Traveling waves for the nonlocal diffusive single species model with Allee effect

Accepted Manuscript Traveling Waves for the Nonlocal Diffusive Single Species Model with Allee Effect

Bang-Sheng Han, Zhi-Cheng Wang, Zhaosheng Feng

PII: DOI: Reference:

S0022-247X(16)30173-1 http://dx.doi.org/10.1016/j.jmaa.2016.05.031 YJMAA 20449

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

25 November 2015

Please cite this article in press as: B.-S. Han et al., Traveling Waves for the Nonlocal Diffusive Single Species Model with Allee Effect, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.05.031

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Traveling Waves for the Nonlocal Diffusive Single Species Model with Allee Effect∗ Bang-Sheng Han and Zhi-Cheng Wang School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China Zhaosheng Feng† Department of Mathematics, University of Texas–Rio Grande Valley, Edinburg, TX 78539, USA

Abstract In this paper, we study a nonlocal diffusive single species model with Allee effect. We prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed √ c ≥ 2 r, where r > 0 is the intrinsic rate of increase of the species. For the sufficient large wave speed c, we show that the unknown steady state is the unique positive equilibrium. For two types of convolution kernel functions, we investigate the change of the wave profile as the non-locality increases, and illustrate that the unknown steady state may be a positive periodic solution. Keywords: reaction-diffusion equation; Allee effect; Leray-Schauder; traveling waves; non-locality degree; steady state. AMS Subject Classification (2010): 35C07; 35K57; 45K05; 35Q92.

1

Introduction

In this paper, we are concerned with the following nonlocal diffusive single species model with Allee effect   ∂u ∂2u 2 = + ru 1 − a (φ ∗ u) − a (φ ∗ u) in R × [0, ∞), 1 2 ∂t ∂x2 where r, a1 , a2 > 0, and (φ ∗ u)(x) := ∗ †

 R

φ(x − y)u(y, t)dy,

x ∈ R.

This work is supported by NSF of China under (No. 11371179). Corresponding author: [email protected]; fax: +1 (956) 665-5091.

1

(1.1)

2

Traveling Waves for Nonlocal Diffusive Single Species Model

The kernel φ(x) is a bounded function and satisfies  φ(x) ≥ 0, φ(0) > 0 and φ(x)dx = 1. R

Equation (1.1) has three constant equilibria   −a1 − a21 + 4a2 −a1 + a21 + 4a2 < 0, u0 := 0 and u+ := > 0. u− := 2a2 2a2 The pioneering work of equation (1.1) could be originated from Gopalsamy and Ladas [16], and Ruan [23], in which the following delayed equation was proposed dx = x(t)[a + bx(t − τ ) − cx2 (t − τ )], dt

(1.2)

where a, b, c, τ are real constants with a, c > 0 and τ ∈ [0, +∞). The model (1.2) describes the growth of a single species population which exhibits the Allee effect. That is, intraspecific mutualism dominates at low densities and intraspecific competition dominates √ 2 at higher densities. Equation (1.2) admits a unique positive equilibrium u∗ = b+ b2c+4ac . It was showed that the equilibrium u∗ attracts all positive solutions of equation (1.2) if τ u∗ (2cu∗ − b) ≤ 32 , and solutions of equation (1.2) oscillate near the equilibrium u∗ if the delay τ is sufficiently large. However, the model (1.2) ignored the spatial movement of individuals in reality. To take into account such movement and to account for the drift of individuals to their present position from all possible positions at previous times, an integro-differential reactiondiffusion equation was proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase [9]. Song et al. [24] proposed the following equation   ∂u ∂2u 2 = + ru(x, t) 1 − a (f ∗ u)(x, t) − a (f ∗ u) (x, t) in R × [0, ∞), 1 2 ∂t ∂x2 ∞∞ where f ∈ L1 (R × [0, ∞), R+ ), 0 −∞ f (y, s)dyds = 1 and  (f ∗ u)(x, t) =

t −∞



∞ −∞

(1.3)

f (x − y, t − s)u(y, s)dyds.

By using the linear train technique and the geometric singular perturbation theory [12], t they considered equation (1.3) with the kernel functions f (x, t) = τ12 e− τ δ(x) and f (x, t) = 2

x t √ 1 e− 4t 1 e− τ τ 4πt

respectively, where δ(x) is the Dirac delta function. For the sufficiently √ small τ , it was shown that there exists a speed c ≥ 2 r such that equation (1.3) has a traveling wave front with the speed c connecting 0 to u+ .

B.-S. Han, Z.-C. Wang and Z. Feng

3

It is notable that equation (1.1) can be incorporated into equation (1.3) when f (x, t) = φ(x)δ(t). For such type of the kernel function, the method described in [24] becomes invalid, but similar results can be established by using the method proposed by Wang 2 √ −x et al. [25] when the nonlocality is sufficiently weak, for example, φ(x) = 4πρe 4ρ with ρ > 0 being small enough. Nevertheless, the existence of traveling wave fronts of equation (1.1) is still unknown for more general kernel functions. In particular, some complex phenomenon of traveling wave fronts of equation (1.1) can be expected due to the lack of quasi-monotonicity. Recently, there were some great progress on traveling wave solutions of equation (1.1) with a1 < 0 and a2 = 0. In this case equation (1.1) reduces to ∂ ∂2 u= u + μu(1 − φ ∗ u), (x, t) ∈ R × (0, +∞), ∂t ∂x2

(1.4)

where μ > 0. We note that such equations with nonlocal reactions arise in several areas, for example, in evolution of biological species [5], ecology [14], and adaptive dynamics [15] as well. Berestycki et al. [8] proved that equation (1.4) admits traveling wave solutions √ connecting 0 to an unknown positive state for all speeds c ≥ c∗ = 2 μ and there exists √ no such traveling wave solution with the wave speed c < 2 μ. Then, Nadin et al. [21] showed that the unknown state is just the equilibrium 1 for some traveling wave solutions. Fang and Zhao [10] further gave a sufficient and necessary condition for the existence of monotone traveling waves of equation (1.4) connecting two equilibria 0 and 1. More recently, Alfaro and Coville [2] pointed out that for any kernel and any slope at the origin, equation (1.4) admits rapid traveling wave solutions connecting two equilibria 0 and 1. In particular, this includes the case where 1 is unstable in the sense of Turing. For more detailed results on traveling wave solutions of equation (1.4), we refer to [1, 3–6, 11, 13, 17– 20, 22]. The present paper is devoted to traveling wave solutions of equation (1.1) with a more general kernel function φ. Firstly, since the comparison principle can not be applied to equation (1.1) directly, the Leray-Schauder degree [4, 8] is used to indicate that equation (1.1) admits traveling wave solutions connecting 0 to an unknown positive steady state. Secondly, we show that the unknown steady state is just the equilibrium u+ for a large c. Finally, we reveal that the unknown steady state might be a periodic steady state through numerical simulations for some kernel functions. Here we summarize our main results regarding the existence of traveling wave solutions of equation (1.1) as follows. √ Theorem 1.1 For any c ≥ c∗ = 2 r, there exists a traveling wave solution u(x − ct)

