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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 714248, 18 pages
A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Received 13 December 2010; Revised 13 February 2011; Accepted 24 February 2011
Academic Editor: Elena Braverman
Copyright © 2011 Jiebao Sun 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 consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.
In this paper, we consider the following two-species cooperative system: where , , , , , , , , , are strictly positive smooth functions and periodic in time with period and and are nonnegative functions and satisfy .
In dynamics of biological groups, the system (1.1)-(1.2) can be used to describe the interaction of two biological groups. The diffusion terms and represent the effect of dispersion in the habitat, which models a tendency to avoid crowding and the speed of the diffusion is rather slow. The boundary conditions (1.3) indicate that the habitat is surrounded by a totally hostile environment. The functions and represent the spatial densities of the species at time and are their respective net birth rate. The functions and are intra-specific competitions, whereas and are those of interspecific competitions.
As famous models for dynamics of population, two-species cooperative systems like (1.1)-(1.2) have been studied extensively, and there have been many excellent results, for detail one can see [1–6] and references therein. As a special case, men studied the following two-species Lotka-Volterra cooperative system of ODEs: For this system, Lu and Takeuchi  studied the stability of positive periodic solution and Cui  discussed the persistence and global stability of it.
For this system, Lin et al.  showed the existence and asymptotic behavior of - periodic solutions when are all smooth positive and periodic in time with period . When are all positive constants, Pao  proved that the Dirichlet boundary value problem of this system admits a unique solution which is uniformly bounded when , while the blowup solutions are possible when the two species are strongly mutualistic . For the homogeneous Neumann boundary value problem of this system, Lou et al.  proved that the solution will blow up in finite time under a sufficient condition on the initial data. When and , from (1.1) we get the single degenerate equation For this equation, Sun et al.  established the existence of nontrivial nonnegative periodic solutions by monotonicity method and showed the attraction of nontrivial nonnegative periodic solutions.
In the recent years, much attention has been paid to the study of periodic boundary value problems for parabolic systems; for detail one can see [9–15] and the references therein. Furthermore, many researchers studied the periodic boundary value problem for degenerate parabolic systems, such as [16–19]. Taking into account the impact of periodic factors on the species dynamics, we are also interested in the existence of the nontrivial periodic solutions of the cooperative system (1.1)-(1.2). In this paper, we first show the existence of the global generalized solution of the initial boundary value problem (1.1)–(1.4). Then under the condition that where , we show that the generalized solution is uniformly bounded. At last, by the method of monotone iteration, we establish the existence of the nontrivial periodic solutions of the system (1.1)-(1.2), which follows from the existence of a pair of large periodic supersolution and small periodic subsolution. At last, we show the existence and the attractivity of the maximal periodic solution.
Our main efforts center on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem (1.1)–(1.4).
Definition 1.1. A nonnegative and continuous vector-valued function is said to be a generalized solution of the problem (1.1)–(1.4) if, for any and any functions with , and
Similarly, we can define a weak supersolution (subsolution ) if they satisfy the inequalities obtained by replacing “=” with “≤” (“≥”) in (1.3), (1.4), and (1.9) and with an additional assumption .
Definition 1.2. A vector-valued function is said to be a -periodic solution of the problem (1.1)–(1.3) if it is a solution in such that , in . A vector-valued function is said to be a -periodic supersolution of the problem (1.1)–(1.3) if it is a supersolution in such that in . A vector-valued function is said to be a -periodic subsolution of the problem (1.1)–(1.3), if it is a subsolution in such that in .
This paper is organized as follows. In Section 2, we show the existence of generalized solutions to the initial boundary value problem and also establish the comparison principle. Section 3 is devoted to the proof of the existence of the nonnegative nontrivial periodic solutions by using the monotone iteration technique.
2. The Initial Boundary Value Problem
To solve the problem (1.1)–(1.4), we consider the following regularized problem: where , are nonnegative bounded smooth functions and satisfy The standard parabolic theory (cf. [20, 21]) shows that (2.1)–(2.4) admits a nonnegative classical solution . So, the desired solution of the problem (1.1)–(1.4) will be obtained as a limit point of the solutions of the problem (2.1)–(2.4). In the following, we show some important uniform estimates for .
Lemma 2.1. Let be a solution of the problem (2.1)–(2.4). (1) If , then there exist positive constants and large enough such that where is a positive constant only depending on , , and .(2) If , then (2.7) also holds when is small enough.
Proof. Multiplying (2.1) by and integrating over , we have that
By Poincaré’s inequality, we have that
where is a constant depending only on and and becomes very large when the measure of the domain becomes small. Since , Young's inequality shows that
For convenience, here and below, denotes a positive constant which is independent of and may take different values on different occasions. Complying (2.8) with (2.9) and (2.10), we obtain
As a similar argument as above, for and positive constant , we have that
Thus we have that
For the case of , there exist large enough such that By Young's inequality, we have that where Together with (2.13), we have that where Furthermore, by Hölder's and Young's inequalities, from (2.17) we obtain Then by Gronwall's inequality, we obtain
Now we consider the case of . It is easy to see that there exist positive constants large enough such that Due to the continuous dependence of upon in (2.9), from (2.13) we have that when is small enough. Then by Young's and Gronwall's inequalities we can also obtain (2.20), and thus we complete the proof of this lemma.
Taking , as the test functions, we can easily obtain the following lemma.
