Invasion traveling wave solutions of a competitive system with dispersal

Boundary Value Problems20122012:120

DOI: 10.1186/1687-2770-2012-120

Received: 31 January 2012

Accepted: 8 October 2012

Published: 24 October 2012

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:
{ u 1 ( x , t ) t = d 1 Δ u 1 ( x , t ) + r 1 u 1 ( x , t ) [ 1 u 1 ( x , t ) b 1 u 2 ( x , t ) ] , u 2 ( x , t ) t = d 2 Δ u 2 ( x , t ) + r 2 u 2 ( x , t ) [ 1 u 2 ( x , t ) b 2 u 1 ( x , t ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ1_HTML.gif
(1.1)

in which all the parameters are positive and x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq2_HTML.gif, u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq3_HTML.gif, u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq4_HTML.gif 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 [116].

In particular, if b 1 < 1 < b 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq5_HTML.gif holds in (1.1), then the corresponding reaction system has a stable equilibrium ( 1 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq6_HTML.gif and an unstable one ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq7_HTML.gif. With the condition b 1 < 1 < b 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq5_HTML.gif, many papers including [2, 3, 5, 6, 8, 16] studied the traveling wave solutions connecting ( 1 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq6_HTML.gif with ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq7_HTML.gif. These traveling wave solutions can formulate the spatial exclusive process between the resident u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq4_HTML.gif and the invader u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq3_HTML.gif 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]):
{ u 1 ( x , t ) t = d 1 [ R J 1 ( x y ) u 1 ( y , t ) d y u 1 ( x , t ) ] u 1 ( x , t ) t = + r 1 u 1 ( x , t ) [ 1 u 1 ( x , t ) b 1 u 2 ( x , t ) ] , u 2 ( x , t ) t = d 2 [ R J 2 ( x y ) u 2 ( y , t ) d y u 2 ( x , t ) ] u 2 ( x , t ) t = + r 2 u 2 ( x , t ) [ 1 u 2 ( x , t ) b 2 u 1 ( x , t ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ2_HTML.gif
(1.2)
in which x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq2_HTML.gif, u 1 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq8_HTML.gif and u 2 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq9_HTML.gif denote the densities of two competitors at time t and location x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq10_HTML.gif, all the parameters are positive and
b 1 < 1 < b 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ3_HTML.gif
(1.3)

J i : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq11_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif, are probability functions formulating the random dispersal of individuals and satisfy the following assumptions:

(J1) J i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq13_HTML.gif is nonnegative and Lebesgue measurable for each i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif;

(J2) for any λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq14_HTML.gif, R J i ( y ) e λ y d y < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq15_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif;

(J3) R J i ( y ) d y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq16_HTML.gif, J i ( y ) = J i ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq17_HTML.gif, y R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq18_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif.

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, 2132] and the references cited therein.

Hereafter, a traveling wave solution of (1.2) is a special solution of the form
u 1 ( x , t ) = ϕ 1 ( ξ ) , u 2 ( x , t ) = ϕ 2 ( ξ ) , ξ = x + c t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equa_HTML.gif
where c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq19_HTML.gif is the wave speed at which the wave profile ( ϕ 1 , ϕ 2 ) C 1 ( R , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq20_HTML.gif propagates in spatial media ℝ. Thus, ( ϕ 1 , ϕ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq21_HTML.gif with c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq19_HTML.gif must satisfy
{ c ϕ 1 ( ξ ) = d 1 [ R J 1 ( ξ y ) ϕ 1 ( y ) d y ϕ 1 ( ξ ) ] c ϕ 1 ( ξ ) = + r 1 ϕ 1 ( ξ ) [ 1 ϕ 1 ( ξ ) b 1 ϕ 2 ( ξ ) ] , ξ R , c ϕ 2 ( ξ ) = d 2 [ R J 2 ( ξ y ) ϕ 2 ( y ) d y ϕ 2 ( ξ ) ] c ϕ 2 ( ξ ) = + r 2 ϕ 2 ( ξ ) [ 1 ϕ 2 ( ξ ) b 2 ϕ 1 ( ξ ) ] , ξ R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ4_HTML.gif
(1.4)
Moreover, we also require the following asymptotic boundary conditions:
lim ξ ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) = ( 0 , 1 ) , lim ξ ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) = ( 1 , 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ5_HTML.gif
(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 x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq10_HTML.gif, only u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq4_HTML.gif (the resident) can be found long time ago ( t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq22_HTML.gif such that x + c t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq23_HTML.gif), but after a long time ( t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq24_HTML.gif such that x + c t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq25_HTML.gif), only u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq3_HTML.gif (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 J 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq26_HTML.gif, d 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq27_HTML.gif, r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq28_HTML.gif and b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq29_HTML.gif, 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 b 1 , b 2 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq30_HTML.gif hold in (1.2), Yu and Yuan [19] established the existence of traveling wave solutions connecting ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq31_HTML.gif with
( 1 b 2 1 b 1 b 2 , 1 b 1 1 b 1 b 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equb_HTML.gif