4

Traveling Waves for Nonlocal Diffusive Single Species Model

satisfying   −u (x) − cu (x) = ru(x) 1 − a1 (φ ∗ u)(x) − a2 (φ ∗ u)2 (x) , x ∈ R,

(1.5)

under the boundary conditions lim inf u(x) > 0 and x→−∞

lim u(x) = 0.

x→+∞

(1.6)

√ In addition, there is no such a traveling wave solution u(x − ct) with the speed c < 2 r. The following result indicates that for a large c in Theorem 1.1 there holds u(−∞) = u+ . Theorem 1.2 Let  

√  √ a1 2 3 , c = max 4r + 1, 2rn2 K ra1 K + ra2 K + r n2 K 2K + a2 where

⎞−1 ⎛   1 2r 4 n2 = x2 φ(x)dx and K = u+ ⎝ φ(z)dz ⎠ . 3 R 0 

Then the traveling wave solution described in Theorem 1.1 with the speed c > c actually satisfies u(−∞) = u+ . This paper is organized as follows. In Section 2, we prove Theorem 1.1. Section 3 is dedicated to the proof of Theorem 1.2. In Section 4, we perform a new numerical simulation way to illustrate the wave connection of 0 to a periodic steady state. Section 5 is a brief discussion.

2

Existence of traveling wave solutions

In this section, we develop the method described in [4, 8] to prove Theorem 1.1. In Subsection 2.1, we give some priori estimates to explore the existence of solutions in a finite domain. In Subsection 2.2, we consider the existence of traveling wave solution on √ √ R with c = 2 r and the non-existence while c < 2 r. In Subsection 2.3, we prove the √ existence of traveling wave solutions with the speed c > 2 r.

2.1

Existence of solutions in a finite domain

B.-S. Han, Z.-C. Wang and Z. Feng

5

In this subsection we show that for any a > 0, 0 ≤ τ ≤ 1 and ε ∈ (0, u4+ ), there exist a constant c = caτ,ε ∈ R and a function u = uaτ,ε ∈ C 2 ([−a, a], R) such that ⎧   ⎨−u − cu = τ 1 2 in (−a, a), {u≥0} ru 1 − a1 (φ ∗ u) − a2 (φ ∗ u) Tτ,ε (a) : ⎩u(−a) = u , u(0) = ε , u(a) = 0, +

2

⎧ ⎪ ⎪ u , in(−∞, −a), ⎪ ⎨ +

where u=

u, in(−a, a), ⎪ ⎪ ⎪ ⎩0, in(a, ∞).

Here we present a homotopy from a local problem T0,ε (a) to a nonlocal problem T1,ε (a) and then get a solution of T1,ε (a) by using the Leray-Schauder degree. For the sake of convenience, in the sequel we just write u in place of u. If a pair (c, u) is a solution of Tτ,ε (a) attaining a negative minimum of u at xl , then xl ∈ (−a, a) and −u − cu = 0 on a neighborhood of xl . By the maximum principle, we can get u ≡ u(xl ), which is impossible. Consequently, for any solution of Tτ,ε (a) satisfying u ≥ 0, by the maximum principle we have   u > 0 and − u − cu = τ ru 1 − a1 (φ ∗ u) − a2 (φ ∗ u)2 in (−a, a).

(2.1)

The following lemma is regarding the priori estimates for u. Lemma 2.1 There exist M > u+ (only depends on the kernel φ and the constants r, a1 , a2 ) and a0 > 0 (only depends on the constant r) such that for all 0 ≤ τ ≤ 1, a > a0 and ε ∈ (0, u4+ ), each solution (c, u) of Tτ,ε (a) satisfies 0 ≤ u(x) ≤ M,

∀x ∈ [−a, a].

Proof. If τ = 0, that is ⎧ ⎨−u − cu = 0 in (−a, a), T0,ε (a) : ⎩u(−a) = u , u(0) = ε , u(a) = 0. +

2

 := It is easy to see that 0 ≤ u(x) ≤ u+ ≤ M . For 0 < τ ≤ 1, we assume that M maxx∈[−a,a] u(x) > u+ (otherwise, the conclusion holds). From the boundary conditions, . By evaluating (2.1) at xm , we find there exists an xm ∈ (−a, a) such that u(xm ) = M that 1 − a1 (φ ∗ u)(xm ) − a2 (φ ∗ u)2 (xm ) ≥ 0,

6

Traveling Waves for Nonlocal Diffusive Single Species Model

which implies (φ ∗ u)(xm ) ≤ u+ . Because of u ≥ 0, we have   −u − cu = τ ru 1 − a1 (φ ∗ u) − a2 (φ ∗ u)2 ≤ ru , ∀ x ∈ (−a, a). ≤ rM

(2.2)

Let us first consider the case of c < 0. Assume that c < 0. Multiplying both sides of (2.2) by e−|c|x yields e−|c|x , ∀ x ∈ (−a, a). (u e−|c|x ) ≥ −rM Integrating the inequality from x (< xm ) to xm gives u (x) ≤

  rM 1 − e|c|(x−xm ) , ∀ x ∈ (−a, xm ), |c|

, integrating both sides of the last inequality due to the fact u (xm ) = 0. Since u(xm ) = M from x to xm leads to      r M r M |c|(x −x) m  1+ (x − xm ) + 2 1 − e u(x) ≥M |c| |c|    1 − r(x − xm )2 g(|c|(xm − x)) , ≥M where g(y) :=

e−y +y−1 . y2

It is clear that 0 ≤ g(y) ≤

1 2

   1 − r (x − xm )2 , u(x) ≥ M 2

for y ≥ 0, which implies ∀x ∈ [−a, xm ].

