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Invasion traveling wave solutions of a competitive system with dispersal

Shuxia Pan1* and Guo Lin2

Author affiliations

1 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

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Citation and License

Boundary Value Problems 2012, 2012:120  doi:10.1186/1687-2770-2012-120


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/120


Received:31 January 2012
Accepted:8 October 2012
Published:24 October 2012

© 2012 Pan and Lin; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the invasion traveling wave solutions of a Lotka-Volterra 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 waves

1 Introduction

In the past decades, much attention has been paid to the spatial propagation modes of the following Lotka-Volterra type diffusion system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M1">View MathML</a>

(1.1)

in which all the parameters are positive and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5">View MathML</a> are two competitors. Many investigators considered its traveling wave solutions connecting different spatial homogeneous steady states such as the existence, monotonicity, minimal wave speed and stability; see [1-16].

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M6">View MathML</a> holds in (1.1), then the corresponding reaction system has a stable equilibrium <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M7">View MathML</a> and an unstable one <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M8">View MathML</a>. With the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M6">View MathML</a>, many papers including [2,3,5,6,8,16] studied the traveling wave solutions connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M8">View MathML</a>. These traveling wave solutions can formulate the spatial exclusive process between the resident <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5">View MathML</a> and the invader <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4">View MathML</a> so that the minimal wave speed reflecting the invasion speed of the invader becomes a hot topic in these works; we refer to Shigesada and Kawasaki [17] for some examples of the corresponding biological records and the literature importance of invasion speed. Moreover, the similar problem was also discussed in different spatial media such as the lattice differential systems in Guo and Liang [4], Guo and Wu [18].

In this paper, we consider the minimal wave speed of traveling wave solutions in the following nonlocal dispersal system (see Yu and Yuan [19]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M14">View MathML</a>

(1.2)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M18">View MathML</a> denote the densities of two competitors at time t and location <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M19">View MathML</a>, all the parameters are positive and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M20">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>, are probability functions formulating the random dispersal of individuals and satisfy the following assumptions:

(J1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M23">View MathML</a> is nonnegative and Lebesgue measurable for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>;

(J2) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>;

(J3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>.

In (1.2), the spatial migration of individuals is formulated by the so-called 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 [13,21-32] and the references cited therein.

Hereafter, a traveling wave solution of (1.2) is a special solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M33">View MathML</a> is the wave speed at which the wave profile <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M34">View MathML</a> propagates in spatial media ℝ. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M33">View MathML</a> must satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M37">View MathML</a>

(1.4)

Moreover, we also require the following asymptotic boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M38">View MathML</a>

(1.5)

From the viewpoint of ecology, a traveling wave solution satisfying (1.4)-(1.5) can model the population invasion process: at any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M19">View MathML</a>, only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M5">View MathML</a> (the resident) can be found long time ago (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M41">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M42">View MathML</a>), but after a long time (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M44">View MathML</a>), only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M4">View MathML</a> (the invader) can be seen. Therefore, we call a traveling wave solution satisfying (1.4)-(1.5) an invasion traveling wave solution.

To obtain the existence of (1.4)-(1.5) if the wave speed is larger than a threshold depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M49">View MathML</a>, we construct proper upper and lower solutions and use the results in Pan et al.[33]. If the wave speed is the threshold, the existence of traveling wave solutions is proved by passing to a limit function. Finally, when the wave speed is smaller than the threshold, the nonexistence of traveling wave solutions is established by the theory of asymptotic spreading developed by Jin and Zhao [34]. For more results on the traveling wave solutions of evolutionary systems with nonlocal dispersal, we refer to Bates et al.[22], Coville and Dupaigne [35,36], Li et al.[37], Lv [38], Pan [39], Pan et al.[33,40], Sun et al.[41], Wu and Liu [42], Xu and Weng [43], Zhang et al.[44]. In particular, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M50">View MathML</a> hold in (1.2), Yu and Yuan [19] established the existence of traveling wave solutions connecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M51">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M52">View MathML</a>

In addition, Li and Lin [45] and Zhang et al.[46] investigated the existence of positive traveling wave solutions of (1.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M55">View MathML</a>, respectively.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M56">View MathML</a>. Moreover, denote

then X is a Banach space equipped with the standard supremum norm. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M58">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M59">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M60">View MathML</a>

In order to apply the comparison principle, we first make a change of variables to obtain a cooperative system. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M62">View MathML</a>, and drop the star for the sake of convenience, then (1.4) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M63">View MathML</a>

(2.1)

At the same time, (1.5) will be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M64">View MathML</a>

(2.2)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M65">View MathML</a> and

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M67">View MathML</a> is monotone in the functional sense if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M68">View MathML</a>. Applying these notations, we further define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M69">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M70">View MathML</a>

Clearly, a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M71">View MathML</a> in X satisfies (2.1), and a solution of (2.1) is also a fixed point of F. To continue our discussion, we also introduce the following definition.

Definition 2.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M72">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M74">View MathML</a> are differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M75">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M76">View MathML</a> contains finite points, and the derivatives are essentially bounded so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M77">View MathML</a>

(2.3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M78">View MathML</a>, then it is an upper (a lower) solution of (2.1).

Using Pan et al.[33], Theorem 3.2, we obtain the following conclusion.

Lemma 2.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M79">View MathML</a>is an upper solution of (2.1), while<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M80">View MathML</a>is a lower solution of (2.1). Also, suppose that

(P1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M81">View MathML</a>;

(P2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M83">View MathML</a>;

(P3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M84">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M87">View MathML</a>.