Proof. For a positive constant , multiplying (2.1) by and integrating the results over , we have that
where and is the characteristic function of . Let
then is absolutely continuous on . Denote by the point where takes its maximum. Assume that , for a sufficient small positive constant . Taking , in (2.25), we obtain
we have that
Letting , we have that
Denote and ; then
By Sobolev's theorem,
where , and denotes various positive constants independent of . By Hölder’s inequality, it yields
On the other hand, for any and , we have that
Combined with (2.35), it yields
It is easy to see that
Then by the De Giorgi iteration lemma , we have that
where . That is,
It is the same for the second inequality of (2.24). The proof is completed.
Proof. Multiplying (2.1) by and integrating over , by (2.3), (2.4) and Young's inequality we have that which together with the bound of shows that where is a positive constant independent of . Noticing the bound of , we have that It is the same for the second inequality. The proof is completed.
From the above estimates of , we have the following results.
Proof. By Lemmas 2.2, 2.3, and 2.4, we can see that there exist subsequences of (denoted by themselves for simplicity) and functions such that as . Then a rather standard argument as  shows that is a generalized solution of (1.1)–(1.4) in the sense of Definition 1.1.
Lemma 2.6. Let be a subsolution of the problem (1.1)–(1.4) with the initial value and a supersolution with a positive lower bound of the problem (1.1)–(1.4) with the initial value . If , , then , on .
Proof. Without loss of generality, we might assume that , , , where is a positive constant. By the definitions of subsolution and supersolution, we have that
Take the test function as
where is a monotone increasing smooth approximation of the function defined as follows:
It is easy to see that as . Since , the test function is suitable. By the positivity of we have that
where is a positive constant depending on . Letting and noticing that
we arrive at
Let be a supsolution with a positive lower bound . Noticing that
with , we have that
where is a positive constant depending upon .
Similarly, we also have that Combining the above two inequalities, we obtain By Gronwall's lemma, we see that . The proof is completed.
Proof. The uniqueness comes from the comparison principle immediately. In order to prove that the solution is global, we just need to construct a bounded positive supersolution of (1.1)–(1.4).
Let and , since ; then and satisfy Let , where is a constant such that ; then we have that That is, is a positive supersolution of (1.1)–(1.4). Since are global and uniformly bounded, so are and .
3. Periodic Solutions
In order to establish the existence of the nontrivial nonnegative periodic solutions of the problem (1.1)–(1.3), we need the following lemmas. Firstly, we construct a pair of -periodic supersolution and -periodic subsolution as follows.
Proof. We first construct a -periodic subsolution of (1.1)–(1.3). Let be the first eigenvalue and be the uniqueness solution of the following elliptic problem:
then we have that
where is a small constant to be determined. We will show that is a (time independent, hence -periodic) subsolution of (1.1)–(1.3).
Taking the nonnegative function as the test function, we have that Similarly, for any nonnegative test function , we have that We just need to prove the nonnegativity of the right-hand side of (3.4) and (3.5).
Since on , then there exists such that where . Choosing then we have that which shows that is a positive (time independent, hence -periodic) subsolution of (1.1)–(1.3) on .
Moreover, we can see that, for some , Choosing then on , that is These relations show that is a positive (time independent, hence -periodic) subsolution of (1.1)–(1.3).
Letting , where are taken as those in Corollary 2.7, it is easy to see that is a positive (time independent, hence -periodic) subsolution of (1.1)–(1.3).
Obviously, we may assume that , by changing , appropriately.
Lemma 3.3 (see ). Define a Poincaré mapping where is the solution of (1.1)–(1.4) with initial value . According to Lemmas 2.6 and 3.2 and Theorem 2.5, the map has the following properties: (i) is defined for any and order preserving;(ii) is order preserving;(iii) is compact.
Observe that the operator is the classical Poincaré map and thus a fixed point of the Poincaré map gives a -periodic solution setting. This will be made by the following iteration procedure.
Theorem 3.4. Assume that and there exists a pair of nontrivial nonnegative -periodic subsolution and -periodic supersolution of the problem (1.1)–(1.3) with ; then the problem (1.1)–(1.3) admits a pair of nontrivial nonnegative periodic solutions such that
Proof. Taking , as those in Lemma 3.1 and choosing suitable , , , and , we can obtain . By Lemma 2.6, we have that . Hence by Definition 1.2 we get , which implies for any . Similarly we have that , and hence for any . By Lemma 2.6, we have that for any . Then exist for almost every . Since the operator is compact (see Lemma 3.3), the above limits exist in , too. Moreover, both and are fixed points of . With the similar method as , it is easy to show that the even extension of the function , which is the solution of the problem (1.1)–(1.4) with the initial value , is indeed a nontrivial nonnegative periodic solution of the problem (1.1)–(1.3). It is the same for the existence of . Furthermore, by Lemma 2.6, we obtain (3.16) immediately, and thus we complete the proof.
Furthermore, by De Giorgi iteration technique, we can also establish a prior upper bound of all nonnegative periodic solutions of (1.1)–(1.3). Then with a similar method as , we have the following remark which shows the existence and attractivity of the maximal periodic solution.
Remark 3.5. If , the problem (1.1)–(1.3) admits a maximal periodic solution . Moreover, if is the solution of the initial boundary value problem (1.1)–(1.4) with nonnegative initial value , then, for any , there exists depending on , , and , such that
This work was supported by NSFC (10801061), the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2009049), Natural Sciences Foundation of Heilongjiang Province (Grant no. A200909), and also the 985 project of Harbin Institute of Technology.
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