Specifically, since u(−a) = u+ , there holds    1 − r (a + xm )2 . u+ ≥ M 2  1 We take a0 = 2r and let  1 . x0 := 2r If xm ∈ (−a, −a + x0 ), inequality (2.4) implies that −1

(a + xm )2  M ≤ u+ 1 − r 2

(2.3)

(2.4)

B.-S. Han, Z.-C. Wang and Z. Feng

7

 r −1 ≤ u+ 1 − x20 2 4 u+ . = 3 If xm ∈ [−a + x0 , a), using (2.3) yields u+ ≥ (φ ∗ u)(xm )  x0 φ(z)u(xm − z)dz ≥ 0  x0  r   φ(z) 1 − z 2 dz. ≥ M 2 0 In view of the definition of x0 , we get ⎛  ⎞−1  1 2r 4u ≤ + ⎝ φ(z)dz ⎠ . M 3 0  Choose M =

4u+ 3





0

1 2r

−1 φ(z)dz

. Then we have completed the proof for the case of

c < 0. For the case of c > 0, we can process in an analogous way by performing integration on [xm , a] instead of [−a, xm ]. For the case of c = 0, integrating the inequality −u ≤ 2M 2 on [x, xm ] twice, we can obtain (2.3) immediately. The rest of the discussions is similar to the above.  Now, we show the priori estimates for c.  Lemma 2.2 For any ε ∈ 0, u4+ , there exists an a0 (ε) > 0 such that for all 0 ≤ τ ≤ 1 √ and a ≥ a0 , each solution (c, u) of Tτ,ε (a) satisfies c ≤ 2 r =: cmax . Proof. Since u ≥ 0, the function u satisfies −u − cu = τ ru[1 − a1 φ ∗ u − a2 (φ ∗ u)2 ] ≤ ru.

(2.5)

√ By way of contradiction, we assume that c > 2 r. Define √

ϕZ (x) := Ze−

rx

,

which satisfies −cϕZ − ϕZ > rϕZ .

(2.6)

Because of u(x) ∈ L∞ (−a, a), we have u(x) < ϕZ (x) when Z > 0 is sufficiently large and u(x) > ϕZ (x) when Z < 0. Then, we define Z0 = inf {Z : ϕZ (x) > u(x) for all x ∈ [−a, a]} .

8

Traveling Waves for Nonlocal Diffusive Single Species Model

It is easy to see that there exists an x0 ∈ [−a, a] such that ϕZ0 (x0 ) = u(x0 ) and Z0 > 0. Using (2.5), (2.6) and the maximum principle, we have that x0 ∈ / (−a, a). Since Z0 > 0, √ we have x0 = −a. Combining it with ϕZ0 (−a) = u+ , we have Z0 = u+ e− ra . However, we know that √ ε u(0) ≤ ϕZ0 (0) = u+ e− ra < u(0) = 2 when a > √1r (ln(2u+ ) − ln ε). This yields a contradiction. Thus, when a > √1r (ln(2u+ ) − ln ε), √ there must be c ≤ 2 r. Consequently, by choosing a0 = √1r (ln(2u+ ) − ln ε), we see that the proof is completed.  Lemma 2.3 For any a > 0 and ε ∈ (0, u4+ ), there exists a cmin (a, ε) > 0 such that for all 0 ≤ τ ≤ 1, each solution (c, u) of Tτ,ε (a) satisfies c ≥ −cmin (a, ε). Since the solution (c, u) of Tτ,ε (a) satisfies −u − cu + r(a1 M + a2 M 2 )u ≥ 0, where M is defined by Lemma 2.1, the proof is closely similar to that of [4, Lemma 2.3] and we omit it. Lemma 2.4 There exist cmin > 0 and a0 > 0 such that for all a ≥ a0 and ε ∈ (0, u4+ ), each solution (c, u) of T1,ε (a) satisfies c ≥ −cmin . Proof. Assume that c ≤ −1 (otherwise, the conclusion is true), and define M > 0 as in Lemma 2.1. It suffices to show that u is uniformly bounded. Since (ecx u (x)) = ecx (u (x)+cu (x)), for x < y we have  x  cx  cy  e u (x) − e u (y) = − ru(z) 1 − a1 (φ ∗ u)(z) − a2 (φ ∗ u)2 (z) ecz dz. y

From Lemma 2.1, we have !  ! !ru(z) 1 − a1 (φ ∗ u)(z) − a2 (φ ∗ u)2 (z) ! ≤ rM (1 + a1 M + a2 M 2 ) =: P. It follows that u (y)e|c|(x−y) − ≤ u (x) ≤ u (y)e|c|(x−y) +

P |c|(x−y) e |c| P |c|(x−y) e , |c|

∀x, y ∈ [−a, a] and x > y.

(2.7)

By choosing x = a and using the fact u (a) ≤ 0, we find u (y) ≤

2P , |c|

∀y ∈ (−a, a).

(2.8)

B.-S. Han, Z.-C. Wang and Z. Feng Define 1 ln L0 := 2 max c≤−1 |c| We claim that for all c ≤ −1 and a ≥ a0 := −

2P ≤ u (x), |c|



L0 2 ,

9

 M c2 +1 . P

there holds

∀x ∈ (−a, a − L0 ].

(2.9)

 By way of contradiction, we assume that − 2P |c| > u (y) for some y ∈ (−a, a − L0 ]. It follows from (2.7) that P u (x) ≤ − e|c|(x−y) for x > y. |c|

Integrating both sides from y to a and using u(a) = 0 gives M ≥ u(y) ≥

 P   P  |c|(a−y) |c|L0 − 1 ≥ − 1 , e e c2 c2

which contradicts the definition of L0 . Since φ ∈ L1 (R), there exists a R > 0 such that  u+ − ε 3u+ ≤ . M φ≤ 32 8 [−R,R]c We know that u(−a) = u+ and u(0) = 2ε in T1 (a), so we can define x0 < 0 as the largest negative real number such that u(x0 ) = 12 (u+ + 2ε ). Then by (2.9), for x ∈ " [x0 − R, x0 + 2R] [−a, a], we have u(x) ≥

ε 2P u+ 3ε u+ + − 2R ≥ + , 2 4 |c| 4 8

(2.10)

32P R . Similarly, using (2.8) leads to where c ≤ − 2u + −ε

u(x) ≤

ε 2P 3u+ ε u+ + + 2R ≤ + . 2 4 |c| 4 8

(2.11)

From (2.10) and (2.11), we know that [x0 − R, x0 + 2R] ⊂ (−a, 0) if −c is large enough. Then for x ∈ [x0 , x0 + R], we deduce that   (φ ∗ u)(x) ≤ φ(y)u(x − y)dy + φ(y)u(x − y)dy [−R, R] [−R, R]c  u+M φ ≤ max [x0 −R, x0 +2R]

≤

7u+ , 8

where c is the same as given in (2.10).

