Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 725495, 12 pages http://dx.doi.org/10.1155/2013/725495
Research Article Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion Xiaojing Yu, Peixuan Weng, and Yehui Huang School of Mathematics, South China Normal University, Guangzhou 510631, China Correspondence should be addressed to Peixuan Weng;
[email protected] Received 17 November 2012; Revised 11 March 2013; Accepted 21 March 2013 Academic Editor: Dumitru Baleanu Copyright Β© 2013 Xiaojing Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study a competing pioneer-climax species model with nonlocal diffusion. By constructing a pair of upper-lower solutions and using the iterative technique, we establish the existence of traveling wavefronts connecting the pioneer-existence equilibrium and the coexistence equilibrium. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts as π = π₯ + ππ‘ β ββ.
1. Introduction As we know, the interactions among species are important in determining the process of evolution for the ecosystem, and the modeling accompanied with the mathematical analysis of the models can help people to understand and control the propagation of species. In general, the per capital growth rate (i.e., fitness) for a species in the model is assumed to be a function of a weighted total density of all interacting species. A well-known example is the standard Lotka-Volterra model; its fitness of a species is a linear function. It is natural to consider other kinds of fitness functions other than the linear one, because of the various species and interaction rules. In this paper, we will analyze a reaction-diffusion model describing pioneer and climax species. This model describes interaction among species with peculiar fitness functions. A species is called a pioneer species if it thrives best at lower density but its fitness decreases monotonically with total population density for overcrowded. Thus, the fitness function of a pioneer species is assumed to be a decreasing function. Pine and yellow poplar are the species of this type. A species is called a climax species if its fitness increases up to a maximum value and then decreases of its total density. Hence, a climax population is assumed to have a nonmonotone, βonehumpedβ smooth fitness function. Oak and maple are the climax species.
A typical reaction-diffusion model for a pioneer-climax species is given by the following system: π2 π’ ππ’ = π1 2 + π’π (π11 π’ + π12 V) , ππ‘ ππ₯ πV π2 V = π2 2 + Vπ (π21 π’ + π22 V) , ππ‘ ππ₯
(1)
where π’ and V represent densities of the pioneer and climax species, respectively. π and π denote the pioneer fitness function and climax fitness function, respectively, πππ > 0 (π, π = 1, 2). By making changes of variables π’Μ = π21 π’, ΜV = π12 V, system (1) changes into the form (the tildes of π’Μ and ΜV are dropped out) π2 π’ ππ’ = π1 2 + π’π (π11 π’ + V) , ππ‘ ππ₯ πV π2 V = π2 2 + Vπ (π’ + π22 V) , ππ‘ ππ₯
(2)
where we still use π11 and π22 as the new coefficients without confusing. From the previous introduction, we assume that the pioneer fitness function π satisfies πσΈ (π§) < 0,
π (π§0 ) = 0
(3)
2
Abstract and Applied Analysis
π()
π(π’)
π§0 π
π’
π
(a)
π€1
π€2
(b)
Figure 1: Typical fitness functions for pioneer species and climax species.
for some π§0 > 0, and the climax fitness function π satisfies π (π€1 ) = π (π€2 ) = 0,
0 < π€1 < π€2 ,
(π€β β π€) πσΈ (π€) > 0 for π€ =ΜΈ π€β β (π€1 , π€2 ) .
(4)
β π, Hassell Ricker [1] used the fitness function π(π’) = π and Comins [2] used the fitness function π(π’) = (π/(1 + ππ’)π ) β π, and Cushing [3] used the fitness function π(π’) = π’ππ(1βπ’) β π. It is obvious that these π and π have the curves in Figure 1. There are some existing results about the stability and traveling wave solutions for (2) ([4β6]). About traveling wave solutions, Brown et al. [4] studied the traveling wave of (1) connecting two boundary equilibria by singular perturbation technique, and Yuan and Zou [6] obtained the existence of traveling wave solutions connecting a monoculture state and a coexistence state by upper-lower solution method combined with the Schauder fixed point theorem. Also see van Vuuren [7] for the existence of traveling plane waves in a general class of competition-diffusion systems and Murray [8] for more biological description of traveling wave solutions. For system without spatial diffusion, the model will be πV = Vπ (π’ + π22 V) . ππ‘
+β ππ’ (π‘, π₯) = π1 β« π½ (π¦ β π₯) [π’ (π‘, π¦) β π’ (π‘, π₯)] ππ¦ ππ‘ ββ
+ π’ (π‘, π₯) π (π11 π’ + V) , +β πV (π‘, π₯) = π2 β« π½ (π¦ β π₯) [V (π‘, π¦) β V (π‘, π₯)] ππ¦ ππ‘ ββ
(6)
+ V (π‘, π₯) π (π’ + π22 V) ,
π(1βπ’)
ππ’ = π’π (π11 π’ + V) , ππ‘
by the variation of the population near π₯. As we know that the individuals can move freely, then the movement of individuals is bounded to affect the other individuals. So, the Laplacian operator may have some shortage to describe the diffusion. One way to deal with this problem is to replace the Laplacian operator with a convolution diffusion β term β«ββ π½(π₯ β π¦)[π’(π‘, π¦) β π’(π‘, π₯)]ππ¦. This implies that the probability distribution function for the population at location π¦ moving to the location π₯ is π½(π₯ β π¦). At time π‘, the total individuals that move from the whole space into the β location π₯ will be β«ββ π½(π₯ β π¦)[π’(π‘, π¦) β π’(π‘, π₯)]ππ¦. Therefore, one may call it as a nonlocal diffusion, and, correspondingly, call π2 π’(π‘, π₯)/ππ₯2 as a local diffusion. During the recent years, the models with the nonlocal diffusion have been attracted much more attentions (see [15β18]). In this paper, instead of (2), we will concentrate on the following pioneer-climax species with nonlocal diffusion:
(5)
Selgrade and Roberds [9], Sumner [10] analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong [11], Sumner [12] considered the stable periodic behavior of (5). Because of the existence of rich equilibria and the various ranges of parameters, the dynamics of ordinary differential system (5) are complex, and a detailed review of all equilibrium types can be found in Buchanan [13, 14]. Although the Laplacian operator Ξ := π2 /ππ₯2 is always used to model the diffusion of the species, it suggests that the population at the location π₯ can only be influenced
where π1 , π2 are positive constants accounting for the diffusivity, π½(π§) is a kernel function which is continuous on R satisfying β« π½ (π₯) ππ₯ = 1, R
π½ (π₯) = π½ (βπ₯) β« π½ (π₯) π]π₯ ππ₯ < β R
π½ (π₯) β₯ 0, for π₯ β R,
(A1)
for any fixed ] β [0, β) .
