Skip to main content

Invasion traveling wave solutions of a competitive system with dispersal

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.

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 ) ] ,
(1.1)

in which all the parameters are positive and xR, t>0, u 1 , u 2 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 holds in (1.1), then the corresponding reaction system has a stable equilibrium (1,0) and an unstable one (0,1). With the condition b 1 <1< b 2 , many papers including [2, 3, 5, 6, 8, 16] studied the traveling wave solutions connecting (1,0) with (0,1). These traveling wave solutions can formulate the spatial exclusive process between the resident u 2 and the invader u 1 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 ) ] ,
(1.2)

in which xR, t>0, u 1 (x,t) and u 2 (x,t) denote the densities of two competitors at time t and location xR, all the parameters are positive and

b 1 <1< b 2 ,
(1.3)

J i :RR, i=1,2, are probability functions formulating the random dispersal of individuals and satisfy the following assumptions:

(J1) J i is nonnegative and Lebesgue measurable for each i=1,2;

(J2) for any λR, R J i (y) e λ y dy<, i=1,2;

(J3) R J i (y)dy=1, J i (y)= J i (y), yR, i=1,2.

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+ct,

where c>0 is the wave speed at which the wave profile ( ϕ 1 , ϕ 2 ) C 1 (R, R 2 ) propagates in spatial media . Thus, ( ϕ 1 , ϕ 2 ) with c>0 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 .
(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).
(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 xR, only u 2 (the resident) can be found long time ago (t such that x+ct), but after a long time (t such that x+ct), only u 1 (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 , d 1 , r 1 and b 1 , 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) hold in (1.2), Yu and Yuan [19] established the existence of traveling wave solutions connecting (0,0) with

( 1 b 2 1 b 1 b 2 , 1 b 1 1 b 1 b 2 ) .

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, b 2 <0 and b 1 b 2 <0, 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 . Moreover, denote

then X is a Banach space equipped with the standard supremum norm. If a,b R 2 with ab, then

X [ a , b ] = { u X : a u ( ξ ) b , ξ R } .

In order to apply the comparison principle, we first make a change of variables to obtain a cooperative system. Let ϕ 1 = ϕ 1 , ϕ 2 =1 ϕ 2 , 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 ) ] .
(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).
(2.2)

Take β=2( d 1 + d 2 + r 1 + r 2 +1)(1+ b 1 + b 2 ) and

then H i is monotone in the functional sense if ( ϕ 1 , ϕ 2 ) X [ 0 , 1 ] . Applying these notations, we further define an operator F=( F 1 , F 2 ): X [ 0 , 1 ] X [ 0 , 1 ] as follows:

F i ( ϕ 1 , ϕ 2 )(ξ)= 1 c ξ e β c ( ξ s ) H i ( ϕ 1 , ϕ 2 )(s)ds,i=1,2.

Clearly, a fixed point of ( F 1 , F 2 ) 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 ] . If ρ 1 , ρ 2 are differentiable on RT, here T 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 ) ]
(2.3)

for ξRT, 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)) is an upper solution of (2.1), while ( ϕ ̲ 1 (ξ), ϕ ̲ 2 (ξ)) is a lower solution of (2.1). Also, suppose that

(P1) ( ϕ ̲ 1 (ξ), ϕ ̲ 2 (ξ))( ϕ ¯ 1 (ξ), ϕ ¯ 2 (ξ));

(P2) lim ξ ( ϕ ¯ 1 (ξ), ϕ ¯ 2 (ξ))=(0,0), lim ξ ( ϕ ¯ 1 (ξ), ϕ ¯ 2 (ξ))=(1,1);

(P3) sup s < ξ ϕ ̲ i (s) inf s > ξ ϕ ¯ i (s) for all ξR, i=1,2, and sup ξ R ϕ ̲ 1 (ξ)>0.

Then (2.1)-(2.2) has a positive monotone solution ( ϕ 1 (ξ), ϕ 2 (ξ)) such that

( ϕ ̲ 1 ( ξ ) , ϕ ̲ 2 ( ξ ) ) ( ϕ 1 ( ξ ) , ϕ 2 ( ξ ) ) ( ϕ ¯ 1 ( ξ ) , ϕ ¯ 2 ( ξ ) ) .