[−R, R]c

10

Traveling Waves for Nonlocal Diffusive Single Species Model

If u is not non-increasing on [x0 , x0 + R], from the definition of x0 there exist an x # ∈ (x0 , x0 + R) such that u(x) attains the local minimum at x #. From (2.1), we have 32P R 128P R (φ ∗ u)(# x) ≥ u+ , which is only possible when c > − 2u+ −ε ≥ − 7u+ . If u is non-increasing on [x0 , x0 + R] and c ≤ 0, for x ∈ [x0 , x0 + R] we have u (x) ≤ u (x) + cu (x) = −ru[1 − a1 φ ∗ u − a2 (φ ∗ u)2 ]

  

u+ 3ε 7u+ 2 7a1 u+ ≤ −r + 1 − a1 − a2 4 8 8 8  2 

ru+ 7a1 u+ 7u+ ≤− 1− − a2 . 4 8 8 It follows that

 2 

7u R 7a u ru + 1 + + 1 − a1 − a2 . u (x0 ) − u (x0 + R) ≥ 4 8 8

Combining (2.8) and (2.9) yields c≥−

ru+ R 1 −

16P a1 7a18u+

− a2



7u+ 8

2  .

⎧ ⎪ ⎪ ⎨

Let cmin := max

⎫ ⎪ ⎪ 128P R ⎬

16P

 2  , 7u ⎪ . ⎪ + ⎪ 7u+ 7a1 u+ ⎪ ⎩ ru+ R 1 − a1 8 − a2 8 ⎭

One can see that the proof is completed.  Now we are in the position to construct a solution (c, u) of T1,ε (a) by using the LeraySchauder topological degree argument [7].  Proposition 2.5 For each ε ∈ 0, u4+ , there exists a0 (ε) > 0 such that for all a ≥ a0 , we can find a solution (c, u) of T1,ε (a) which satisfies ⎧ ⎪ ⎪ −u − cu = ru(1 − a1 (φ ∗ u) − a2 (φ ∗ u)2 ) in (−a, a), ⎪ ⎨ u(−a) = u+ , u(0) = 2ε , u(a) = 0, ⎪ ⎪ ⎪ ⎩u > 0, on (−a, a), and uC 2 (−a,a) ≤ K,

−cmin ≤ c ≤ cmax ,

where K is a constant, which is independent of ε and a.

B.-S. Han, Z.-C. Wang and Z. Feng

11

 Proof. Let ε ∈ 0, u4+ . Given the function v ≥ 0 defined on (−a, a) with v(−a) = u+ and v(a) = 0, we consider a family of linear problems ⎧ ⎨−u − cu = τ rv(1 − a φ ∗ v − a (φ ∗ v)2 ) in (−a, a), 1 2 Sτc (a) : ⎩u(−a) = u , u(a) = 0. +

We denote by Fτ the mapping of the Banach space X := R × C 1,α (−a, a), equipped with the norm (c, v)X := max(|c|, vC 1,α ) onto itself defined by Fτ : (c, v) →

ε 2

 − v(0) + c, ucτ := the solution of Sτc (a) .

Constructing a solution (c, u) of T1,ε (a) is equivalent to proving that the kernel of Id − F1 is nontrivial. We are able to apply the Leray-Schauder topological theory because Fτ is compact and continuous dependence of the parameter 0 ≤ τ ≤ 1. Define E := {(c, v) : −cmin (a, ε) − 1 < c < cmax + 1, v > 0, vC 1,α < M + 1} ⊂ X, where cmin (a, ε), cmax and M are defined as in Lemma 2.3, Lemma 2.2 and Lemma 2.1, respectively. From Lemmas 2.1-2.3, it is easy to see that there exists an a0 (ε) > 0 such that for any a ≥ a0 and any τ ∈ [0, 1], the operator Id−Fτ can not vanish on the boundary ∂E. Thus, by the homotopy invariance of the degree we obtain deg(Id − F1 , E, 0) = deg(Id − F0 , E, 0). In addition, a direct calculation gives uc0 (x) = u+

e−cx − e−ca if c = 0 eca − e−ca

uc0 (x) = −

u+ u+ x+ if c = 0. 2a 2

and

In particular, uc0 (x) is decreasing with regard to c. Hence, by using two additional homotopies (see [8, p.2834] or [3, 4] for details), we have deg(Id − F0 , E, 0) = −1, and then deg(Id − F1 , E, 0) = −1. This implies that there is a (c, u) ∈ E solving T1,ε (a). Consequently, it follows from Lemma 2.2 and Lemma 2.4 that −cmin ≤ c ≤ cmax . The existence  of the constant K > 0, which is independent of ε ∈ 0, u4+ , directly follows from the standard elliptic estimates. This completes the proof. 

12

2.2

Traveling Waves for Nonlocal Diffusive Single Species Model

Existence of solution on R

In this subsection we establish the existence of solutions of the problem (1.5)-(1.6) √ with c = 2 r. Here we first prove a lemma which play a key role in this subsection.  > 0 and ξ < η, there exists an ε = ε(M , ξ, η) > 0 with ε ∈ Lemma 2.6 For all M  u+  and 0, 4 such that if u(x) is a solution of equation (1.5) with c ∈ [ξ, η], 0 ≤ u ≤ M inf x∈R u(x) > 0, then there holds inf x∈R u(x) > ε. Proof. On the contrary, we assume that (cn , un ) is a sequence of solutions of equation (1.5) which satisfies ηn := inf un > 0 with ηn → 0 as n → +∞. x∈R

Let vn = un /ηn . Since inf x∈R vn = 1 for all n, there exists an xn ∈ R such that vn (xn ) ≤ 1 + n1 . Let wn (x) = vn (x + xn ). Then we get −wn − cn wn = rwn (1 − a1 (φ ∗ un ) − a2 (φ ∗ un )2 ),  and the coefficients above are uniformly where un = u(x + xn ). Since supx∈R un (x) ≤ M bounded with regard to n, one can extract a subsequence of wn which converges locally uniformly to a function w∞ (x) and cn → c ∈ [ξ, η]. In addition, because of un (0) ≤ ηn (1 + n1 ), it follows from the Harnack inequality that un (x) → 0 locally uniformly in x ∈ R. Thus we have   −w∞ (x) − cw∞ (x) = rw∞ (x) in x ∈ R.

