Abstract
This paper is concerned with the invasion traveling wave solutions of a LotkaVolterra type competition system with nonlocal dispersal, the purpose of which is to formulate the dynamics between the resident and the invader. By constructing upper and lower solutions and passing to a limit function, the existence of traveling wave solutions is obtained if the wave speed is not less than a threshold. When the wave speed is smaller than the threshold, the nonexistence of invasion traveling wave solutions is proved by the theory of asymptotic spreading.
MSC: 35C07, 35K57, 37C65.
Keywords:
comparison principle; asymptotic spreading; upper and lower solutions; invasion waves1 Introduction
In the past decades, much attention has been paid to the spatial propagation modes of the following LotkaVolterra type diffusion system:
in which all the parameters are positive and
In particular, if
In this paper, we consider the minimal wave speed of traveling wave solutions in the following nonlocal dispersal system (see Yu and Yuan [19]):
in which
(J1)
(J2) for any
(J3)
In (1.2), the spatial migration of individuals is formulated by the socalled dispersal operator, which has significant sense in population dynamics. For example, in the patch models of population dynamics [20], the rate of immigration into a patch from a particular other patch is usually taken as proportional to the local population, and the dispersal can be regarded as the extension of these ideas to a continuous media model. Such a diffusion mechanism also arises from physics processes with long range effect and other disciplines [13], and the dynamics of evolutionary systems with dispersal effect has been widely studied in recent years; we refer to [13,2132] and the references cited therein.
Hereafter, a traveling wave solution of (1.2) is a special solution of the form
where
Moreover, we also require the following asymptotic boundary conditions:
From the viewpoint of ecology, a traveling wave solution satisfying (1.4)(1.5) can
model the population invasion process: at any fixed
To obtain the existence of (1.4)(1.5) if the wave speed is larger than a threshold
depending on
In addition, Li and Lin [45] and Zhang et al.[46] investigated the existence of positive traveling wave solutions of (1.2) for
The rest of this paper is organized as follows. In Section 2, we give some preliminaries. By constructing upper and lower solutions and using a limit process, the existence of traveling wave solutions is established in Section 3. In the last section, we obtain the nonexistence of traveling wave solutions.
2 Preliminaries
In this paper, we shall use the standard partial order in
then X is a Banach space equipped with the standard supremum norm. If
In order to apply the comparison principle, we first make a change of variables to
obtain a cooperative system. Let
At the same time, (1.5) will be
Take
then
Clearly, a fixed point of
Definition 2.1 Assume that
for
Using Pan et al.[33], Theorem 3.2, we obtain the following conclusion.
Lemma 2.2Assume that
(P1)
(P2)
(P3)
Then (2.1)(2.2) has a positive monotone solution
We now consider the following initial value problem:
where J satisfies (J1) to (J3),
In addition, let
In Jin and Zhao [34], the authors investigated the asymptotic spreading of a periodic population model with spatial dispersal. Note that the parameters in (2.4) are positive constants, then [34], Theorem 2.1, implies the following result.
Lemma 2.3Assume that
In particular, if
Furthermore, we can also apply the results of Jin and Zhao [34], Theorem 3.5, since the assumptions (H1) and (H2) of [34] are clear. Define
Then Jin and Zhao [34], Theorem 3.5, indicates the following conclusion.
Lemma 2.4Assume that
where
3 Existence of traveling wave solutions
In this section, we shall prove the existence of positive solutions of (2.1)(2.2). Let
for any
Lemma 3.1There exists a constant
(1) For each
(2) If
(3) If
The above result is clear and we omit the proof here. Using these constants, we can prove the following conclusion.
Theorem 3.2Assume that
(1)
(2)
Then (2.1)(2.2) has a monotone solution.
Proof Define continuous functions as follows:
Claim A:
Moreover, let
and
Evidently,
If
such that
which completes the proof on
We now consider
and
Therefore, (3.1) leads to
If
Therefore, Claim A is true. The proof is complete. □
Theorem 3.3Assume that one of the following items holds.
(1)
(2)
Then (2.1)(2.2) has a monotone solution with
Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence
for any n. By the AscoliArzela lemma and a standard nested subsequence argument (see, e.g., Thieme and Zhao [47]), there exists a subsequence of
and the convergence in s is uniform for
(T1)
(T2)
(T3)
The items (T1) to (T3) further indicate that
From (T1), it is clear that
If
Using the dominated convergence theorem in
(L1)
(L2)
If (L1) is true, then the dominated theorem in
which implies a contradiction. If (L2) is true, then
which is also a contradiction. What we have done implies that
If
has a monotone solution, which is impossible. Therefore,
Thus,
4 Nonexistence of traveling wave solutions
In this section, we shall formulate the nonexistence of invasion traveling wave solutions of (1.2) by the theory of asymptotic spreading. Before this, we first present a comparison principle formulated by Jin and Zhao [34], Theorem 2.3.
Lemma 4.1Assume that
then
We now give the main result of this section.
Theorem 4.2If
Proof Define
Then
If (2.1)(2.2) has a positive solution
implies that
with the following asymptotic boundary condition:
Recalling the definition of traveling wave solutions, we see that
and
Using Lemmas 2.4 and 4.1, we see that
since
However, the boundary condition (4.3) indicates that
and
which implies a contradiction between (4.6) and (4.7). The proof is complete. □
Remark 4.3 Under proper assumptions, we have obtained the threshold of the existence of positive solutions to (2.1)(2.2).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main results in this article were derived by SP and GL. All authors read and approved the final manuscript.
Acknowledgements
The authors express their thanks to the referees for their helpful comments and suggestions on the manuscript. This work was partially supported by the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (1010ZCX019), NSF of China (11101094) and FRFCU (lzujbky2011k27).
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