We are interested in traveling wavefronts accounting for a mildinvasion of the two species (traveling wavefronts connecting a boundary equilibrium and the coexistence equilibrium). For system (2), the sufficient condition for (2) to have a mildinvasion is π2 /π1 β₯ 1/2 (see [6]), but our condition in this paper for system (6) is π2 /π1 β₯ 1, which reveals a fact that the nonlocal diffusion of either the pioneer species or the climax species did affect the climax invasion and wave propagation. Please see Section 5 for the discussion. The remaining of this paper is organized as follows. In Section 2, there are some preliminaries about the equilibria and the system is transformed into a cooperative one. In Section 3, we prove the existence of traveling wavefronts by using an iteration scheme combined with a pair of admissible upper and lower solutions, which can be constructed obviously, and thus a criterion of the existence for traveling wavefronts is obtained. We also give a discussion on asymptotic behavior for the traveling wavefront tail as π = π₯ + ππ‘ β ββ in Section 4. At last, we give some concluding discussions in Section 5.
Abstract and Applied Analysis
3
2. Preliminaries
It is evident that (0, 0) is a trivial equilibrium of (6). The system (6) has at least four equilibria and at most six equilibria. The existence of nonnegative steady states depends on the locations of the three nullclines: π11 π’ + V = π§0 ,
π’ + π22 V = π€1 ,
π’ + π22 V = π€2 .
(7)
The long-term behavior of solutions to (6) can be qualitatively different caused by the different number, distribution, and types of equilibria. The dynamics of the system (6) are of course very rich and complex. However, in this paper, we will only consider the following case: π§ π€1 < 0 < π€2 . π11
π€ π§0 > 2 , π22
π22 π§0 β π€2 , π11 π22 β 1
Vβ =
π11 π€2 β π§0 . π11 π22 β 1
(9)
It is obvious that π’β < π§0 /π11 . We further assume that β
π€ β€π’
β
+ V (π‘, π₯) π (
π€1 /π22
π’ + π22 = π€2
π’ + π22 = π€1
π€1
(π’β , β )
+ π€β
π’ π§0 /π11
π€2
Figure 2: One coexistence state for the two species with π€β β€ π’β .
wavefront is a traveling wave solution (π(π ), π(π )) which has finite limits (π(Β±β), π(Β±β)). Denoting the traveling wave coordinate π₯ + ππ‘ still by π‘, we derive the wave profile system from (11): +β
ββ
(10)
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘)
+ (π (π‘) β σΈ
ππ (π‘) = π2 β«
+β
ββ
π§0 ) π (π§0 β π11 π (π‘) + π (π‘)) , π11
(12)
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘)
+ π (π‘) π (
π§0 β π (π‘) + π22 π (π‘)) . π11
Associated with (12), we consider its solutions subject to the following boundary value conditions:
+β ππ’ (π‘, π₯) = π1 β« π½ (π¦ β π₯) π’ (π‘, π¦) ππ¦ β π1 π’ (π‘, π₯) ππ‘ ββ
+β πV (π‘, π₯) = π2 β« π½ (π¦ β π₯) V (π‘, π¦) ππ¦ β π2 V (π‘, π₯) ππ‘ ββ
π€2 /π22
ππσΈ (π‘) = π1 β«
for the technical reason. See Figure 2 for this situation. As mentioned in the introduction, we are interested in the coexistence of the two species. That means that we will seek traveling wavefronts connecting equilibrium (π§0 /π11 , 0) and equilibrium (π’β , Vβ ). By making changes of variables π’Μ = π§0 /π11 β π’, ΜV = V and dropping the tildes, system (6) becomes
π§ + (π’ (π‘, π₯) β 0 ) π (π§0 β π11 π’ + V) , π11
π11 π’ + = π§0
(8)
The condition π11 π22 > 1 follows as a sequence. Under the previous assumption, (6) has four nontrivial equilibria: (π§0 /π11 , 0), (0, π€1 /π22 ), (0, π€2 /π22 ), and (π’β , Vβ ) except for (0, 0), where π’β =
π§0
lim (π (π‘) , π (π‘)) = (0, 0) ,
π‘ β ββ
lim (π (π‘) , π (π‘)) = (π’+ , V+ ) .
π‘ β +β
(11)
π§0 β π’ + π22 V) , π11
and the equilibria (π§0 /π11 , 0), (π’β , Vβ ) are changed into (0, 0), (π’+ , V+ ), respectively, where π’+ = π§0 /π11 β π’β , V+ = Vβ .
3. Existence of Traveling Wavefronts A traveling wavefront of (6) connecting equilibria (π§0 /π11 , 0) and (π’β , Vβ ) can be changed into a traveling wavefront of (11) connecting (0, 0) and (π’+ , V+ ). Therefore, we consider the system (11) hereby. A traveling wave solution of (11) is a solution with the form π’(π‘, π₯) = π(π₯ + ππ‘) = π(π ) and V(π‘, π₯) = π(π₯ + ππ‘) = π(π ), where π = π₯ + ππ‘ and π > 0 is a wave speed. A traveling
(13)
For π = (π1 , π2 ), π = (π1 , π2 ) β R2 , π β€ π implies ππ β€ ππ (π = 1, 2); π < π implies π β€ π but π =ΜΈ π; π βͺ π implies ππ < ππ (π = 1, 2). Furthermore, the norm β β
β in R2 is the Euclidean norm. Define D = {(π, π) β πΆ (R, R2 ) | 0 β€ π (π‘) β€ π’+ , 0 β€ π (π‘) β€ V+ , π‘ β R} .
(14)
For some constants π½1 , π½2 , letting π1 = π½1 β π1 , π2 = π½2 β π2 , we define an operator π» = (π»1 , π»2 ) : D β πΆ(R, R2 ) by π»1 (π, π) (π‘) = π1 β«
+β
ββ
π½ (π¦ β π‘) π (π¦) ππ¦
+ (π (π‘) β
π§0 ) π (π§0 β π11 π (π‘) + π (π‘)) + π1 π (π‘) , π11
4
Abstract and Applied Analysis π»2 (π, π) (π‘)
π»2 (π1 , π2 ) (π‘) β π»2 (π1 , π1 ) (π‘) +β
= π2 β«
ββ
= π2 β«
π½ (π¦ β π‘) π (π¦) ππ¦
+ π (π‘) π (
ββ
π§0 β π (π‘) + π22 π (π‘)) + π2 π (π‘) . π11
(19)
Then, (12) can be written as an equivalent form:
ππσΈ (π‘) = βπ½2 π (π‘) + π»2 (π, π) (π‘) .