By the definition of w∞ (x), we have w∞ (0) = 1 and w∞ ≥ 1. However, using the strong maximum principle leads to w∞ ≡ 1, which is a contradiction because r = 0. So we complete the proof.  Remark 2.7 Let M > 0, cmax > 0 and cmin > 0 be defined in Lemmas 2.1, 2.2 and 2.4, respectively. It is clear that they are independent of ε ∈ (0, u4+ ). It follows that there  exists ε = ε(M, −cmin , cmax ) with ε ∈ 0, u4+ so that Lemma 2.6 holds. In the following we prove the existence of solutions of the problem (1.5)-(1.6) with √ c = 2 r. Let ε = ε(M, −cmin , cmax ) ∈ (0, u4+ ) be determined by Remark 2.7. It follows from Proposition 2.5 that for any a > a0 (ε ), Problem T1,ε (a) admits a solution (ca , ua ) satisfying −cmin ≤ ca ≤ cmax , ua C 2 (−a,a) ≤ K and ua C 0 [−a,a] ≤ M , where K > 0 is independent of a > a0 (ε ). Taking a sequence {an } with an → +∞ and passing to the

B.-S. Han, Z.-C. Wang and Z. Feng

13

limit of (can , uan ) as n → +∞, we know that there exist a positive function u ∈ Cb2 (R) with 0 ≤ u ≤ M and a speed c ∈ R with −cmin ≤ c ≤ cmax such that ⎧ ⎨−cu − u = ru 1 − a (φ ∗ u) − a (φ ∗ u)2 in R, 1 2 (2.12)  ⎩u(0) = ε . 2

To complete the proof of the main result of this subsection, it suffices to show that u(x) √ satisfies (1.6) and c = 2 r. Lemma 2.8 There exists a sequence xn such that |xn | → +∞ and u(xn ) → 0 as n → +∞. The proof of this lemma is straightforward by way of contradiction. Conversely, assume that it has inf R u > 0. By Lemma 2.6 and the definitions of c, u and ε , we know that  inf R u ≥ ε . It is apparently a contradiction with u(0) = ε2 . By virtue of Lemma 2.8, we conclude that there exists a sequence {xn } satisfying either lim xn = +∞ and

n→+∞

lim u(xn ) = 0,

(2.13)

lim u(xn ) = 0.

(2.14)

n→+∞

or lim xn = −∞ and

n→+∞

n→+∞

The following lemma shows that in the first case of (2.13), u(x) is monotonically decreasing for sufficiently large x and limx→+∞ u(x) = 0. Lemma 2.9 There exists a Z0 > 0 such that u(x) is monotonically decreasing for x > Z0 and limx→+∞ u(x) = 0. Proof. The proof can be processed straightforwardly by way of contradiction. Assume that u(x) is not eventually monotonic. Then there exists a sequence zn → +∞ such that u(x) attains a local minimum at zn and u(zn ) → 0. From (2.12), we know that (φ ∗ u)(zn ) ≥ u+ .

(2.15)

On the other hand, since u(x) is bounded in C 2 (R), by the Harnack inequality we know  that, for any Z > 0 and any δ ∈ 0, u4+ there exists an N > 0 such that u(x) ≤ δ for all x ∈ (zn − Z, zn + Z) for n > N . However, it is impossible for any sufficiently large Z due to (2.15).  √ Lemma 2.10 There holds c ≥ 2 r.

14

Traveling Waves for Nonlocal Diffusive Single Species Model

Proof. Let vn (x) = u(x + xn )/u(xn ). Since u satisfies (2.15), we have  #n ) − a2 (φ ∗ u #n )2 in R, −vn − cvn = rvn 1 − a1 (φ ∗ u #n converges to zero where u #n (x) = u(x + xn ). By the Harnack inequality, we see that u locally uniformly in x as n → +∞. Assume that the sequence vn converges to a function v(x) which satisfies −v  − cv  = rv in R. (2.16) √ Note that v is non-negative and v(0) = 1. We see that c ≥ 2 r since equation (2.16) √ admits such a solution if and only if c ≥ 2 r.  Lemma 2.11 There holds lim inf u(x) > 0. x→−∞

Proof. We prove this lemma by way of contradiction. On the contrary, we assume that there exists a sequence yn → −∞ as n → +∞, such that u(yn ) → 0. Take u #(x) = u(−x) √ and # c = −c, then u #(−yn ) → 0. By using Lemmas 2.9 and 2.10, it has # c ≥ 2 r, which √ implies c ≤ −2 r. So this is a contradiction.  For the second case of (2.14), using the transforms of u #(x) = u(−x) and # c = −c, by a closely similar argument to that of Lemmas 2.9-2.11 for (# c, u #), one can complete the √ proof of Theorem 1.1 for c = 2 r. In addition, the proof of Lemma 2.10 implies that the √ problem (1.5)-(1.6) with c < 2 r admits no positive solutions.

2.3

√ Traveling wave solutions with speeds c > 2 r

In this section, we prove the second part of Theorem 1.1, namely, we establish the √ existence of solutions of (1.5) and (1.6) for c > 2 r. Our method is to first consider a two-point boundary value problem on a finite interval and then take the limit of solutions of the problem as the interval passes to the whole line. Specifically, the solutions of the two-point boundary value problem are obtained by constructing super- and subsolutions and using Schauder fixed point theorem. Supersolution. Let q c (x) = e−λc x with λc > 0 being the smaller root of λ2c − cλc + r = 0. Then one has  −cq c = q c + rq c ≥ q c + rq c 1 − a1 (φ ∗ q c ) − a2 (φ ∗ q c )2 ,

B.-S. Han, Z.-C. Wang and Z. Feng where

15



φ(y)e−λc (x−y) dy  −λc x = e φ(y)eλc y dy

(φ ∗ q c )(x) =

R

= Zc e

R −λc x

.

This implies that q c is a super-solution. Subsolution. Let

1 −λc x − e−(λc +ε)x , e A where ε ∈ (0, λc ) is small enough such that pc (x) =

ϑc = c(λc + ε) − (λc + ε)2 − r > 0, and A > 1 is large enough such that ln A 1 > ln ε λc − ε



Then for x satisfying pc (x) > 0, namely, x >

a1 rZc + a2 rZc2 ϑc ln A ε ,

 .

we have

− cpc − pc − rpc + a1 rq c (φ ∗ q c ) + a2 rq c (φ ∗ q c )2 = − ϑc e−(λc +ε)x + a1 rZc e−2λc x + a2 rZc2 e−3λc x   =e−(λc +ε)x −ϑc + a1 rZc e−(λc −ε)x + a2 rZc2 e−(2λc −ε)x   ≤e−(λc +ε)x −ϑc + a1 rZc e−(λc −ε)x + a2 rZc2 e−(λc −ε)x <0. Let pc (x) = max(0, pc (x)), x ∈ R. We then find −cpc ≤ pc + r · pc − a1 rpc (φ ∗ q c ) − a2 rpc (φ ∗ q c )2 , ∀x =

ln A . ε

√ A two-point boundary value problem. For c > 2 r, we consider the problem in a finite domain (−a, a):  −cu − u = ru 1 − a1 (φ ∗ u) − a2 (φ ∗ u)2 , (2.17) u(±a) = pc (±a),

16

Traveling Waves for Nonlocal Diffusive Single Species Model

where a >

ln A ε .