In addition, Li and Lin [45] and Zhang et al. [46] investigated the existence of positive traveling wave solutions of (1.2) for b 1 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq32_HTML.gif, b 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq33_HTML.gif and b 1 b 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq34_HTML.gif, 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 R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq35_HTML.gif. Moreover, denote
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equc_HTML.gif
then X is a Banach space equipped with the standard supremum norm. If a , b R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq36_HTML.gif with a b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq37_HTML.gif, then
X [ a , b ] = { u X : a u ( ξ ) b , ξ R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equd_HTML.gif
In order to apply the comparison principle, we first make a change of variables to obtain a cooperative system. Let ϕ 1 = ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq38_HTML.gif, ϕ 2 = 1 ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq39_HTML.gif, and drop the star for the sake of convenience, then (1.4) becomes
{ c ϕ 1 ( ξ ) = d 1 [ R J 1 ( ξ y ) ϕ 1 ( y ) d y ϕ 1 ( ξ ) ] + r 1 ϕ 1 ( ξ ) [ 1 b 1 ϕ 1 ( ξ ) + b 1 ϕ 2 ( ξ ) ] , c ϕ 2 ( ξ ) = d 2 [ R J 2 ( ξ y ) ϕ 2 ( y ) d y ϕ 2 ( ξ ) ] + r 2 [ 1 ϕ 2 ( ξ ) ] [ b 2 ϕ 1 ( ξ ) ϕ 2 ( ξ ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ6_HTML.gif
(2.1)
At the same time, (1.5) will be
lim ξ ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) = ( 0 , 0 ) , lim ξ ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) = ( 1 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ7_HTML.gif
(2.2)
Take β = 2 ( d 1 + d 2 + r 1 + r 2 + 1 ) ( 1 + b 1 + b 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq40_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Eque_HTML.gif
then H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq41_HTML.gif is monotone in the functional sense if ( ϕ 1 , ϕ 2 ) X [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq42_HTML.gif. Applying these notations, we further define an operator F = ( F 1 , F 2 ) : X [ 0 , 1 ] X [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq43_HTML.gif as follows:
F i ( ϕ 1 , ϕ 2 ) ( ξ ) = 1 c ξ e β c ( ξ s ) H i ( ϕ 1 , ϕ 2 ) ( s ) d s , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equf_HTML.gif

Clearly, a fixed point of ( F 1 , F 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq44_HTML.gif 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 ( ρ 1 , ρ 2 ) X [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq45_HTML.gif. If ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq46_HTML.gif, ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq47_HTML.gif are differentiable on R T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq48_HTML.gif, here T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq49_HTML.gif contains finite points, and the derivatives are essentially bounded so that
{ c ρ 1 ( ξ ) ( ) d 1 [ R J 1 ( ξ y ) ρ 1 ( y ) d y ρ 1 ( ξ ) ] c ρ 1 ( ξ ) ( ) + r 1 ρ 1 ( ξ ) [ 1 b 1 ρ 1 ( ξ ) + b 1 ρ 2 ( ξ ) ] , c ρ 2 ( ξ ) ( ) d 2 [ R J 2 ( ξ y ) ρ 2 ( y ) d y ρ 2 ( ξ ) ] c ρ 2 ( ξ ) ( ) + r 2 [ 1 ρ 2 ( ξ ) ] [ b 2 ρ 1 ( ξ ) ρ 2 ( ξ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ8_HTML.gif
(2.3)

for ξ R T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq50_HTML.gif, 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 2.2 Assume that ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq51_HTML.gif is an upper solution of (2.1), while ( ϕ ̲ 1 ( ξ ) , ϕ ̲ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq52_HTML.gif is a lower solution of (2.1). Also, suppose that