(16)
π‘ 1 π2 (π, π) (π‘) = πβ(π½2 /π)π‘ β« π(π½2 /π)π π»2 (π, π) (π ) ππ . π ββ
(17)
Lemma 1. Assume that (A1) holds, for sufficiently large π½1 , π½2 > 0 and (ππ , ππ ) β D, π = 1, 2 with π1 (π ) β€ π2 (π ) and π1 (π ) β€ π2 (π ), π β R; one has π»2 (π1 , π2 )(π‘) β₯ π»2 (π1 , π1 )(π‘), π»2 (π2 , π1 )(π‘) β₯ π»2 (π1 , π1 )(π‘)
Proof. In order to prove (i), let π1 (π₯, π¦) = (π₯ β π§0 /π11 )π(π§0 β π11 π₯ + π¦) + π1 π₯, π1 (π₯, π¦) = π¦π(π§0 /π11 β π₯ + π22 π¦) + π2 π¦. Then ππ1 = π (π§0 β π11 π₯ + π¦) ππ₯ (18)
π§0 β π₯ + π22 π¦) + π½2 β π2 . π11
For 0 β€ π₯ β€ π’+ , 0 β€ π¦ β€ V+ and sufficiently large π½1 , π½2 > 0, it follows that ππ1 /ππ₯ β₯ 0, ππ1 /ππ¦ β₯ 0. Thus, if π1 (π ) β€ π2 (π ) and π1 (π ) β€ π2 (π ) for π β R, we have
ββ
π½ (π¦ β π‘) [π2 (π¦) β π1 (π¦)] ππ¦
+ π1 (π2 , π1 ) β π1 (π1 , π1 ) β₯ 0,
π§0 π§ π§ β ( 0 β π’β ) β€ 0 β π2 (π‘) + π22 π1 (π‘) π11 π11 π11
π§ β€ 0 β π1 (π‘) + π22 π1 (π‘) . π11
(21)
Note that πσΈ (π€) < 0 for π€ > π€β . This leads to
π§0 β π2 (π‘) + π22 π1 (π‘)) π11
(22)
π§0 β π1 (π‘) + π22 π1 (π‘))] β₯ 0. π11
The proof is complete. The conclusion of the following lemma is direct. Lemma 2. Assume that π½1 > 0, π½2 > 0 are sufficiently large. For (π, π) β D with π(π‘), π(π‘) nondecreasing on π‘ β R, π»1 (π, π)(π‘), π»2 (π, π)(π‘) are also nondecreasing on π‘ β R.
+ (π§0 β π11 π₯) πσΈ (π§0 β π11 π₯ + π¦) + π½1 β π1 ,
+β
π€β β€ π’β =
βπ (
for all π‘ β R.
= π1 β«
From 0 β€ π1 (π‘) β€ V+ = Vβ , 0 β€ π1 (π‘) β€ π2 (π‘) β€ π’+ = π§0 /π11 β π’β , we have
= π1 (π‘) [π (
(ii) π»1 (π1 , π2 )(π‘) β₯ π»1 (π1 , π1 )(π‘),
π»1 (π2 , π1 ) (π‘) β π»1 (π1 , π1 ) (π‘)
π§0 ) [π (π§0 β π11 π1 (π‘) + π2 (π‘)) π11
π»2 (π2 , π1 ) (π‘) β π»2 (π1 , π1 ) (π‘)
(i) π»1 (π2 , π1 )(π‘) β₯ π»1 (π1 , π1 )(π‘),
+ π22 π¦πσΈ (
π»1 (π1 , π2 ) (π‘) β π»1 (π1 , π1 ) (π‘)
βπ (π§0 β π11 π1 (π‘) + π1 (π‘))] β₯ 0. (20)
It is obvious that a traveling wave solution of the problem (12) and (13) is a fixed point of π and vice verse. The following lemma states the monotone property of π».
ππ1 π§ = π ( 0 β π₯ + π22 π¦) ππ¦ π11
For (ii), we know that 0 β€ π1 (π‘) β€ π’+ = π§0 /π11 β π’β β€ π§0 /π11 and πσΈ (π¦) < 0 for π¦ β R. It follows that
= (π1 (π‘) β
Denote π = (π1 , π2 ) by π‘ 1 π1 (π, π) (π‘) = πβ(π½1 /π)π‘ β« π(π½1 /π)π π»1 (π, π) (π ) ππ , π ββ
π½ (π¦ β π‘) [π2 (π¦) β π1 (π¦)] ππ¦
+ π1 (π1 , π2 ) β π1 (π1 , π1 ) β₯ 0. (15)
ππσΈ (π‘) = βπ½1 π (π‘) + π»1 (π, π) (π‘) ,
+β
We can easily see that π = (π1 , π2 ) also enjoys the same properties as those for π» = (π»1 , π»2 ) settled in Lemmas 1 and 2. Let π β (0, min {π½1 /π, π½2 /π}) and σ΅¨ σ΅¨ π΅π (R, R2 ) = {π β πΆ (R, R2 ) | sup σ΅¨σ΅¨σ΅¨π (π‘)σ΅¨σ΅¨σ΅¨ πβπ|π‘| < β} . π‘βR
(23) It is clear that (π΅π (R, R2 ), | β
|π ) is a Banach space equipped with the norm | β
|π defined by |π|π = supπ‘βR |π(π‘)|πβπ|π‘| . Definition 3. A pair of continuous functions Ξ¦(π‘) = (π(π‘), π(π‘)), Ξ¦(π‘) = (π(π‘), π(π‘)) is called an upper solution and
Abstract and Applied Analysis
5
a lower solution of (12), respectively, if there exists a set Ξ = {ππ β R, π = 1, . . . , π} such that Ξ¦ and Ξ¦ are differentiσΈ able in R \ Ξ and the essential bounded functions Ξ¦ , Ξ¦σΈ satisfy π1 β«
+β
ββ
+β
ββ
π1 β«
ββ
+β
ββ
ππ (π‘) = π1 (ππβ1 , ππβ1 ) (π‘) , ππ (π‘) = π2 (ππβ1 , ππβ1 ) (π‘) , π (π‘) = π1 (π π
π§0 ) π (π§0 β π11 π (π‘) + π (π‘)) β€ 0, π11
π§0 β π (π‘) + π22 π (π‘)) β€ 0, π11
π
πβ1
,π
πβ1
,π
πβ1
(26)
) (π‘) .
π
(i) (ππ (π‘), ππ (π‘)) β Ξ; (24)
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππ (π‘) π§0 ) π (π§0 β π11 π (π‘) + π (π‘)) β₯ 0, π11
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππσΈ (π‘) π§0 β π (π‘) + π22 π (π‘)) β₯ 0 π11
for π‘ β R \ Ξ.
(ii) (π
πβ1
(π‘), π
πβ1
(π‘)) β€ (ππ (π‘), π (π‘)) β€ (ππ (π‘), ππ (π‘)) β€
(ππβ1 (π‘), ππβ1 (π‘)) for π‘ β R;
π
(iii) (ππ (π‘), ππ (π‘)) and (π (π‘), π (π‘)) is a pair of upper and π π lower solutions of (12); (iv) (ππ (π‘), ππ (π‘)) and (π (π‘), π (π‘)) are continuously differπ π entiable on π‘ β R. Proof. We only give the argument for π = 2, and the situation for π β₯ 3 can be obtained by mathematical induction. From Definition 3, we obtain σΈ
π»1 (π1 , π1 ) (π‘) β€ ππ1 (π‘) + π½1 π1 (π‘)
In what follows, we assume that (12) has an upper solution (π(π‘), π(π‘)) and a lower solution (π(π‘), π(π‘)), such that (P1) (0, 0) βͺ (π(π‘), π(π‘)) β€ (π(π‘), π(π‘)) β€ (π’+ , V+ ) for π‘ β R; (P2) limπ‘ β ββ (π(π‘), π(π‘)) = (0, 0), limπ‘ β +β (π(π‘), π(π‘)) = (π’+ , V+ );
π2 (π‘) = π1 (π1 , π1 ) (π‘) = β€
are nondecreasing for π‘ β R.} (25)
(28)
where π = 1, 2, . . . , π + 1. By similar arguments, we can get (π (π‘) , π (π‘)) β€ (π (π‘) , π (π‘)) β€ (π2 (π‘) , π2 (π‘)) 1
(ii) π (π‘) , π (π‘)
1 π‘ β(π½1 /π)(π‘βπ ) π»1 (π1 , π1 ) (π ) ππ β« π π ββ
} σΈ Γ [ππ1 (π ) + π½1 π1 (π )] ππ } = π1 (π‘) , }
Ξ (Ξ¦, Ξ¦)
π (π‘) β€ π (π‘) β€ π (π‘) ,
(27)
π‘ 1 { πβ1 ππ ( + ) πβ(π½1 /π)(π‘βπ ) β« β β« π { π=1 ππβ1 ππβ1 {
Define the following profile set Ξ = Ξ(Ξ¦, Ξ¦) by
= {(π, π) β D | (i) π (π‘) β€ π (π‘) β€ π (π‘) ,
for π‘ β R \ Ξ.