To establish the existence of the problem (2.17), we consider the following two-point boundary problem  −cu = u + ru0 − ru a1 (φ ∗ u0 ) + a2 (φ ∗ u0 )2 ,

(2.18)

u(±a) = pc (±a), where u0 ∈ Ma and the convex set Ma is defined as Ma = {u ∈ C[−a, a] : pc (x) ≤ u(x) ≤ q c (x)}. Let Ψa be the solution mapping of the problem (2.18). That is, Ψa u0 = u. It is clear that a solution of the problem (2.17) is a fixed point of the problem (2.18). It is easy to see that Ψa is compact. It suffices to show that the set Ma is invariant for the mapping Ψa . Given u0 ∈ Ma , since u ≡ 0 is a subsolution of the problem (2.18), we have u(x) > 0 for any x ∈ (−a, a). Hence, we get − u − cu + ru(a1 (φ ∗ u0 ) + a2 (φ ∗ u0 )2 ) =ru0 ≤rq c = − q c − cq c

 ≤ − q c − cq c + rq c a1 (φ ∗ u0 ) + a2 (φ ∗ u0 )2 , where u(±a) = pc (±a) ≤ q c (±a). By the maximum principle, we know that u(x) ≤ q c (x) for all x ∈ (−a, a). On the other hand, we have  − cu − u + ru a1 (φ ∗ u0 ) + a2 (φ ∗ u0 )2 =ru0 ≥rpc

 ≥ − pc − cpc + rpc a1 (φ ∗ q c ) + a2 (φ ∗ q c )2 

 ln A   2 ≥ − pc − cpc + rpc a1 (φ ∗ u0 ) + a2 (φ ∗ u0 ) , in x ∈ ,a . ε where u(±a) = pc (±a). Using the maximum principle again leads to u(x) ≥ pc (x) for all  x ∈ lnεA , a . Thus, we see that the set Ma is invariant.

By the Schauder fixed point theorem, we know that Ψa has a fixed point ua in Ma , which is just the solution of (2.17). In addition, we have the following lemma.

B.-S. Han, Z.-C. Wang and Z. Feng

17

√ Lemma 2.12 There exists a constant M0 which does not depend on c > c∗ = 2 r such that each solution of the problem (2.17) satisfies 0 ≤ ua (x) ≤ M0 for all a > lnεA and all x ∈ (−a, a). The proof of this lemma is similar to that of Lemma 2.1, so we omit the details. Take the limit of ua as a → +∞. From Lemma 2.12 and the standard elliptic estimates, we know that there exists K > 0 such that ua C 2,α (− a , a ) ≤ K for any a > lnεA , where 2 2 α ∈ (0, 1) is some constant. Letting a → +∞ (possibly along a subsequence), then we 2 (R) and u(x) satisfies have ua → u in Cloc  −cu = u + ru 1 − a1 (φ ∗ u) − a2 (φ ∗ u)2 , x ∈ R. Moreover, we know that pc (x) ≤ u(x) ≤ min {M0 , q c (x)}, which implies lim u(x) = 0.

x→+∞

Using an analogous argument as the proof of Lemma 2.11, we can get lim inf u(x) > 0. x→−∞

√ Thus, we have established the existence of (1.5) and (1.6) for c > 2 r. Consequently, the proof of Theorem 1.1 is completed.

3

Rapid waves connecting u ≡ 0 with u ≡ u+

In the last section, we have proved that equation (1.5) admits traveling wave solutions connecting the equilibrium 0 to an unknown positive steady state u∞ (x) for all speeds √ c ≥ 2 r. It follows from [24] that the unknown positive steady state should be the equilibrium u+ if the nonlocality is sufficiently weak. In this section we further show that for a more general kernel function φ, the unknown positive steady state is also the equilibrium u+ if the wave speed c is larger than some positive constant. For convenience of our statement, we define  ni := |z|i φ(z)dz, i = 1, 2. R

Proof of Theorem 1.2 Assume that (c, u) satisfies the problem (1.5)-(1.6), where √ c > 2 r. It suffices to show that limx→−∞ u(x) exists and is equal to u+ if 

 √  √ a1 2 3 , 4r + 1, 2rn2 K ra1 K + ra2 K + r n2 K 2K + c > max a2

18

Traveling Waves for Nonlocal Diffusive Single Species Model

where

⎛  ⎞−1  1 2r 4 ⎝ φ(z)dz ⎠ . K = u+ 3 0

We divide our discussions into three steps. Step 1. uL∞ and u L∞ is bounded. It follows from Lemmas 2.1 and 2.12 that uL∞ ≤ K. In addition, we know that u(x) satisfies  ∞ 1 u(x) = [eλ1 (x−y) − eλ2 (x−y) ][ra1 u(y)(φ ∗ u)(y) + ra2 u(y)(φ ∗ u)2 (y)]dy, λ2 − λ1 x where λ1 < λ2 < 0 are two negative roots of the characteristic equation λ2 + cλ + r = 0. So there holds  ∞ 1  u (x) = [λ1 eλ1 (x−y) − λ2 eλ2 (x−y) ][ra1 u(y)(φ ∗ u)(y) + ra2 u(y)(φ ∗ u)2 (y)]dy, λ2 − λ 1 x which implies |u (x)| ≤ √

 2 ra1 K 2 + ra2 K 3 =: K  , ∀x ∈ R. c2 − 4r

Step 2. u ∈ L2 (R) and limx→±∞ u (x) = 0 if  

√  √ a1 c > max . 4r + 1, 2rn2 K ra1 K 2 + ra2 K 3 + r n2 K 2K + a2 Define W  (x) = x(u+ − x)(x − u− ). We rewrite equation (1.5) as cu = − u − ru (φ ∗ u − u− ) (u+ − φ ∗ u) = − u − ru (φ ∗ u − u + u − u− ) (u+ − u + u − φ ∗ u) = − u − ru(u − u− )(u+ − u) − ru(u − φ ∗ u)(2u − u− − u+ ) + ru(u − φ ∗ u)2 . We derive that  c

!' ! B ! (B !! ! 1 ! ! ! ! 2 2  ! ! + r u dx ≤ ! − u − rW (u) u u (u − φ ∗ u) dx ! ! ! ! 2 −A −A −A ! ! B ! ! !  ! +r! uu (2u − u− − u+ )(u − φ ∗ u)dx!! B

2

−A



B

≤K + 2rW L∞ (−K,K) + rKK (u − φ ∗ u)2 dx −A ! B ! ! !  ! +r! uu (2u − u− − u+ )(u − φ ∗ u)dx!! . 2



−A

(3.1)

B.-S. Han, Z.-C. Wang and Z. Feng Define

 ΥA,B =

Since u+ + u− =

− aa12 ,

B

19

uu (2u − u− − u+ )(u − φ ∗ u)dx.