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 ,
(2.4)

where J satisfies (J1) to (J3), d>0 and r>0 are constants, and the initial value ϕ(x)C(R,R) with

In addition, let C + be a subset of C defined by

C + = { ϕ C : ϕ ( x ) 0 , x R } .

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 + . Then (2.4) has a unique solution u(x,t) such that

u(x,t)0,xR,t>0.

In particular, if ϕ(x) C [ 0 , a ] with some a1, then

0u(x,t)a,xR,t>0.

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 .

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

Lemma 2.4 Assume that ϕ(x) C + admits nonempty support. Then

lim inf t inf | x | < c t u(x,t)= lim sup t sup | x | < c t u(x,t)=1for any c< c 1 ,

where u(x,t) 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 ] cλ+ r 1 (1 b 1 )

for any λ0, c>0.

Lemma 3.1 There exists a constant c >0 such that the following items hold.

  1. (1)

    For each c> c , Δ 1 (λ,c)=0 has two positive real roots λ 1 (c)< λ 2 (c).

  2. (2)

    If c= c , then there exists λ( c )>0 such that Δ 1 (λ( c ), c )=0 and Δ 1 (λ, c )>0 for any λλ( c ).

  3. (3)

    If c< c , then Δ 1 (λ,c)>0 for any λ0.

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 and one of the following two items holds.

  1. (1)

    b 1 b 2 >1 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.
    (3.1)
  2. (2)

    b 1 b 2 1 and

    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 (c)0.
    (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 } .

Claim A: ( ϕ ¯ 1 (ξ), ϕ ¯ 2 (ξ)) is an upper solution to (2.1).

Moreover, let ϕ ̲ 2 (ξ)=0 hold and ϕ ̲ 1 (ξ) 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 ) ]

and

lim ξ ϕ 1 (ξ) e λ 1 ( c ) ξ =1.

Evidently, ( ϕ ̲ 1 (ξ), ϕ ̲ 2 (ξ)) is a lower solution to (2.1) (for the existence of ϕ ̲ 1 (ξ) and ϕ ̲ 1 (ξ)min{ e λ 1 ( c ) ξ ,1 b 1 }, 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)). Now, it suffices to prove Claim A.

If ϕ ¯ 1 (ξ)=1 or ϕ ¯ 2 (ξ)=1, the result is clear. If ξ0, then

ϕ ¯ 2 (ξ) e λ 1 ( c ) ξ / b 1

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 ,

which completes the proof on ϕ ¯ 1 (ξ) for ξ0.

We now consider ϕ ¯ 2 (ξ)<1 with ξ<0. If b 1 b 2 1, then b 2 e λ 1 ( c ) ξ e λ 1 ( c ) ξ / b 1 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

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 ] .

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 .

If b 1 b 2 <1, then b 2 ϕ ¯ 1 (ξ) ϕ ¯ 2 (ξ)0 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 .

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 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.
    (3.3)
  2. (2)

    b 1 b 2 1 and

    d 2 [ R J 2 ( y ) e λ 1 ( c ) y d y 1 ] c λ 1 ( c ) <0.
    (3.4)

Then (2.1)-(2.2) has a monotone solution with c= c .

Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence { c n } n = 1 with c n c , n such that for each c n , (2.1)-(2.2) has a positive monotone solution ( ϕ 1 n , ϕ 2 n ). 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
(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 , which is still denoted by { c n } n = 1 without confusion, such that ( ϕ 1 n (ξ), ϕ 2 n (ξ)) converges uniformly on every bounded interval, and hence pointwise on to a continuous function ( ϕ ˆ 1 (ξ), ϕ ˆ 2 (ξ)). Moreover, for each c n , we have

1 c n e β c n ( ξ s ) 1 c e β c ( ξ s ) for any ξR,sξ,

and the convergence in s is uniform for sξ. Letting n and using the dominated convergence theorem in ( F 1 , F 2 ), we know that ( ϕ ˆ 1 (ξ), ϕ ˆ 2 (ξ)) also satisfies (2.1) with c= c . In addition, the following items are also clear.

(T1) ϕ ˆ 2 (0)=1/2 (by (3.5));

(T2) ϕ ˆ 1 (ξ), ϕ ˆ 2 (ξ) are nondecreasing in ξ;

(T3) 0 ϕ ˆ 1 (ξ), ϕ ˆ 2 (ξ)1, ξR.