(P1) ( ϕ ̲ 1 ( ξ ) , ϕ ̲ 2 ( ξ ) ) ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq53_HTML.gif;

(P2) lim ξ ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq54_HTML.gif, lim ξ ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) = ( 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq55_HTML.gif;

(P3) sup s < ξ ϕ ̲ i ( s ) inf s > ξ ϕ ¯ i ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq56_HTML.gif for all ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq57_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif, and sup ξ R ϕ ̲ 1 ( ξ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq58_HTML.gif.

Then (2.1)-(2.2) has a positive monotone solution ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq59_HTML.gif such that
( ϕ ̲ 1 ( ξ ) , ϕ ̲ 2 ( ξ ) ) ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equg_HTML.gif
We now consider the following initial value problem:
{ u ( x , t ) t = d [ R J ( x y ) u ( y , t ) d y u ( x , t ) ] + r u ( x , t ) [ 1 u ( x , t ) ] , u ( x , 0 ) = ϕ ( x ) , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ9_HTML.gif
(2.4)
where J satisfies (J1) to (J3), d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq60_HTML.gif and r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq61_HTML.gif are constants, and the initial value ϕ ( x ) C ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq62_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equh_HTML.gif
In addition, let C + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq63_HTML.gif be a subset of C defined by
C + = { ϕ C : ϕ ( x ) 0 , x R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equi_HTML.gif

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 2.3 Assume that ϕ ( x ) C + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq64_HTML.gif. Then (2.4) has a unique solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq65_HTML.gif such that
u ( x , t ) 0 , x R , t > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equj_HTML.gif
In particular, if ϕ ( x ) C [ 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq66_HTML.gif with some a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq67_HTML.gif, then
0 u ( x , t ) a , x R , t > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equk_HTML.gif
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
c 1 = inf λ > 0 d [ R J ( y ) e λ y d y 1 ] + r λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equl_HTML.gif

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

Lemma 2.4 Assume that ϕ ( x ) C + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq64_HTML.gif admits nonempty support. Then
lim inf t inf | x | < c t u ( x , t ) = lim sup t sup | x | < c t u ( x , t ) = 1 for any  c < c 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equm_HTML.gif

where u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq65_HTML.gif 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
Δ 1 ( λ , c ) = d 1 [ R J 1 ( y ) e λ y d y 1 ] c λ + r 1 ( 1 b 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equn_HTML.gif

for any λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq68_HTML.gif, c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq69_HTML.gif.

Lemma 3.1 There exists a constant c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq70_HTML.gif such that the following items hold.
  1. (1)

    For each c > c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq71_HTML.gif, Δ 1 ( λ , c ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq72_HTML.gif has two positive real roots λ 1 ( c ) < λ 2 ( c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq73_HTML.gif.

     
  2. (2)

    If c = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq74_HTML.gif, then there exists λ ( c ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq75_HTML.gif such that Δ 1 ( λ ( c ) , c ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq76_HTML.gif and Δ 1 ( λ , c ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq77_HTML.gif for any λ λ ( c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq78_HTML.gif.

     
  3. (3)

    If c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq79_HTML.gif, then Δ 1 ( λ , c ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq80_HTML.gif for any λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq81_HTML.gif.

     

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

Theorem 3.2 Assume that c > c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq71_HTML.gif and one of the following two items holds.
  1. (1)
    b 1 b 2 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq82_HTML.gif and
    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) + r 2 ( b 1 b 2 1 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ10_HTML.gif
    (3.1)
     
  2. (2)
    b 1 b 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq83_HTML.gif and
    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ11_HTML.gif
    (3.2)
     

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

Proof Define continuous functions as follows:
ϕ ¯ 1 ( ξ ) = min { e λ 1 ( c ) ξ , 1 } , ϕ ¯ 2 ( ξ ) = min { e λ 1 ( c ) ξ / b 1 , 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equo_HTML.gif

Claim A: ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq84_HTML.gif is an upper solution to (2.1).