Let π0 = ββ, ππ+1 = +β. For any π‘ β R, there exists some π such that π‘ β (ππβ1 , ππ ] (π = 1, 2, β
, π) or π‘ β (ππ , β). We then derive
(P3) π(π‘) and π(π‘) are nondecreasing.
It is obvious that Ξ(Ξ¦, Ξ¦) is nonempty.
) (π‘) ,
Lemma 4. For π β₯ 2, the functions ππ (π‘), ππ (π‘) and π (π‘), π π (π‘) defined by (26) satisfy
σΈ
+ π (π‘) π (
πβ1
π (π‘) = π2 (π
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππσΈ (π‘)
+ (π (π‘) β π2 β«
1
σΈ
+ π (π‘) π ( +β
1
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππ (π‘)
+ (π (π‘) β π2 β«
For π‘ β R and π β₯ 2, define (π1 , π1 )(π‘) = (π, π)(π‘), (π , π )(π‘) = (π, π)(π‘) and
1
2
2
β€ (π1 (π‘) , π1 (π‘))
(29)
for π‘ β R.
The previous arguments implies that the conclusion (ii) holds. From the monotone property of π, we can easily obtain that π2 (π‘), π2 (π‘) are nondecreasing for π‘ β R, and therefore the conclusion (i) holds.
6
Abstract and Applied Analysis For π2 (π‘) = π1 (π1 , π1 )(π‘), by Lemma 1, we have σΈ
β π½1 π2 (π‘) β ππ2 (π‘) + π»1 (π2 , π2 ) (π‘) σΈ
β€ βπ½1 π2 (π‘) β ππ2 (π‘) + π»1 (π1 , π1 ) (π‘) = 0
(30) for π‘ β R \ Ξ.
In a similar way, we can prove that βπ½2 π2 (π‘) β ππσΈ (π‘) + π»2 (π2 , π2 ) (π‘) β€ 0, 2
βπ½1 π (π‘) β 2
σΈ ππ2
(π‘) + π»1 (π , π ) (π‘) β₯ 0, 2
2
(31)
βπ½2 π (π‘) β ππσΈ (π‘) + π»2 (π , π ) (π‘) β₯ 0 2
2
2
(36)
2
for π‘ β R\Ξ. This indicates that (π2 (π‘), π2 (π‘)) and (π (π‘), π (π‘)) 2 2 are a pair of upper and lower solutions of (12). The conclusion (iv) is obvious. The proof is complete. Lemma 5. limπ β β (ππ (π‘), ππ (π‘)) = (πβ (π‘), πβ (π‘)) β Ξ, and the convergence is uniform with respect to the decay norm | β
|π . Proof. We have from Lemma 4 that the following limit exists: lim (ππ (π‘) , ππ (π‘)) = (πβ (π‘) , πβ (π‘)) .
πββ
(32)
It is easy to know that Ξ is a closed and convex set. By the nondecreasing property of (ππ (π‘), ππ (π‘)), we have (πβ (π‘), πβ (π‘)) β Ξ. In the following, we prove that the convergence is uniform with respect to the decay norm. Since σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨(ππ (π‘) , ππ (π‘)) β (πβ (π‘) , πβ (π‘))σ΅¨σ΅¨σ΅¨ β€ 2 (π’+ + V+ ) for π β₯ 1, σ΅¨ σ΅¨ (33) for any π > 0, there exists a π = π(π) > 0, such that for all π β₯ 1, σ΅¨ σ΅¨ sup σ΅¨σ΅¨σ΅¨σ΅¨(ππ (π‘) , ππ (π‘)) β (πβ (π‘) , πβ (π‘))σ΅¨σ΅¨σ΅¨σ΅¨ πβπ|π‘| < π. (34) π‘>|π| β
Now, we consider the sequences {(ππ , ππ )}π=1 for π‘ β [βπ, π]. Note ππ (π‘) is nondecreasing on π‘, and thus σΈ
0 β€ ππ (π‘) = π1σΈ (ππβ1 , ππβ1 ) (π‘) =
π½ 1 π» (π , π ) (π‘) β 21 πβ(π½1 /π)π‘ π 1 πβ1 πβ1 π Γβ«
π‘
ββ
β€
π(π½1 /π)π π»1 (ππβ1 , ππβ1 ) (π ) ππ
1 1 π» (π , π ) (π‘) β€ π»1 (π1 , π1 ) (π‘) . π 1 πβ1 πβ1 π
Furthermore, we have σ΅¨ σ΅¨ sup σ΅¨σ΅¨σ΅¨σ΅¨(ππ (π‘) , ππ (π‘)) β (πβ (π‘) , πβ (π‘))σ΅¨σ΅¨σ΅¨σ΅¨ πβπ|π‘| < π
π‘β[βπ,π]
(37)
for π > πβ . Summarizing the previous arguments, for π > πβ we have σ΅¨ σ΅¨ supσ΅¨σ΅¨σ΅¨σ΅¨(ππ , ππ ) β (πβ , πβ )σ΅¨σ΅¨σ΅¨σ΅¨π π‘βR σ΅¨ σ΅¨ = sup σ΅¨σ΅¨σ΅¨σ΅¨(ππ (π‘) , ππ (π‘)) β (πβ (π‘) , πβ (π‘))σ΅¨σ΅¨σ΅¨σ΅¨ πβπ|π‘| < π. π‘βR
(38) The proof is complete. Theorem 6. Assume that (A1) holds; if (12) has a pair of upper and lower solutions that satisfy (P1)β(P3), then the system (11) has a traveling wavefront satisfying (13). Proof. By the Lebesgueβs dominated convergence theorem and the iteration scheme (26), we have (πβ (π‘) , πβ (π‘)) = (π1 (πβ (π‘) , πβ (π‘)) , π2 (πβ (π‘) , πβ (π‘))) . (39) Therefore, (πβ (π‘), πβ (π‘)) is a fixed point of π, which also satisfies (12). Furthermore, (P2) indicates that (πβ (π‘), πβ (π‘)) satisfy limπ‘ β ββ (πβ (π‘), πβ (π‘)) = (0, 0). On the other hand, we have from the monotonicity of (πβ (π‘), πβ (π‘)) that limπ‘ β ββ (πβ (π‘), πβ (π‘)) = (π’, V) exists. Furthermore, since (0, 0) βͺ (π(π‘), π(π‘)) β€ (πβ (π‘), πβ (π‘)), we know that (π’, V) β« (0, 0). By using LβHΛospitalβs rule, we obtain π’ = lim πβ (π‘) = lim π1 (πβ (π‘) , πβ (π‘)) π‘ββ