−A

by the Cauchy-Schwarz inequality we get

Υ2A,B

2  B a1 u u 2u + ≤ dx (u − φ ∗ u)2 dx a2 −A −A

   B a1 2 B 2 2 u dx (u − φ ∗ u)2 dx. ≤K 2K + a2 −A −A 

B







(3.2)

For a given x, we have  φ(x − y)(u(x) − u(y))dy   1 φ(x − y)(x − y)u (x + t(y − x))dtdy. =

(u − φ ∗ u) (x) =

R

R

0

Using the Cauchy-Schwarz inequality again yields   1   1 2 2 φ(x − y)(x − y) dtdy φ(x − y)u2 (x + t(y − x))dtdy (u − φ ∗ u) (x) ≤ R 0 R 0   1 φ(−z)u2 (x + tz)dzdt. ≤n2 0

R

It further gives 

B −A



2

(u − φ ∗ u) (x) ≤ n2

Since |u | ≤ K  , it has  B+tz u

2



≤

−A

2 .

2

−A+tz  B 2

u2 (y)dydzdt.



B



2

B+tz

u + −A

u2

B

u + 2t|z|K 2 .

−A



≤ n2 −A u

B+tz

−A+tz

R

u +

2

B

0

−A

(u − φ ∗ u) (x) ≤ n2

Let HA,B :=

 φ(−z)

=

−A+tz

We further find  B

1

B

−A  B −A

2

u + 2n2 K

2



1

tφ(−z)|z|dzdt 0

R

u2 + n1 n2 K 2 .

From (3.1)-(3.3) we see that

|c|HA,B ≤K 2 + 2rW L∞ (−K,K) + rKK  (n2 HA,B + n1 n2 K 2 )

(3.3)

20

Traveling Waves for Nonlocal Diffusive Single Species Model

  a1  + rK 2K + HA,B n2 HA,B + n1 n2 K 2 . a2  √ B If c > rKK  n2 + rK 2K + aa12 n2 , then HA,B := −A u2 is bounded, which explicitly implies u ∈ L2 . Since u is uniformly continuous on R, it has lim u (x) = 0.

x→±∞

Step 3. limx→−∞ u(x) exists and is equal to u+ if  

√  √ a1 2 3 c > max . 4r + 1, 2rn2 K ra1 K + ra2 K + r n2 K 2K + a2 Define a set Γ as limit points of u at −∞. Because u is bounded, we know that Γ is not empty. Let ξ ∈ Γ. There exists a sequence xn → −∞ such that u(xn ) → ξ. Thus, vn (x) = u(x + xn ) satisfies  vn + cvn = −rvn 1 − a1 (φ ∗ vn ) − a2 (φ ∗ vn )2 in R. From the interior elliptic estimates and the Sobolev embedding theorem, one can extract 1,β a subsequence of vn , still denoted by vn , satisfying vn → v strongly in Cloc (R) and weakly 2,p in Wloc (R), respectively. Then by Step 2, we have v  (x) = lim u (x + xn ) = 0, ∀x ∈ R. n→∞

In addition, v satisfies  v  + cv  = −rv 1 − a1 (φ ∗ v) − a2 (φ ∗ v)2 on R, which means v ≡ 0 or v ≡ u+ . Since v(0) = limn→∞ u(xn ) = ξ, then ξ ∈ {0, u+ }. Because u is continuous and Γ is connected, we know that Γ = {0} or Γ = {u+ }. Combining it with (1.6), we know that lim u(x) = u+ . x→−∞

Consequently, we have completed the proof of Theorem 1.2.

4

Numerical simulations

In Section 2, we prove that equation (1.1) admits traveling wave solutions connecting √ u = 0 to an unknown positive steady state for all speeds c ≥ 2 r. In the preceding section we show that this unknown steady state is u(−∞) ≡ u+ for a sufficiently large c. Nevertheless, very little has been known on the qualitative properties of slow traveling wave

B.-S. Han, Z.-C. Wang and Z. Feng

21

solutions such as the shape of wave profiles and the steady state at positive infinity etc. In the following we explore these features of traveling wave solutions based on numerical simulations and provide some biological explanations on those simulation results. Here we mainly consider two specific kernel functions. |x| 1 − σ (I) φ(x) = φσ (x) = 2σ e , where σ > 0 is a constant. Let v(x, t) = (φσ ∗ u)(x, t). Its second order derivative with respect to x is vxx = −

1 (u − v) . σ2

Then equation (1.1) can be rewritten as ⎧ ⎨u = u + ru(1 − a v − a v 2 ), t xx 1 2 ⎩0 = v + 1 (u − v) , xx

(4.1)

σ2

for (x, t) ∈ R × (0, ∞). Before performing numerical simulations, we define the initial condition of u(x, t) as ⎧ ⎨u , for x ≤ L , + 0 u(x, 0) = (4.2) ⎩0, for x > L0 . Since v(x, 0) is determined by  v(x, 0) = R

we have

1 − |x−y| e σ u(y, 0)dy, 2σ

⎧ 0 ⎨u − u+ e x−L σ , for x ≤ L0 , + 2 v(x, 0) = x−L0 u − ⎩ +e σ , for x > L0 . 2

(4.3)

Note that the zero-flux boundary condition is considered here. Along with (4.2) and (4.3), we perform simulations on system (4.1) by using the Matlab pdepe solver (see Figure 1). Figure 1 indicates that the solution of equation (1.1) possesses a ‘hump’ as σ increases, but the equilibrium point u = u+ is stable. This phenomenon can be explained as follows. Plugging u = u # + u+ and v = v# + u+ into system (4.1), we get a linearized system ⎧ ⎨u #t = u #xx + r(1 − a1 u+ − a2 u2+ )# u + ru+ (−a1 − 2a2 u+ )# v, (4.4) ⎩0 = v# + 1 (# u − v#). xx

σ2

Choose the test function of the form     ∞ ) Ck1 u # eλt+ikx , = 2 Ck v# k=1

(4.5)

22

Traveling Waves for Nonlocal Diffusive Single Species Model

σ=1

σ=5/3

2 species u

species u

2

1

1

0 15

0 15

10

40

10 time t

40

20

5 0

5 time t

distance x

σ=50/29

0

distance x σ=20/11

2

2 species u

species u

20

1

0 15

1

0 15 10

40 5

time t

distance x

20

5

20 0

40

10 time t

0 0

distance x

Figure 1: The time and space evolution for nonlocal equation (1.1) with the kernel |x| 1 − σ φσ (x) = 2σ e . The computational domain is x ∈ [0, 40] and t ∈ [0, 15]. The values of corresponding parameters are L0 = 10, r = 1, a1 = a2 = 12 , and σ takes the values 20 of 1, 53 , 50 29 and 11 , respectively.