Moreover, let ϕ ̲ 2 ( ξ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq85_HTML.gif hold and ϕ ̲ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq86_HTML.gif satisfy
c ϕ 1 ( ξ ) = d 1 [ R J 1 ( ξ y ) ϕ 1 ( y ) d y ϕ 1 ( ξ ) ] + r 1 ϕ 1 ( ξ ) [ 1 b 1 ϕ 1 ( ξ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equp_HTML.gif
and
lim ξ ϕ 1 ( ξ ) e λ 1 ( c ) ξ = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equq_HTML.gif

Evidently, ( ϕ ̲ 1 ( ξ ) , ϕ ̲ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq87_HTML.gif is a lower solution to (2.1) (for the existence of ϕ ̲ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq86_HTML.gif and ϕ ̲ 1 ( ξ ) min { e λ 1 ( c ) ξ , 1 b 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq88_HTML.gif, we refer to Pan et al. [33]). By Lemma 2.2, we see that (2.1)-(2.2) has a monotone solution ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq89_HTML.gif. Now, it suffices to prove Claim A.

If ϕ ¯ 1 ( ξ ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq90_HTML.gif or ϕ ¯ 2 ( ξ ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq91_HTML.gif, the result is clear. If ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq92_HTML.gif, then
ϕ ¯ 2 ( ξ ) e λ 1 ( c ) ξ / b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equr_HTML.gif
such that
d 1 [ R J 1 ( ξ y ) ϕ ¯ 1 ( y ) d y ϕ ¯ 1 ( ξ ) ] c ϕ ¯ 1 ( ξ ) + r 1 ϕ ¯ 1 ( ξ ) [ 1 b 1 ϕ ¯ 1 ( ξ ) + b 1 ϕ ¯ 2 ( ξ ) ] d 1 [ R J 1 ( ξ y ) e λ 1 ( c ) y d y e λ 1 ( c ) ξ ] c λ 1 ( c ) e λ 1 ( c ) ξ + r 1 e λ 1 ( c ) ξ [ 1 b 1 e λ 1 ( c ) ξ + b 1 e λ 1 ( c ) ξ / b 1 ] = e λ 1 ( c ) ξ Δ 1 ( λ 1 ( c ) , c ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equs_HTML.gif

which completes the proof on ϕ ¯ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq93_HTML.gif for ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq94_HTML.gif.

We now consider ϕ ¯ 2 ( ξ ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq95_HTML.gif with ξ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq96_HTML.gif. If b 1 b 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq97_HTML.gif, then b 2 e λ 1 ( c ) ξ e λ 1 ( c ) ξ / b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq98_HTML.gif such that
b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) = b 2 e λ 1 ( c ) ξ e λ 1 ( c ) ξ b 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equt_HTML.gif
and
r 2 [ 1 ϕ ¯ 2 ( ξ ) ] [ b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) ] r 2 [ b 2 e λ 1 ( c ) ξ e λ 1 ( c ) ξ b 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equu_HTML.gif
Therefore, (3.1) leads to
d 2 [ R J 2 ( ξ y ) ϕ ¯ 2 ( y ) d y ϕ ¯ 2 ( ξ ) ] c ϕ ¯ 2 ( ξ ) + r 2 [ 1 ϕ ¯ 2 ( ξ ) ] [ b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) ] d 2 [ R J 2 ( ξ y ) ϕ ¯ 2 ( y ) d y ϕ ¯ 2 ( ξ ) ] c ϕ ¯ 2 ( ξ ) + r 2 [ b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) ] e λ 1 ( c ) ξ b 1 [ d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) + r 2 ( b 1 b 2 1 ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equv_HTML.gif
If b 1 b 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq99_HTML.gif, then b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq100_HTML.gif and (3.2) imply that
d 2 [ R J 2 ( ξ y ) ϕ ¯ 2 ( y ) d y ϕ ¯ 2 ( ξ ) ] c ϕ ¯ 2 ( ξ ) + r 2 [ 1 ϕ ¯ 2 ( ξ ) ] [ b 2 ϕ ¯ 1 ( ξ ) ϕ ¯ 2 ( ξ ) ] d 2 [ R J 2 ( ξ y ) ϕ ¯ 2 ( y ) d y ϕ ¯ 2 ( ξ ) ] c ϕ ¯ 2 ( ξ ) e λ 1 ( c ) ξ b 1 [ d 2 R J 2 ( y ) e λ 1 ( c ) y d y d 2 c λ 1 ( c ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equw_HTML.gif

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

Theorem 3.3 Assume that one of the following items holds.
  1. (1)
    b 1 b 2 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq82_HTML.gif and
    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) + r 2 ( b 1 b 2 1 ) < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ12_HTML.gif
    (3.3)
     
  2. (2)
    b 1 b 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq83_HTML.gif and
    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ13_HTML.gif
    (3.4)
     

Then (2.1)-(2.2) has a monotone solution with c = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq74_HTML.gif.

Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence { c n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq101_HTML.gif with c n c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq102_HTML.gif, n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq103_HTML.gif such that for each c n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq104_HTML.gif, (2.1)-(2.2) has a positive monotone solution ( ϕ 1 n , ϕ 2 n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq105_HTML.gif. Note that a traveling wave solution is invariant in the sense of phase shift, so we can assume that
ϕ 2 n ( 0 ) = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ14_HTML.gif
(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 { c n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq101_HTML.gif, which is still denoted by { c n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq106_HTML.gif without confusion, such that ( ϕ 1 n ( ξ ) , ϕ 2 n ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq107_HTML.gif converges uniformly on every bounded interval, and hence pointwise on ℝ to a continuous function ( ϕ ˆ 1 ( ξ ) , ϕ ˆ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq108_HTML.gif. Moreover, for each c n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq104_HTML.gif, we have
1 c n e β c n ( ξ s ) 1 c e β c ( ξ s ) for any  ξ R , s ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equx_HTML.gif

and the convergence in s is uniform for s ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq109_HTML.gif. Letting n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq110_HTML.gif and using the dominated convergence theorem in ( F 1 , F 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq44_HTML.gif, we know that ( ϕ ˆ 1 ( ξ ) , ϕ ˆ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq111_HTML.gif also satisfies (2.1) with c = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq74_HTML.gif. In addition, the following items are also clear.

(T1) ϕ ˆ 2 ( 0 ) = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq112_HTML.gif (by (3.5));

(T2) ϕ ˆ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq113_HTML.gif, ϕ ˆ 2 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq114_HTML.gif are nondecreasing in ξ;

(T3) 0 ϕ ˆ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq115_HTML.gif, ϕ ˆ 2 ( ξ ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq116_HTML.gif, ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq57_HTML.gif.

The items (T1) to (T3) further indicate that lim ξ ± ϕ ˆ i ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq117_HTML.gif exists for i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq12_HTML.gif. Denote
lim ξ ϕ ˆ i ( ξ ) = ϕ ˆ i , lim ξ ϕ ˆ i ( ξ ) = ϕ ˆ i + , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equy_HTML.gif
From (T1), it is clear that
0 ϕ ˆ 2 1 2 ϕ ˆ 2 + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equz_HTML.gif
If ϕ ˆ 2 ( 0 , 1 / 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq118_HTML.gif, then the dominated convergence theorem in F 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq119_HTML.gif implies that
b 2 ϕ ˆ 1 = ϕ ˆ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equaa_HTML.gif

Using the dominated convergence theorem in F 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq120_HTML.gif for ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq121_HTML.gif, we get the following possible conclusions:

(L1) ϕ ˆ 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq122_HTML.gif;

(L2) 1 b 1 ϕ ˆ 1 + b 1 ϕ ˆ 2 = 1 b 1 ϕ ˆ 1 + b 1 b 2 ϕ ˆ 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq123_HTML.gif.

If (L1) is true, then the dominated theorem in F 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq119_HTML.gif tells us
ϕ ˆ 2 [ 1 ϕ ˆ 2 ] = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equab_HTML.gif
which implies a contradiction. If (L2) is true, then b 1 b 2 > b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq124_HTML.gif leads to
0 = 1 b 1 ϕ ˆ 1 + b 1 b 2 ϕ ˆ 1 > 1 b 1 ϕ ˆ 1 + b 1 ϕ ˆ 1 = ( 1 ϕ ˆ 1 ) ( 1 b 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equac_HTML.gif

which is also a contradiction. What we have done implies that ϕ ˆ 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq125_HTML.gif. Using the dominated convergence theorem in F 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq119_HTML.gif again, we see that b 2 ϕ ˆ 1 = ϕ ˆ 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq126_HTML.gif and ϕ ˆ 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq127_HTML.gif.