Then (2.1)-(2.2) has a positive monotone solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M88">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M89">View MathML</a>

We now consider the following initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M90">View MathML</a>

(2.4)

where J satisfies (J1) to (J3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M92">View MathML</a> are constants, and the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M93">View MathML</a> with

In addition, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M95">View MathML</a> be a subset of C defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M96">View MathML</a>

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97">View MathML</a>. Then (2.4) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M98">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M99">View MathML</a>

In particular, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M100">View MathML</a>with some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M101">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M102">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M103">View MathML</a>

Then Jin and Zhao [34], Theorem 3.5, indicates the following conclusion.

Lemma 2.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97">View MathML</a>admits nonempty support. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M105">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M98">View MathML</a>is defined by (2.4).

3 Existence of traveling wave solutions

In this section, we shall prove the existence of positive solutions of (2.1)-(2.2). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M107">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M109">View MathML</a>.

Lemma 3.1There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M110">View MathML</a>such that the following items hold.

(1) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M112">View MathML</a>has two positive real roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M113">View MathML</a>.

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M115">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M117">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M118">View MathML</a>.

(3) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M119">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M120">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M121">View MathML</a>.

The above result is clear and we omit the proof here. Using these constants, we can prove the following conclusion.

Theorem 3.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M111">View MathML</a>and one of the following two items holds.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M123">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M124">View MathML</a>

(3.1)

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M125">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M126">View MathML</a>

(3.2)

Then (2.1)-(2.2) has a monotone solution.

Proof Define continuous functions as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M127">View MathML</a>

Claim A: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M128">View MathML</a> is an upper solution to (2.1).

Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M129">View MathML</a> hold and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M130">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M132">View MathML</a>

Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M133">View MathML</a> is a lower solution to (2.1) (for the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M135">View MathML</a>, we refer to Pan et al.[33]). By Lemma 2.2, we see that (2.1)-(2.2) has a monotone solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M136">View MathML</a>. Now, it suffices to prove Claim A.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M137">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M138">View MathML</a>, the result is clear. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M139">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M140">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M141">View MathML</a>

which completes the proof on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M142">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M143">View MathML</a>.

We now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M144">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M145">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M146">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M147">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M148">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M149">View MathML</a>

Therefore, (3.1) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M150">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M151">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M152">View MathML</a> and (3.2) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M153">View MathML</a>

Therefore, Claim A is true. The proof is complete. □

Theorem 3.3Assume that one of the following items holds.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M123">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M155">View MathML</a>

(3.3)

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M125">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M157">View MathML</a>

(3.4)

Then (2.1)-(2.2) has a monotone solution with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114">View MathML</a>.

Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M159">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M161">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M162">View MathML</a>, (2.1)-(2.2) has a positive monotone solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M163">View MathML</a>. Note that a traveling wave solution is invariant in the sense of phase shift, so we can assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M164">View MathML</a>

(3.5)

for any n. By the Ascoli-Arzela lemma and a standard nested subsequence argument (see, e.g., Thieme and Zhao [47]), there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M159">View MathML</a>, which is still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M166">View MathML</a> without confusion, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M167">View MathML</a> converges uniformly on every bounded interval, and hence pointwise on ℝ to a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M168">View MathML</a>. Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M162">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M170">View MathML</a>

and the convergence in s is uniform for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M171">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M172">View MathML</a> and using the dominated convergence theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M71">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M174">View MathML</a> also satisfies (2.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M114">View MathML</a>. In addition, the following items are also clear.

(T1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M176">View MathML</a> (by (3.5));

(T2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M178">View MathML</a> are nondecreasing in ξ;

(T3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M85">View MathML</a>.

The items (T1) to (T3) further indicate that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M182">View MathML</a> exists for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M22">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M184">View MathML</a>

From (T1), it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M185">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M186">View MathML</a>, then the dominated convergence theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M188">View MathML</a>

Using the dominated convergence theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M189">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M190">View MathML</a>, we get the following possible conclusions:

(L1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M191">View MathML</a>;

(L2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M192">View MathML</a>.

If (L1) is true, then the dominated theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187">View MathML</a> tells us

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M194">View MathML</a>

which implies a contradiction. If (L2) is true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M195">View MathML</a> leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M196">View MathML</a>

which is also a contradiction. What we have done implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M197">View MathML</a>. Using the dominated convergence theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M187">View MathML</a> again, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M200">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M201">View MathML</a>, then a discussion similar to that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M202">View MathML</a> can be presented and we omit it here. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M203">View MathML</a>, then the dominated convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M189">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M205">View MathML</a> indicates that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M206">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M207">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M206">View MathML</a> is true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M209">View MathML</a> holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M210">View MathML</a>

has a monotone solution, which is impossible. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M211">View MathML</a> holds.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M174">View MathML</a> is a positive monotone solution of (2.1)-(2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M213">View MathML</a>, the proof is complete. □

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M97">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3">View MathML</a>is bounded such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M218">View MathML</a>

(4.1)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M3">View MathML</a>.

We now give the main result of this section.

Theorem 4.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M222">View MathML</a>, then (2.1)-(2.2) has no positive solutions.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M223">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M224">View MathML</a> is evident.

If (2.1)-(2.2) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M136">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M226">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M227">View MathML</a>

implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M228">View MathML</a> also satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M229">View MathML</a>

(4.2)

with the following asymptotic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M230">View MathML</a>

(4.3)

Recalling the definition of traveling wave solutions, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M231">View MathML</a> also satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M232">View MathML</a>

(4.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M233">View MathML</a>

(4.5)

Using Lemmas 2.4 and 4.1, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M234">View MathML</a>

(4.6)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M235">View MathML</a>.

However, the boundary condition (4.3) indicates that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M236">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/120/mathml/M237">View MathML</a>

(4.7)

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 (lzujbky-2011-k27).

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