The items (T1) to (T3) further indicate that lim ξ ± ϕ ˆ i (ξ) exists for i=1,2. Denote

lim ξ ϕ ˆ i (ξ)= ϕ ˆ i , lim ξ ϕ ˆ i (ξ)= ϕ ˆ i + ,i=1,2.

From (T1), it is clear that

0 ϕ ˆ 2 1 2 ϕ ˆ 2 + 1.

If ϕ ˆ 2 (0,1/2], then the dominated convergence theorem in F 2 implies that

b 2 ϕ ˆ 1 = ϕ ˆ 2 .

Using the dominated convergence theorem in F 1 for ξ, we get the following possible conclusions:

(L1) ϕ ˆ 1 =0;

(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.

If (L1) is true, then the dominated theorem in F 2 tells us

ϕ ˆ 2 [ 1 ϕ ˆ 2 ] =0,

which implies a contradiction. If (L2) is true, then b 1 b 2 > b 1 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 ),

which is also a contradiction. What we have done implies that ϕ ˆ 2 =0. Using the dominated convergence theorem in F 2 again, we see that b 2 ϕ ˆ 1 = ϕ ˆ 2 =0 and ϕ ˆ 1 =0.

If ϕ ˆ 2 + [1/2,1), then a discussion similar to that on ϕ ˆ 2 can be presented and we omit it here. Because ϕ ˆ 2 + =1, then the dominated convergence in F 1 as ξ+ indicates that ϕ ˆ 1 + =0 or ϕ ˆ 1 + =1. If ϕ ˆ 1 + =0 is true, then ϕ 1 (ξ)0 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

has a monotone solution, which is impossible. Therefore, ϕ ˆ 1 + =1 holds.

Thus, ( ϕ ˆ 1 (ξ), ϕ ˆ 2 (ξ)) is a positive monotone solution of (2.1)-(2.2) with c= c , 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 + . If w(x,t)0, xR, t>0 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 ,
(4.1)

then w(x,t)()u(x,t), xR, t>0.

We now give the main result of this section.

Theorem 4.2 If c< c , 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 } .

Then c 2 = c is evident.

If (2.1)-(2.2) has a positive solution ( ϕ 1 (ξ), ϕ 2 (ξ)) for some c= c ¯ < c , then

ϕ 2 (ξ)= ϕ 2 (x+ c ¯ t)0,xR,t>0,ξR

implies that ϕ 1 (ξ) 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 ) ]
(4.2)

with the following asymptotic boundary condition:

lim ξ ϕ 1 (ξ)=0, lim ξ ϕ 1 (ξ)=1.
(4.3)

Recalling the definition of traveling wave solutions, we see that w(x,t)= ϕ 1 (x+ c ¯ t) 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 ) ]
(4.4)

and

0w(x,t)1,xR,t0, lim x w(x,0)=1.
(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
(4.6)

since c ¯ + c <2 c .

However, the boundary condition (4.3) indicates that

ξ=x+ c ¯ twith 2x= ( c ¯ + c ) t,t

and

lim t , 2 x = ( c ¯ + c ) t w(x,t)=0,
(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).

References

  1. Ahmad S, Lazer AC, Tineo A: Traveling waves for a system of equations. Nonlinear Anal. TMA 2008, 68: 3909-3912. 10.1016/j.na.2007.04.029

    Article  MathSciNet  Google Scholar 

  2. Fei N, Carr J: Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system. Nonlinear Anal., Real World Appl. 2003, 4: 503-524. 10.1016/S1468-1218(02)00077-9

    Article  MathSciNet  Google Scholar 

  3. Gourley SA, Ruan S: Convergence and traveling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J. Math. Anal. 2003, 35: 806-822. 10.1137/S003614100139991

    Article  MathSciNet  Google Scholar 

  4. Guo JS, Liang X: The minimal speed of traveling fronts for the Lotka-Volterra competition system. J. Dyn. Differ. Equ. 2011, 23: 353-363. 10.1007/s10884-011-9214-5

    Article  MathSciNet  Google Scholar 

  5. Hosono Y: The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model. Bull. Math. Biol. 1998, 60: 435-448. 10.1006/bulm.1997.0008

    Article  Google Scholar 

  6. Huang W: Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model. J. Dyn. Differ. Equ. 2010, 22: 285-297. 10.1007/s10884-010-9159-0

    Article  Google Scholar 

  7. Kan-on Y, Fang Q: Stability of monotone travelling waves for competition-diffusion equations. Jpn. J. Ind. Appl. Math. 1996, 13: 343-349. 10.1007/BF03167252