π‘ββ
π‘
(35)
By (π1 (π‘), π1 (π‘)) β π·, there exists a positive constant π1 , such σΈ
there exists a positive number π2 , such that |πσΈ π (π‘)| β€ π2 for π‘ β [βπ, π]. From the previous estimates, we know that {(ππ (π‘), ππ (π‘))}β π=1 is equicontinuous on [βπ, π] with respect to the supremum norm. On the other hand, we have from Lemma 4 β (ii) that {(ππ (π‘), ππ (π‘))}π=1 is uniformly bounded. By ArzΒ΄elaβ Ascoli theorem, there exist subsequences of {(ππ (π‘), ππ (π‘))}π=1 which are uniformly convergent in π‘ β [βπ, π]. Without loss of generality, we still express this subsequence as {(ππ (π‘), β ππ (π‘))}β π=1 . Thus, there exists a positive integer π > 2, such that σ΅¨ σ΅¨ sup σ΅¨σ΅¨σ΅¨σ΅¨(ππ (π‘) , ππ (π‘)) β (πβ (π‘) , πβ (π‘))σ΅¨σ΅¨σ΅¨σ΅¨ < π for π > πβ . π‘β[βπ,π]
that |ππ (π‘)| β€ π1 for π‘ β [βπ, π]. Similarly, we can prove that
= lim
(1/π) β«ββ π(π½1 /π)π π»1 (πβ , πβ ) (π ) ππ
π‘ββ
=
1 π» (π’, V) . π½1 1
π(π½1 /π)π‘
(40)
Abstract and Applied Analysis
7
Similarly, one can obtain V = (1/π½2 )π»2 (π’, V). That is, (π’, V) is an equilibrium of (12). Note that the assumption (10) implies that there is only one positive equilibrium (π’+ , V+ ) of (12) satisfying (0, 0) < (π’, V) β€ (π’+ , V+ ). Therefore, (π’, V) = (π’+ , V+ ) and (πβ (π‘), πβ (π‘)) is a traveling wavefront satisfying (13). The proof is complete. In order to construct a pair of admissible upper-lower solutions for (12), we linearize (12) at (0, 0) and obtain +β
π1 β«
ββ
π2 β«
ββ
σΈ
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππ (π‘)
+ π (π‘) π (
πΉ (π, π) := π2 β«
ββ
π (π‘) = min {ππ 1 π‘ , π’+ } ,
(41)
π§0 ) = 0. π11
π§ π½ (π¦) π ππ¦ β π2 β ππ + π ( 0 ) = 0. π11 (42) ππ¦
Then for π2 β₯ π1 , (π(π‘), π(π‘)) is an upper solution of (12).
π1 β«
+β
ββ
σΈ
β€ π1 β«
+β
ββ
(by (π΄1) and ππ¦ + πβπ¦ β₯ 2) ,
π§0 ) > 0, π11
= (π1 β«
+β
ββ
+β ππΉ (π, π) = π2 β« π¦π½ (π¦) πππ¦ ππ¦ β π, ππ ββ +β π2 πΉ (π, π) = π π¦2 π½ (π¦) πππ¦ ππ¦ > 0 for any π β R, β« 2 ππ2 ββ
(43)
(i) There exists a (πβ , πβ ), πβ > 0, πβ > 0 such that πΉ (πβ , πβ ) = 0,
ππΉ (π, π) σ΅¨σ΅¨σ΅¨σ΅¨ = 0. σ΅¨ ππ σ΅¨σ΅¨σ΅¨(πβ ,πβ )
β€β
+β
ββ
πΉ (π, π) < 0 for π 1 < π < π 2 .
(45)
π§ π1 π π ( 0 ) + 1 ππ 1 β ππ 1 ) ππ 1 π‘ π2 π11 π2
π§ π1 π ( 0 ) ππ 1 π‘ < 0. π2 π11
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππσΈ (π‘)
+ π (π‘) π (
(ii) For 0 < π < πβ , πΉ(π, π) > 0 for π β R. (iii) For π > πβ , the equation πΉ(π, π) = 0 has two zeros 0 < π 1 < π 2 such that
(48)
Similarly, we have from the fact π(π‘) β€ π11 ππ 1 π‘ for π‘ β R, π11 π22 > 1 and πσΈ (π€) < 0 for π€ > π€β that π2 β«
(44)
π½ (π¦) ππ 1 π¦ ππ¦ β π1 β ππ 1 ) ππ 1 π‘
+ππ 1 ) β ππ 1 ] ππ 1 π‘ = (β
Lemma 7. The following conclusions are true.
π§0 ) π (π§0 ) π11
+β π1 (π2 β« π½ (π¦) ππ 1 π¦ ππ¦ β π2 β ππ 1 π2 ββ
=[
The convex property of πΉ leads to πΉ(+β, π) = +β for any π > 0. In view of the previous observation, we have the following lemma directly.
π§0 ) π (π§0 β π11 π (π‘) + π (π‘)) π11
π½ (π¦ β π‘) ππ 1 π¦ ππ¦ β π1 ππ 1 π‘ β ππ 1 ππ 1 π‘
+ (ππ 1 π‘ β
πΉ (π, +β) = ββ for any π > 0,
ππΉ (π, π) = βπ < 0 for any π > 0. ππ
(47)
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππ (π‘) + (π (π‘) β
for any π > 0
πΉ (0, π) = π (
π11 ππ 1 π‘2 = V+ .
Notice that V+ = π11 π’+ , we have π‘0 := π‘1 = π‘2 = (1/π 1 ) ln (V+ /π11 ). If π‘ < π‘0 , π(π‘) = ππ 1 π‘ , π(π‘) = π11 ππ 1 π‘ , we have from the fact π(π‘) β€ ππ 1 π‘ for π‘ β R and π€1 < π€β β€ π’β < π§0 /π11 < π€2 that
Note that πΉ (π, 0) > 0
π (π‘) = min {π11 ππ 1 π‘ , V+ } . (46)
ππ 1 π‘1 = π’+ ,
Thus, we consider the following characteristic equation: +β
Lemma 8. Define
Proof. Let π‘1 , π‘2 be such that
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππσΈ (π‘) = 0,
+β
Now we are ready to construct the upper solution of (12).
