotone increasing;
(b2) , ;
(b3) there exists such that and
(b4) if , then , .
Therefore, satisfies (U1)-(U4) and Lemma 3.1 implies that
In a similar way, we have
The proof is complete. □
Theorem 3.3 Assume that (A3)-(A7) hold. If is given by Theorem 2.7, then (1.5) is true.
Proof By what we have done, we see that
Applying the dominated convergence theorem in P when , then the monotone condition (A7) implies that
By (3.2) and (3.4), we have
Furthermore, (3.3) and (3.5) lead to
Since (A6) implies that , then we have
by (3.6) and (3.7). Again by (A6), we see that
The proof is complete. □
Remark 3.4 Although we did not construct the contracting rectangle (see ) in this paper, the proof of Theorem 3.3 was motivated by Lin [, Sections 4-5]. Of course, if a model involves more unknown functions, it is difficult to obtain the asymptotic behavior of traveling wave solutions by the inequalities similar to (3.6) and (3.7).
4 Nonexistence of monotone traveling wave solutions
In this section, we confirm that is the minimal wave speed of monotone invasion traveling wave solutions by presenting the following nonexistence of monotone traveling wave solutions.
Theorem 4.1 Assume that (A3)-(A6) hold. If , then there is no strict monotone satisfying (1.3)-(1.4) and
Proof Without loss of generality, we assume that and
Were the statement false, then there exists such that there is satisfying (1.3)-(1.4) and (4.1).
Since , are monotone increasing, then
are one to one and there is a continuous mapping h such that
in which satisfying
(H2) is continuous and strictly monotone decreasing;
At the same time, h also leads to
Let , then
Note that is a special solution to
Define , then is monotone and continuous for . Moreover, applying Lemma 3.1 to (4.2), we see that . Again by Lemma 3.1, we obtain
A contradiction occurs. The proof is complete. □
Before ending this paper, we make the following remark.
Remark 4.2 By what we have done, is the minimal wave speed of monotone traveling wave solutions of (1.1). However, when the wave speed is , the existence and nonexistence of nontrivial traveling wave solutions remain open. At the same time, if we remove the monotonicity of traveling wave solutions in Theorem 4.1, then we believe that the result still holds. Clearly, for these two problems, we cannot discuss them directly by the methods similar to those in this paper, and we shall consider these problems in our future studies.
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The first author was supported by NSF of Gansu Province of China (No. 1208RJYA004) and the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (No. 1010ZCX019), the second author was supported by National Undergraduate Training Programs for Innovation and Entrepreneurship (No. 201310730086).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.