If ϕ ˆ 2 + [ 1 / 2 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq128_HTML.gif, then a discussion similar to that on ϕ ˆ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq129_HTML.gif can be presented and we omit it here. Because ϕ ˆ 2 + = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq130_HTML.gif, then the dominated convergence in F 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq120_HTML.gif as ξ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq131_HTML.gif indicates that ϕ ˆ 1 + = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq132_HTML.gif or ϕ ˆ 1 + = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq133_HTML.gif. If ϕ ˆ 1 + = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq132_HTML.gif is true, then ϕ 1 ( ξ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq134_HTML.gif holds and
{ c ϕ 2 ( ξ ) = d 2 [ R J 2 ( ξ y ) ϕ 2 ( y ) d y ϕ 2 ( ξ ) ] r 2 ϕ 2 ( ξ ) [ 1 ϕ 2 ( ξ ) ] , lim ξ ϕ 2 ( ξ ) = 0 , lim ξ ϕ 2 ( ξ ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equad_HTML.gif

has a monotone solution, which is impossible. Therefore, ϕ ˆ 1 + = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq135_HTML.gif holds.

Thus, ( ϕ ˆ 1 ( ξ ) , ϕ ˆ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq111_HTML.gif is a positive monotone solution of (2.1)-(2.2) with c = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq136_HTML.gif, 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 4.1 Assume that ϕ ( x ) C + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq64_HTML.gif. If w ( x , t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq137_HTML.gif, x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq2_HTML.gif is bounded such that
{ w ( x , t ) t ( ) d [ R J ( x y ) w ( y , t ) d y w ( x , t ) ] + r w ( x , t ) [ 1 w ( x , t ) ] , w ( x , 0 ) ( ) ϕ ( x ) , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ15_HTML.gif
(4.1)

then w ( x , t ) ( ) u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq138_HTML.gif, x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq2_HTML.gif.

We now give the main result of this section.

Theorem 4.2 If c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq139_HTML.gif, then (2.1)-(2.2) has no positive solutions.

Proof Define
c 2 = inf λ > 0 { d 1 [ R J 1 ( y ) e λ y d y 1 ] + r 1 ( 1 b 1 ) λ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equae_HTML.gif

Then c 2 = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq140_HTML.gif is evident.

If (2.1)-(2.2) has a positive solution ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq89_HTML.gif for some c = c ¯ < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq141_HTML.gif, then
ϕ 2 ( ξ ) = ϕ 2 ( x + c ¯ t ) 0 , x R , t > 0 , ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equaf_HTML.gif
implies that ϕ 1 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq142_HTML.gif also satisfies
c ¯ ϕ 1 ( ξ ) d 1 [ R J 1 ( ξ y ) ϕ 1 ( y ) d y ϕ 1 ( ξ ) ] + r 1 ϕ 1 ( ξ ) [ 1 b 1 ϕ 1 ( ξ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ16_HTML.gif
(4.2)
with the following asymptotic boundary condition:
lim ξ ϕ 1 ( ξ ) = 0 , lim ξ ϕ 1 ( ξ ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ17_HTML.gif
(4.3)
Recalling the definition of traveling wave solutions, we see that w ( x , t ) = ϕ 1 ( x + c ¯ t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq143_HTML.gif also satisfies
w ( x , t ) t d 1 [ R J 1 ( x y ) w ( y , t ) d y w ( x , t ) ] + r 1 w ( x , t ) [ 1 b 1 w ( x , t ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ18_HTML.gif
(4.4)
and
0 w ( x , t ) 1 , x R , t 0 , lim x w ( x , 0 ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ19_HTML.gif
(4.5)
Using Lemmas 2.4 and 4.1, we see that
lim t inf 2 | x | = ( c ¯ + c ) t w ( x , t ) 1 b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ20_HTML.gif
(4.6)

since c ¯ + c < 2 c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_IEq144_HTML.gif.

However, the boundary condition (4.3) indicates that
ξ = x + c ¯ t with  2 x = ( c ¯ + c ) t , t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equag_HTML.gif
and
lim t , 2 x = ( c ¯ + c ) t w ( x , t ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-120/MediaObjects/13661_2012_Article_244_Equ21_HTML.gif
(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).

Declarations

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).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Lanzhou University of Technology
(2)
School of Mathematics and Statistics, Lanzhou University

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