β€ π2 β«
+β
ββ
π§0 β π (π‘) + π22 π (π‘)) π11
π½ (π¦ β π‘) π11 ππ 1 π¦ ππ¦ β π2 π11 ππ 1 π‘ β ππ11 π 1 ππ 1 π‘
+ π11 ππ 1 π‘ π (
π§0 β ππ 1 π‘ + π11 π22 ππ 1 π‘ ) π11
8
Abstract and Applied Analysis +β
< (π2 β«
ββ
Let π := minπ€β[π§0 /π11 ,π§0 /π11 +π22 V+ ] πσΈ (π€) < 0. By the fact that
π½ (π¦) ππ 1 π¦ ππ¦ β π2 β ππ 1
π11 π22 (ππ 1 π‘ β ππππ 1 π‘ ) β₯ 0
π§ +π ( 0 )) π11 ππ 1 π‘ = 0. π11 If π‘ > π‘0 , π(π‘) = π’+ , π(π‘) = V+ , we have +β
ββ
π2 β«
ββ
(50)
σΈ
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππ (π‘)
+ π (π‘) π ( β€ V+ π (
π(
π§0 + π11 π22 (ππ 1 π‘ β ππππ 1 π‘ )) π11
= [π (
σΈ
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππ (π‘)
π§ + (π (π‘) β 0 ) π (π§0 β π11 π (π‘) + π (π‘)) β€ 0; π11 +β
β₯ ππ11 π22 (ππ 1 π‘ β ππππ 1 π‘ ) + π (
π§0 β π (π‘) + π22 π (π‘)) π11
π§0 β π’+ + π22 V+ ) = V+ π (π€2 ) = 0. π11
ββ
π½ (π¦) πππ 1 π¦ ππ¦ β π2 β πππ 1 + π (
(ππ 1 π‘ β ππππ 1 π‘ ) β€ π2π 1 π‘ ,
(ππ 1 π‘ β ππππ 1 π‘ ) π (
π§0 ) < 0. π11
π (π‘) = 0, β
π (π‘) = max {π11 (π β
ππ 1 π‘
β ππ
(52)
) , 0} ,
(π (π‘) , π (π‘)) := (π1 (π (π‘) , π (π‘)) , π2 (π (π‘) , π (π‘))) β β β β (54)
(π (π‘) , π (π‘)) β« (0, 0)
for π‘ β R.
(55)
Proof. Let π‘3 be such that π11 (ππ 1 π‘ β ππππ 1 π‘ ) = 0, it follows that 1 1 ln < 0. (π β 1) π 1 π
(56)
If π‘ < π‘3 < 0, π (π‘) = 0, π (π‘) = π11 (ππ 1 π‘ β ππππ 1 π‘ ), and β
π1 β«
+β
ββ
β
π§ π§0 π 1 π‘ ) π + ππ11 π22 π2π 1 π‘ β π ( 0 ) ππππ 1 π‘ . π11 π11
+β
(61)
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππσΈ (π‘) + π (π‘) β
Γ π(
β
β
π§0 β π (π‘) + π22 π (π‘)) β β π11
β₯ π11 π2 β«
+β
ββ
π1 π‘
Γ (π
β
π½ (π¦ β π‘) [ππ 1 π¦ β ππππ 1 π¦ ] ππ¦ β π2 π11
β ππππ 1 π‘ )
Γ π(
π§0 + π11 π22 (ππ 1 π‘ β ππππ 1 π‘ )) π11 +β
β₯ π11 ππ 1 π‘ [π2 β«
ββ
π½ (π¦) ππ 1 π¦ ππ¦ β π2 β ππ 1 + π (
β π11 πππ 1 π‘ {π [π2 β«
β
+π (
β
π§ + (π (π‘) β 0 ) π (π§0 β π11 π (π‘) + π (π‘)) β β β π11 π§ π (π§0 ) π§ = β 0 π (π§0 + π (π‘)) β₯ β 0 = 0. β π11 π11
+β
ββ
) for π‘ β R, we have
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππσΈ (π‘) β
π§0 + π11 π22 (ππ 1 π‘ β ππππ 1 π‘ )) π11
β ππ11 (π 1 ππ 1 π‘ β πππ 1 πππ 1 π‘ ) + π11 (ππ 1 π‘ β ππππ 1 π‘ )
is a lower solution of (12) satisfying
π (π‘) β₯ π11 (ππ 1 π‘ β ππ
(60)
Therefore, ββ
where π > 1 is a constant to be chosen later. Then (π (π‘), π (π‘)) β β is a lower solution of (12). Furthermore,
β ππ 1 π‘
β₯ π(
π2 β« (53)
π‘3 =
(59)
and thus
Lemma 9. Define π1 π‘
π§0 ). π11
2
(51)
Let π β (1, min{2, π 2 /π 1 }) satisfying π 1 < ππ 1 < π 2 , we can obtain from (45) that +β
π§ π§ π§0 + π11 π22 (ππ 1 π‘ β ππππ 1 π‘ )) β π ( 0 )] + π ( 0 ) π11 π11 π11
Note that (58) leads to ππ 1 π‘ β ππππ 1 π‘ > 0 for π‘ < π‘3 ; it follows that
The proof is complete.
π2 β«
(58)
we have (49)
π1 β«
for π‘ < π‘3 ,
(57)
β₯ βπ11 πππ 1 π‘ {π [π2 β«
+β
ββ
+π (
π§0 )] π11
π½ (π¦) πππ 1 π¦ ππ¦ β π2 β πππ 1
π§0 )] β ππ11 π22 π(2βπ)π 1 π‘ } π11 π½ (π¦) πππ 1 π¦ ππ¦ β π2 β πππ 1 π§0 )] β ππ11 π22 } . π11
(62)
Abstract and Applied Analysis
9
Let π > 1 sufficiently large; we can have +β
π [π2 β«
ββ
π½ (π¦) πππ 1 π¦ ππ¦ β π2 β πππ 1 (63)
π§ +π ( 0 )] β ππ11 π22 < 0, π11 hence, +β
π2 β«
ββ
π½ (π¦ β π‘) π (π¦) ππ¦ β π2 π (π‘) β ππσΈ (π‘) β
β
β
π§ + π (π‘) π ( 0 β π (π‘) + π22 π (π‘)) β₯ 0. β β β π11
(64)
If π‘ > π‘3 , π (π‘) = 0, π (π‘) = 0, and π (π‘) β₯ 0 for π‘ β R, we β β β have +β
π1 β«
ββ
method, we can obtain a subsequence of {(ππ (π‘), ππ (π‘))}, still denoted by {(ππ (π‘), ππ (π‘))}, such that (ππ (π‘), ππ (π‘)) β (πβ (π‘), πβ (π‘)) uniformly for π‘ in any bounded subset of R, as π β β. Clearly, (πβ (π‘), πβ (π‘)) is nondecreasing and (πβ (0), πβ (0)) = (π’+ /2, V+ /2). By the dominated convergence theorem and (67), it follows that π‘ β β 1 πβ (π‘) = β πβ(π½1 /π )π‘ β« π(π½1 /π )π π»1 (πβ , πβ ) (π ) ππ , π ββ (68) 1 β(π½2 /πβ )π‘ π‘ (π½2 /πβ )π β β β π»2 (π , π ) (π ) ππ . π (π‘) = β π β« π π ββ Since limπ‘ β Β±β πβ (π‘) and limπ‘ β Β±β πβ (π‘) exist, using LβHΛospital rule leads to lim (πβ (π‘) , πβ (π‘)) = (0, 0) ,
π‘ β ββ
lim (πβ (π‘) , πβ (π‘)) = (π’+ , V+ ) .