    Article  MathSciNet  Google Scholar 

  8. Lewis MA, Li B, Weinberger HF: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 2002, 45: 219-233. 10.1007/s002850200144

    Article  MathSciNet  Google Scholar 

  9. Li WT, Lin G, Ruan S: Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems. Nonlinearity 2006, 19: 1253-1273. 10.1088/0951-7715/19/6/003

    Article  MathSciNet  Google Scholar 

  10. Lin G, Li WT: Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays. J. Differ. Equ. 2008, 244: 487-513. 10.1016/j.jde.2007.10.019

    Article  Google Scholar 

  11. Lin G, Li WT, Ma M: Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin. Dyn. Syst., Ser. B 2010, 19: 393-414.

    MathSciNet  Google Scholar 

  12. Lv G, Wang M: Traveling wave front in diffusive and competitive Lotka-Volterra system with delays. Nonlinear Anal., Real World Appl. 2010, 11: 1323-1329. 10.1016/j.nonrwa.2009.02.020

    Article  MathSciNet  Google Scholar 

  13. Murray LD: Mathematical Biology. Heidelberg, Springer; 1989.

    Book  Google Scholar 

  14. Tang MM, Fife P: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 1980, 73: 69-77. 10.1007/BF00283257

    Article  MathSciNet  Google Scholar 

  15. Wang M, Lv G: Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays. Nonlinearity 2010, 23: 1609-1630. 10.1088/0951-7715/23/7/005

    Article  MathSciNet  Google Scholar 

  16. Yuan Z, Zou X: Co-invasion waves in a reaction diffusion model for competing pioneer and climax species. Nonlinear Anal., Real World Appl. 2010, 11: 232-245. 10.1016/j.nonrwa.2008.11.003

    Article  MathSciNet  Google Scholar 

  17. Shigesada N, Kawasaki K: Biological Invasions: Theory and Practice. Oxford University Press, Oxford; 1997.

    Google Scholar 

  18. Guo JS, Wu CH: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 2012, 252: 4367-4391.

    MathSciNet  Google Scholar 

  19. Yu Z, Yuan R: Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications. ANZIAM J. 2009, 51: 49-66. 10.1017/S1446181109000406

    Article  MathSciNet  Google Scholar 

  20. van den Driessche P: Spatial structure: patch models. In Mathematical Epidemiology. Edited by: Brauer F, Driessche P, Wu J. Springer, Berlin; 2008:179-189.

    Chapter  Google Scholar 

  21. Bates PW: On some nonlocal evolution equations arising in materials science. Fields Inst. Commun. 48. In Nonlinear Dynamics and Evolution Equations. Edited by: Brunner H, Zhao X, Zou X. Amer. Math. Soc., Providence; 2006:13-52.

    Google Scholar 

  22. Bates PW, Fife PC, Ren X, Wang X: Traveling waves in a convolution model for phase transition. Arch. Ration. Mech. Anal. 1997, 138: 105-136. 10.1007/s002050050037

    Article  MathSciNet  Google Scholar 

  23. Bates PW, Han J: The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Differ. Equ. 2005, 212: 235-277. 10.1016/j.jde.2004.07.003

    Article  MathSciNet  Google Scholar 

  24. Bates PW, Zhao G: Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 2007, 332: 428-440. 10.1016/j.jmaa.2006.09.007

    Article  MathSciNet  Google Scholar 

  25. Carr J, Chmaj A: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 2004, 132: 2433-2439. 10.1090/S0002-9939-04-07432-5

    Article  MathSciNet  Google Scholar 

  26. Cortázar C, Coville J, Elgueta M, Martínez S: A nonlocal inhomogeneous dispersal process. J. Differ. Equ. 2007, 241: 332-358. 10.1016/j.jde.2007.06.002

    Article  Google Scholar 

  27. Ermentrout B, Mcleod J: Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinb. A 1994, 123: 461-478.

    Article  MathSciNet  Google Scholar 

  28. Fife PC: Some nonclassical trends in parabolic and parabolic-like evolutions. In Trends in Nonlinear Analysis. Edited by: Kirkilionis M, Krömker S, Rannacher R, Tomi F. Springer, Berlin; 2003:153-191.

