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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 725495, 12 pages
Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion
School of Mathematics, South China Normal University, Guangzhou 510631, China
Received 17 November 2012; Revised 11 March 2013; Accepted 21 March 2013
Academic Editor: Dumitru Baleanu
Copyright © 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 .
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, “one-humped” 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: where and represent densities of the pioneer and climax species, respectively. and denote the pioneer fitness function and climax fitness function, respectively, . By making changes of variables , , system (1) changes into the form (the tildes of and are dropped out) where we still use and as the new coefficients without confusing.
From the previous introduction, we assume that the pioneer fitness function satisfies for some , and the climax fitness function satisfies Ricker  used the fitness function , Hassell and Comins  used the fitness function , and Cushing  used the fitness function . It is obvious that these and have the curves in Figure 1.
There are some existing results about the stability and traveling wave solutions for (2) ([4–6]). About traveling wave solutions, Brown et al.  studied the traveling wave of (1) connecting two boundary equilibria by singular perturbation technique, and Yuan and Zou  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  for the existence of traveling plane waves in a general class of competition-diffusion systems and Murray  for more biological description of traveling wave solutions.
For system without spatial diffusion, the model will be Selgrade and Roberds , Sumner  analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong , Sumner  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 is always used to model the diffusion of the species, it suggests that the population at the location can only be influenced by the variation of the population near . 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 term This implies that the probability distribution function for the population at location moving to the location is . At time , the total individuals that move from the whole space into the location will be . Therefore, one may call it as a nonlocal diffusion, and, correspondingly, call as a local diffusion. During the recent years, the models with the nonlocal diffusion have been attracted much more attentions (see [15–18]).
In this paper, instead of (2), we will concentrate on the following pioneer-climax species with nonlocal diffusion: where , are positive constants accounting for the diffusivity, is a kernel function which is continuous on satisfying
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 (see ), but our condition in this paper for system (6) is , 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 in Section 4. At last, we give some concluding discussions in Section 5.
It is evident that 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: 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: The condition follows as a sequence. Under the previous assumption, (6) has four nontrivial equilibria: , , , and except for , where It is obvious that . We further assume that 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 and equilibrium .
By making changes of variables , and dropping the tildes, system (6) becomes and the equilibria , are changed into , , respectively, where , .
3. Existence of Traveling Wavefronts
A traveling wave solution of (11) is a solution with the form and , where and is a wave speed. A traveling wavefront is a traveling wave solution which has finite limits . Denoting the traveling wave coordinate still by , we derive the wave profile system from (11): Associated with (12), we consider its solutions subject to the following boundary value conditions:
For , , implies ; implies but ; implies . Furthermore, the norm in is the Euclidean norm. Define For some constants , , letting , , we define an operator by Then, (12) can be written as an equivalent form: Denote by It is obvious that a traveling wave solution of the problem (12) and (13) is a fixed point of and vice verse.
The following lemma states the monotone property of .
Proof. In order to prove (i), let , . Then
For , and sufficiently large , it follows that , . Thus, if and for , we have
For (ii), we know that and for . It follows that From , , we have Note that for . This leads to The proof is complete.
The conclusion of the following lemma is direct.
Lemma 2. Assume that , are sufficiently large. For with nondecreasing on , , are also nondecreasing on .
Let and It is clear that is a Banach space equipped with the norm defined by .
Definition 3. A pair of continuous functions , is called an upper solution and a lower solution of (12), respectively, if there exists a set such that and are differentiable in and the essential bounded functions satisfy for .
In what follows, we assume that (12) has an upper solution and a lower solution , such that(P1) for ;(P2), ;(P3) and are nondecreasing.
Define the following profile set by It is obvious that is nonempty.
For and , define , and
Proof. We only give the argument for , and the situation for can be obtained by mathematical induction. From Definition 3, we obtain
Let , . For any , there exists some such that or . We then derive
By similar arguments, we can get The previous arguments implies that the conclusion (ii) holds.
From the monotone property of , we can easily obtain that , are nondecreasing for , and therefore the conclusion (i) holds.