π½ (π¦ β π‘) π (π¦) ππ¦ β π1 π (π‘) β ππσΈ (π‘) β
β
(69)
π‘ β +β
β
(65)
Thus, (π (π‘), πβ (π‘)) is a traveling wavefront of the system (11) connecting (0, 0) and (π’+ , V+ ).
From the previous arguments, we obtain that (π (π‘), β π (π‘)) is a lower solution of (12). By Lemma 4, we can get that β (π(π‘), π(π‘)) is also a lower solution. Furthermore, for π‘ < π‘3 , π (π‘) > 0, by direct calculation, we have π»1 (π , π )(π‘) > 0 β β β for π‘ < π‘3 . Therefore, π(π‘) = π1 (π , π ) (π‘) > 0 for π‘ β β β R. Similarly, we have π(π‘) > 0 for π‘ β R. The proof is complete.
Remark 11. We say that the πβ is the minimal wave speed in the sense that (11) has no traveling wavefront with π β (0, πβ ). We could briefly explain this in the following. In fact, the linearization of (12) at zero solution is (41), and the function πΉ(π, π) is obtained by substituting πππ‘ in the second equation of (41). For 0 < π < πβ , we know from (ii) of Lemma 7 that πΉ(π, π) > 0 for any π β R. We have from the second equation of (12) and the second equation of (41) that (12) cannot have a solution (π(π‘), π(π‘)) that satisfies limπ‘ β ββ (π(π‘), π(π‘)) = (0, 0).
π§ + (π (π‘) β 0 ) π (π§0 β π11 π (π‘) + π (π‘)) = 0. β β β π11
Theorem 10. Assume that (A1) and π2 β₯ π1 hold. Then for any π β₯ πβ , the system (11) has a traveling wavefront with speed π, which connects (0, 0) and (π’+ , V+ ). Proof. The conclusion for π > πβ can be obtained from the previous discussions. We only need to establish the existence of wave fronts when π = πβ . Let ππ β (πβ , πβ + 1) with limπ β β ππ = πβ . For ππ > πβ , (12) with π = ππ admits a nondecreasing solution (ππ (π‘), ππ (π‘)) such that lim (ππ (π‘) , ππ (π‘)) = (0, 0) ,
π‘ β ββ
lim (ππ (π‘) , ππ (π‘)) = (π’+ , V+ ) .
(66)
π‘ β +β
Without loss of generality, we assume that (ππ (0), ππ (0)) = (π’+ /2, V+ /2). Obviously, |ππ (π‘)| β€ π’+ , |ππ (π‘)| β€ V+ , and ππ (π‘), ππ (π‘) satisfy ππ (π‘) = ππ (π‘) =
1 β(π½1 /ππ )π‘ π‘ (π½1 /ππ )π π π»1 (ππ , ππ ) (π ) ππ , β« π ππ ββ 1 β(π½2 /ππ )π‘ π‘ (π½2 /ππ )π π π»2 (ππ , ππ ) (π ) ππ . β« π ππ ββ
(67)
As the same argument in Lemma 5, we can obtain that {(ππ (π‘), ππ (π‘))} is uniformly bounded and equicontinuous on R; using ArzΒ΄ela-Ascoli theorem and the standard diagonal
β
Theorem 12. Assume that (A1) and π2 β₯ π1 hold. Then for any π β₯ πβ , the system (6) has a traveling wavefront with speed c, which connects (π§0 /π11 , 0) and (π’β , Vβ ).
4. Asymptotic Behavior for Traveling Wavefronts In this section, we discuss the asymptotic behavior for the traveling wavefronts obtained in the previous section as π‘ β ββ. Theorem 13. Let (π(π‘), π(π‘)) be a traveling wavefront of (11) decided by Theorem 10; then lim (π (π‘) πβπ 1 π‘ , π (π‘) πβπ 1 π‘ )
π‘ β ββ
=(
π2 π§0 πσΈ (π§0 ) ,π ), ππ 1 (π1 β π2 ) β π1 π (π§0 /π11 ) + π2 π§0 πσΈ (π§0 ) 11
lim (πσΈ (π‘) πβπ 1 π‘ , πσΈ (π‘) πβπ 1 π‘ )
π‘ β ββ
=(
π 1 π2 π§0 πσΈ (π§0 ) ,π π ), ππ 1 (π1 β π2 ) β π1 π (π§0 /π11 ) + π2 π§0 πσΈ (π§0 ) 11 1 (70)
where π 1 is the smallest root of the characteristic equation (42).
10
Abstract and Applied Analysis On the other hand, by (76), (15), (42), and (75), noting π(π§0 ) = 0 and using LβHΛospitalβs rule, we get
Proof. Note that π11 (ππ 1 π‘ β ππππ 1 π‘ ) β€ π (π‘) β€ π11 ππ 1 π‘
for π‘ β R.
(71)
lim π (π‘) πβπ 1 π‘
π‘ β ββ
Then we have σ΅¨ σ΅¨ lim σ΅¨σ΅¨σ΅¨π (π‘) πβπ 1 π‘ β π11 σ΅¨σ΅¨σ΅¨σ΅¨ β€ lim π11 ππ(πβ1)π 1 π‘ = 0, π‘ β ββ
π‘ β ββ σ΅¨
lim π (π‘) πβπ 1 π‘ = π11 .