    Chapter  Google Scholar 

  29. Kao CY, Lou Y, Shen W: Random dispersal vs. non-local dispersal. Discrete Contin. Dyn. Syst. 2010, 26: 551-596.

    MathSciNet  Google Scholar 

  30. Shen W, Zhang A: Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J. Differ. Equ. 2010, 249: 747-795. 10.1016/j.jde.2010.04.012

    Article  MathSciNet  Google Scholar 

  31. Yanagida E, Zhang L: Speeds of traveling waves in some integro-differential equations arising from neuronal networks. Jpn. J. Ind. Appl. Math. 2010, 27: 347-373. 10.1007/s13160-010-0021-x

    Article  MathSciNet  Google Scholar 

  32. Zhang G: Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity. Nonlinear Anal. 2011, 74: 6518-6529. 10.1016/j.na.2011.06.035

    Article  MathSciNet  Google Scholar 

  33. Pan S, Li WT, Lin G: Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications. Z. Angew. Math. Phys. 2009, 60: 377-392. 10.1007/s00033-007-7005-y

    Article  MathSciNet  Google Scholar 

  34. Jin Y, Zhao XQ: Spatial dynamics of a periodic population model with dispersal. Nonlinearity 2009, 22: 1167-1189. 10.1088/0951-7715/22/5/011

    Article  MathSciNet  Google Scholar 

  35. Coville J, Dupaigne L: Propagation speed of travelling fronts in nonlocal reaction-diffusion equation. Nonlinear Anal. TMA 2005, 60: 797-819. 10.1016/j.na.2003.10.030

    Article  MathSciNet  Google Scholar 

  36. Coville J, Dupaigne L: On a non-local equation arising in population dynamics. Proc. R. Soc. Edinb. A 2007, 137: 725-755.

    Article  MathSciNet  Google Scholar 

  37. Li WT, Sun Y, Wang ZC: Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal., Real World Appl. 2010, 11: 2302-2313. 10.1016/j.nonrwa.2009.07.005

    Article  MathSciNet  Google Scholar 

  38. Lv G: Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation. Nonlinear Anal. TMA 2010, 72: 3659-3668. 10.1016/j.na.2009.12.047

    Article  Google Scholar 

  39. Pan S: Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity. J. Math. Anal. Appl. 2008, 346: 415-424. 10.1016/j.jmaa.2008.05.057

    Article  MathSciNet  Google Scholar 

  40. Pan S, Li WT, Lin G: Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay. Nonlinear Anal. TMA 2010, 72: 3150-3158. 10.1016/j.na.2009.12.008

    Article  MathSciNet  Google Scholar 

  41. Sun Y, Li WT, Wang ZC: Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity. Nonlinear Anal. TMA 2011, 74: 814-826. 10.1016/j.na.2010.09.032

    Article  MathSciNet  Google Scholar 

  42. Wu S, Liu S: Traveling waves for delayed non-local diffusion equations with crossing-monostability. Appl. Math. Comput. 2010, 217: 1435-1444. 10.1016/j.amc.2009.05.056

    Article  MathSciNet  Google Scholar 

  43. Xu Z, Weng P: Traveling waves in a convolution model with infinite distributed delay and non-monotonicity. Nonlinear Anal., Real World Appl. 2011, 12: 633-647. 10.1016/j.nonrwa.2010.07.006

    Article  MathSciNet  Google Scholar 

  44. Zhang G, Li WT, Wang ZC: Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. J. Differ. Equ. 2012, 252: 5096-5124. 10.1016/j.jde.2012.01.014

    Article  MathSciNet  Google Scholar 

  45. Li X, Lin G: Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays. Appl. Math. Comput. 2008, 204: 738-744. 10.1016/j.amc.2008.07.016

    Article  MathSciNet  Google Scholar 

  46. Zhang G, Li WT, Lin G: Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure. Math. Comput. Model. 2009, 49: 1021-1029. 10.1016/j.mcm.2008.09.007

    Article  MathSciNet  Google Scholar 

  47. Thieme HR, Zhao XQ: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J. Differ. Equ. 2003, 195: 430-470. 10.1016/S0022-0396(03)00175-X

    Article  MathSciNet  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shuxia Pan.

Additional information

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.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Pan, S., Lin, G. Invasion traveling wave solutions of a competitive system with dispersal. Bound Value Probl 2012, 120 (2012). https://doi.org/10.1186/1687-2770-2012-120

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2012-120

Keywords