B.-S. Han, Z.-C. Wang and Z. Feng

23

where k is a real parameter. Substituting (4.5) into (4.4) we find ! ! !−k 2 − λ + r(1 − a u − a u2 ) ru (−a − 2a u )! ! 1 + 2 e + 1 2 + ! ! ! = 0. 1 1 2 ! ! − − k 2 2 σ σ So we have

 λ + k2



1 + k2 σ2

 +

ru+ (a1 + 2a2 u+ ) = 0. σ

(4.6)

For any σ, from (4.6) we see that λ is negative. It implies that (u, v) = (u+ , u+ ) is stable. Thus, the equilibrium state u = u+ of equation (1.1) is stable. Moreover, we can provide a biological interpretation about the hump. It is known that the traveling wave connecting 0 to a positive steady state can be interpreted as the species moving out and colonizing a region of space that was previously uninhabited. It means that each individual is competing with those in front and with those behind at the same time. If the species invade to a previously uninhabited region, they will find no individual ahead of them. So they are only competing with individuals behind, which allows population to raise above the carrying capacity level in a short time, namely, a ‘hump’ in the figure.    (II) φ(x) = φσ (x) =

a |x| A −σ σe

− σ1 e−

|x| σ

, where A =

3a 2

> 0 and a ∈

2 3,

2 3

.

Let φ+ σ (x) = Define

A − a |x| 1 − |x| e σ e σ. and φ− σ (x) = σ σ

 v(t, x) = φ+ σ ∗ u (t, x)

and

 w(t, x) = φ− σ ∗ u (t, x).

Differentiating v(x, t) and w(x, t) with respect to x twice gives vxx = −

1  2 1 3a u − a2 v and wxx = − 2 (−2u − w) . 2 σ σ

So equation (1.1) reduces to ⎧  ⎪ ⎪ u = uxx + ru 1 − a1 (v + w) − a2 (v + w)2 , ⎪ ⎨ t  0 = vxx + σ12 3a2 u − a2 v , ⎪ ⎪ ⎪ ⎩0 = wxx + 1 (−2u − w) . σ2 Similar to the above processes, we set ⎧ ⎨u , for x ≤ L , + 0 u(x, 0) = ⎩0, for x > L0 .

(4.7)

(4.8)

24

Traveling Waves for Nonlocal Diffusive Single Species Model

By the definition of v(x, t), we find that ⎧ 0) ⎨3u − 3 u e a(x−L σ , for x ≤ L0 , + 2 + v(x, 0) = a(x−L0 ) ⎩ 3 u e− σ , for x > L0 . 2 + Simultaneously, w(x, 0) is given by ⎧ 0 ⎨2u − u e x−L σ , for x ≤ L0 , + + w(x, 0) = x−L0 ⎩u e− σ , for x > L0 . +

(4.9)

(4.10)

Using (4.8)-(4.10) and the zero-flux boundary conditions, we perform simulations on system (4.7) by the Matlab pdepe solver (see Figure 2). From Figure 2, it is shown that as σ increases, not only a ‘hump’ occurs, but also the stability of the equilibrium u = u+ will change. In particular, the unknown steady state connected by traveling wave solutions at negative infinity will be a periodic solution around the equilibrium u = u+ . Now we give a fundamental explanation of the phenomenon. Since system (4.7) has three equilibria (0, 0, 0), (u+ , 3u+ , −2u+ ) and (u− , 3u− , −2u− ), we employ the linear analysis of system (4.7) near the equilibrium point (u+ , 3u+ , −2u+ ) as follows: ⎧  ⎪ ⎪ ut = uxx + r 1 − a1 u+ − a2 u2+ u + (−3a1 − 2a2 ) ru2+ v + 2r (a1 − a2 ) u2+ w, ⎪ ⎨  (4.11) 0 = vxx + σ12 3a2 u − a2 v , ⎪ ⎪ ⎪ ⎩0 = wxx + 1 (−2u − w) . σ2 Using the Fourier expansions about the variable states u, v and w gives ⎞ ⎛ ⎛ ⎞ u Ck1 ∞ ) ⎟ ⎜ ⎜ 2 ⎟ λt+ikx , ⎝ v ⎠= ⎝ Ck ⎠ e w

k=1

(4.12)

Ck3

where λ is the growth rate of perturbation in time t, i is the imaginary unit (i2 = −1) and k is the wave number. By (4.12) and (4.11) we have ! ! !r(1 − a1 u+ − a2 u2 ) − k 2 − λ (−a1 − 2a2 u+ )ru+ (−a1 − 2a2 u+ )ru+ ! + ! ! ! ! a2 3a2 2 − − k 0 ! ! = 0, σ2 σ2 ! ! 2 1 2 ! ! − σ2 0 − σ2 − k which is equivalent to 

 

2 a2  2 1 1 a 2 2 2 2 +k + k + ru+ 2 (a1 + 2a2 u+ ) − 2 + (2 − 3a )k = 0. − k +λ σ2 σ2 σ σ This indicates that λ can be positive when σ takes the appropriate values. So the equilibrium point u = u+ of system (1.1) becomes unstable (see Figure 2).

σ

σ

σ

σ

26

5

Traveling Waves for Nonlocal Diffusive Single Species Model

Discussion

In this paper, we dealt with the existence of traveling wave solutions of system (1.1) √ connecting u = 0 to an unknown positive steady state for all speeds c ≥ 2 r. We showed that the unknown steady state is u(−∞) ≡ u+ for a sufficiently large c. Furthermore, numerical simulations indicate that the unknown steady state can be either the positive equilibrium u+ or a positive periodic solution around the positive equilibrium u+ even for slow traveling wave solutions. At this stage we are still working on a rigorous analysis on this problem, which should be submitted in a subsequent work somewhere else.

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