π‘ 1 lim πβ(π 1 +π½1 /π)π‘ β« π(π½1 /π)π π»1 (π, π) (π ) ππ π π‘ β ββ ββ
=
π(π½1 /π)π‘ π»1 (π, π) (π‘) 1 lim π π‘ β ββ (π 1 + π½1 /π) π(π 1 +π½1 /π)π‘
=
+β 1 (π1 lim πβπ 1 π‘ β« π½ (π¦ β π‘) π (π¦) ππ¦ π‘ β ββ (ππ 1 + π½1 ) ββ
(72)
which implies that (73)
π‘ β ββ
=
Note that we have from 0 β€ π(π‘)πβπ 1 π‘ β€ π11 and (A1) that πβπ 1 π‘ β«
+β
ββ
=β«
+β
ββ
β
π½ (π¦ β π‘) π (π¦) ππ¦
+π1 lim π (π‘) πβπ 1 π‘ )
(74) π1 π¦
π½ (π¦) π
βπ 1 (π¦+π‘)
π (π¦ + π‘) π
ππ¦ < β,
π‘ β ββ
=
which is uniformly on π‘. By the second equation of (12), we have from (73), πΉ(π 1 , π) = 0 and convergence theorem that
π§ π 1 [ 1 (π2 + ππ 1 β π ( 0 )) lim π (π‘) πβπ 1 π‘ π‘ β ββ π11 (ππ 1 + π½1 ) π2 β
lim πσΈ (π‘) πβπ 1 π‘
π‘ β ββ
=
π‘ β ββ
+β
ββ
π‘ β ββ
=
π½ (π¦ β π‘) π (π¦) ππ¦]
π§ ππ π π 1 [(π1 + π1 + 1 1 β 1 π ( 0 )) π2 π2 π11 (ππ 1 + π½1 ) Γ lim π (π‘) πβπ 1 π‘ + π‘ β ββ
π§ 1 = [π11 (π ( 0 ) β π2 ) π π11 +π2 lim πβπ 1 π‘ β« π‘ β ββ
=
+β
ββ
+ π2 β«
ββ
π§0 πσΈ (π§0 ) π1
Γ lim πσΈ (π‘) πβπ 1 π‘ β π§0 πσΈ (π§0 ) ] . π‘ β ββ
π½ (π¦) π (π¦ + π‘) ππ¦]
(77) By the first equation of (12), we have from (75) and πΉ(π 1 , π) = 0 that
π§ 1 [π (π ( 0 ) β π2 ) π 11 π11 +β
π§0 πσΈ (π§0 ) βπ11 πσΈ (π‘) + πσΈ (π‘) lim π11 π 1 π‘ β ββ ππ 1 π‘
+π1 lim π (π‘) πβπ 1 π‘ ]
π§ 1 [π11 (π ( 0 ) β π2 ) π π11 +π2 lim πβπ 1 π‘ β«
π (π§0 β π11 π (π‘) + π (π‘)) π§0 lim π11 π‘ β ββ ππ 1 π‘
π lim πσΈ (π‘) πβπ 1 π‘ π‘ β ββ
π½ (π¦) ππ 1 π¦ lim π (π¦ + π‘) πβπ 1 (π¦+π‘) ππ¦] π‘ β ββ
π§ π§ 1 = [π11 (π ( 0 ) β π2 ) + π11 (π2 + ππ 1 β π ( 0 ))] π π11 π11
= π1 lim πβπ 1 π‘ β« π‘ β ββ
= π11 π 1 .
Since π(π‘) = π1 (π, π)(π‘), by (17), we know that
π½ (π¦ β π‘) π (π¦) ππ¦ β π1
=
π (π§0 β π11 π (π‘) + π (π‘)) π§0 lim π11 π‘ β ββ ππ 1 π‘
π§ π1 (π2 + ππ 1 β π ( 0 )) lim π (π‘) πβπ 1 π‘ π‘ β ββ π2 π11 β π1 lim π (π‘) πβπ 1 π‘ + π‘ β ββ
π‘
1 π (π‘) = πβ(π½1 /π)π‘ β« π(π½1 /π)π π»1 (π, π) (π ) ππ . π ββ
ββ
Γ lim π (π‘) πβπ 1 π‘ β π‘ β ββ
(75)
+β
(76)
π§0 πσΈ (π§0 ) π1
Γ lim πσΈ (π‘) πβπ 1 π‘ β π§0 πσΈ (π§0 ) π‘ β ββ
Abstract and Applied Analysis =
11
π§ π1 (ππ 1 β π ( 0 )) lim π (π‘) πβπ 1 π‘ π‘ β ββ π2 π11
The generalized boundary conditions, lim (
π§ πσΈ (π§0 ) + 0 lim πσΈ (π‘) πβπ 1 π‘ β π§0 πσΈ (π§0 ) . π 1 π‘ β ββ
π‘ β ββ
(78) Combining (77) and (78), and noting that π½1 = π1 + π1 , we obtain from a direct calculation that lim π (π‘) πβπ 1 π‘
π‘ β ββ
=
π2 π§0 πσΈ (π§0 ) , ππ 1 (π1 β π2 ) β π1 π (π§0 /π11 ) + π2 π§0 πσΈ (π§0 )
lim πσΈ (π‘) πβπ 1 π‘
(79)
π‘ β ββ
=
π 1 π2 π§0 πσΈ (π§0 ) . ππ 1 (π1 β π2 ) β π1 π (π§0 /π11 ) + π2 π§0 πσΈ (π§0 )
Now, letting (π(π‘), π(π‘)) be a traveling wavefront of (6) decided by Theorem 12, by the equivalence between (6) and (11), we know that (π(π‘), π(π‘)) = (π§0 /π11 β π(π‘), π(π‘)) and thus obtain the following theorem. Theorem 14. Let (π(π‘), π(π‘)) be a traveling wavefront of (6) decided by Theorem 12; then lim (π (π‘) πβπ 1 π‘ , π (π‘) πβπ 1 π‘ ) = (β, π11 ) ,
π‘ β ββ
lim (πσΈ (π‘) πβπ 1 π‘ , πσΈ (π‘) πβπ 1 π‘ )
π‘ β ββ
π 1 π2 π§0 πσΈ (π§0 ) =( ,π π ), ππ 1 (π2 β π1 ) + π1 π (π§0 /π11 ) β π2 π§0 πσΈ (π§0 ) 11 1 (80) where π 1 is the smallest root of the characteristic equation (42).
5. Concluding Discussions We have considered a reaction model with nonlocal diffusion for competing pioneer-climax species. Some recent works (see Brown et al. [4], Yuan and Zou [6]) showed that the model with local diffusion (expressed by Laplacian operator) supports traveling wavefronts connecting two boundary equilibria or one boundary equilibrium and the coexisting equilibrium under some restrictions on the parameters. In Yuan and Zou [6], the sufficient condition for (2) to have the traveling wavefront connecting (π§0 /π11 , 0) and (π’β , Vβ ) is π2 /π1 β₯ 1/2, but ours for (6) with nonlocal diffusion is π2 /π1 β₯ 1. For a fixed π2 , to shift π2 /π1 β₯ 1/2 to π2 /π1 β₯ 1, one needs a smaller π1 . But for a fixed π1 , to shift π2 /π1 β₯ 1/2 to π2 /π1 β₯ 1, one needs a larger π2 . These facts imply that the nonlocal diffusion of the pioneer species accelerates the mild wave propagation, while the nonlocal diffusion of the climax species defers the mild wave propagation. That is to say, the nonlocal diffusion did affect the wave propagation of these two competitive species.
π§0 π§ β π (π‘) , π (π‘)) = ( 0 , 0) , π11 π11
π§ lim ( 0 β π (π‘) , π (π‘)) = (π’β , Vβ ) , π‘ββ π 11
(81)
lead to an explanation that the climax species starts its invasion after that the pioneer species has achieved its steady state, and also the competition between the two species was not intense; they can achieve a coexistence state finally. See [19] for the significance of biological invasion. We believe that this is the first time that the dynamics of the pioneer-climax competition model with nonlocal diffusion are studied. This model has complicated equilibrium structure, and we only considered one possible case about the traveling wavefront connecting one boundary equilibrium and the coexisting equilibrium. Some other situations about the species invasions and propagation of waves would be of great interest for further research.
Acknowledgments The authors are very grateful to the anonymous referees for careful reading and helpful suggestions which led to an improvement of their original paper. Research is supported by the Natural Science Foundation of China (11171120), the Doctoral Program of Higher Education of China (20094407110001), and the Natural Science Foundation of Guangdong Province (10151063101000003).
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