For , by Lemma 1, we have In a similar way, we can prove that for . This indicates that and are a pair of upper and lower solutions of (12).
The conclusion (iv) is obvious. The proof is complete.
Lemma 5. , and the convergence is uniform with respect to the decay norm .
Proof. We have from Lemma 4 that the following limit exists:
It is easy to know that is a closed and convex set. By the nondecreasing property of , we have . In the following, we prove that the convergence is uniform with respect to the decay norm.
Since for any , there exists a , such that for all , Now, we consider the sequences for . Note is nondecreasing on , and thus By , there exists a positive constant , such that for . Similarly, we can prove that there exists a positive number , such that for .
From the previous estimates, we know that is equicontinuous on with respect to the supremum norm. On the other hand, we have from Lemma 4 (ii) that is uniformly bounded. By Arzéla-Ascoli theorem, there exist subsequences of which are uniformly convergent in . Without loss of generality, we still express this subsequence as . Thus, there exists a positive integer , such that Furthermore, we have
Summarizing the previous arguments, for we have The proof is complete.
Proof. By the Lebesgue's dominated convergence theorem and the iteration scheme (26), we have Therefore, is a fixed point of , which also satisfies (12). Furthermore, (P2) indicates that satisfy . On the other hand, we have from the monotonicity of that exists. Furthermore, since , we know that . By using L’Hôspital’s rule, we obtain Similarly, one can obtain . That is, is an equilibrium of (12). Note that the assumption (10) implies that there is only one positive equilibrium of (12) satisfying . Therefore, and 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 and obtain Thus, we consider the following characteristic equation: Note that The convex property of leads to for any . In view of the previous observation, we have the following lemma directly.
Lemma 7. The following conclusions are true. (i)There exists a , , such that (ii)For , for .(iii)For , the equation has two zeros such that
Now we are ready to construct the upper solution of (12).
Lemma 8. Define Then for , is an upper solution of (12).
Proof. Let be such that
Notice that , we have .
If , , , we have from the fact for and that Similarly, we have from the fact for , and for that
If , , , we have The proof is complete.
Let satisfying , we can obtain from (45) that
Proof. Let be such that , it follows that
If , , , and for , we have
Let By the fact that we have Note that (58) leads to for ; it follows that and thus Therefore, Let sufficiently large; we can have hence,
If , , , and for , we have
From the previous arguments, we obtain that is a lower solution of (12). By Lemma 4, we can get that is also a lower solution. Furthermore, for , , by direct calculation, we have for . Therefore, for . Similarly, we have for . The proof is complete.
Theorem 10. Assume that and hold. Then for any , the system (11) has a traveling wavefront with speed , which connects and .
Proof. The conclusion for can be obtained from the previous discussions. We only need to establish the existence of wave fronts when .
Let with . For , (12) with admits a nondecreasing solution such that Without loss of generality, we assume that . Obviously, , , and satisfy
As the same argument in Lemma 5, we can obtain that is uniformly bounded and equicontinuous on ; using Arzéla-Ascoli theorem and the standard diagonal method, we can obtain a subsequence of , still denoted by , such that uniformly for in any bounded subset of , as . Clearly, is nondecreasing and .
By the dominated convergence theorem and (67), it follows that Since and exist, using L’Hôspital rule leads to Thus, is a traveling wavefront of the system (11) connecting and .
Remark 11. We say that the is the minimal wave speed in the sense that (11) has no traveling wavefront with . We could briefly explain this in the following. In fact, the linearization of (12) at zero solution is (41), and the function is obtained by substituting in the second equation of (41). For , we know from (ii) of Lemma 7 that for any . We have from the second equation of (12) and the second equation of (41) that (12) cannot have a solution that satisfies .
Theorem 12. Assume that and hold. Then for any , the system (6) has a traveling wavefront with speed c, which connects and .
4. Asymptotic Behavior for Traveling Wavefronts
In this section, we discuss the asymptotic behavior for the traveling wavefronts obtained in the previous section as .