Qualitative properties of a cooperative degenerate Lotka-Volterra model
© Sun et al.; licensee Springer 2013
Received: 3 April 2013
Accepted: 13 August 2013
Published: 8 October 2013
This paper considers a kind of degenerate parabolic systems. First, we consider the initial boundary value problem of the two-species degenerate parabolic cooperative system. By using the method of a parabolic regularization and energy estimate, we establish the existence of the weak solution of the problem. Then we establish the comparison principle and discuss the uniqueness and the uniform bound. At last, we consider the periodic boundary value problem of the system. By constructing a pair of ordered upper and lower solutions, we establish the existence of nontrivial nonnegative periodic solutions.
where Ω is a bounded domain in with smooth boundary ∂ Ω, , , , , , , , , , are strictly positive smooth functions and periodic in time t with period , and are nonnegative smooth functions.
Our motivation for the present study comes from population dynamics, to be specific, such model can be used to describe the population dynamics behavior. We refer to [1, 2] for a survey on this model. The functions u and v represent the spatial densities of two species at time t, 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 since , the boundary conditions (1.3) describe the living environment at the boundary, a, d are their respective net birth rate, b and f are intra-specific competitions, whereas c and e are those of inter-specific competitions.
Recently, degenerate cooperative systems have been the subject of extensive study, and most of the works are devoted to the existence, uniqueness, regularity properties and some other interesting properties of the weak solutions (one can see [3–8]). Since such models can describe nonlinear diffusion phenomenon, they are introduced into the discussion of population dynamics. For example, Vishnevskiĭ  studied the behavior at large time of solutions to mixed problems for weakly connected cooperative parabolic systems and obtained the monotonicity of the solutions. Pozio, Tesei  investigated the coexistence of prey-predator or competing species, subject to density dependent diffusion in an inhomogeneous habitat. They proved that coexistence arises in suitable domains, where favorable conditions are satisfied and also investigated the support properties and attractivity of the resulting stationary solutions. Later, Delgado and Suarez  studied the stability and uniqueness for a cooperative degenerate Lotka-Volterra model. For the semilinear case with , some results of this kind cooperative systems have already been obtained. The basic questions which have been considered are existence, uniqueness and boundary behavior of solutions, for details, one can see [12–16] and the references therein.
where , . At last, by constructing a pair of ordered upper and lower solutions, we establish the existence of the nontrivial nonnegative periodic solutions and the attractivity of the maximal periodic solution.
Since (1.1), (1.2) are degenerate at points where , , problem (1.1)-(1.4) might not have classical solutions in general. Therefore, we focus our main efforts on the discussion of weak solutions in the sense of the following.
Similarly, we can define a weak supersolution (subsolution ) if they satisfy the inequalities obtained by replacing ‘=’ with ‘≤’ (‘≥’) in (1.5), (1.6) with additional assumptions ().
Definition 1.2 A vector-valued function is said to be a T-periodic solution of the problem (1.1)-(1.3) if it is a solution in such that , in Ω. A vector-valued function is said to be a T-periodic supersolution of problem (1.1)-(1.3), if it is a super-solution in such that , in Ω. A vector-valued function is said to be a T-periodic subsolution of problem (1.1)-(1.3), if it is a subsolution in such that , in Ω.
This paper is organized as follows: In Section 2, we establish the existence and uniqueness of the weak solution of the problem (1.1)-(1.4). In Section 3, we establish the existence of the nontrivial nonnegative periodic solutions by constructing a pair of ordered upper and lower solutions and the method of monotone iteration technique.
2 Initial boundary value problem
By the result of , the regularity problem (2.1)-(2.4) admits a classical solution . So we just need to establish a necessary energy estimate for the classic solution and then establish the existence of weak solution of the initial boundary value problem by letting . For convenience, here and below, C denotes various positive constants independent of ε.
The proof is complete. □
By Lemma 2.1 and choosing , , , as the test functions, we can easily show the following estimates.
In order to obtain the maximum norm estimate of the approximate solution, we introduce the following lemma.
That is, a.e. in .
Similarly, we also have the same results for . The proof is complete. □
Theorem 2.1 The initial boundary value problem (1.1)-(1.4) has a weak solution .
A rather standard argument as that in  shows that , a.e. in . Then we can prove that meets Definition 1.1. Thus we complete the proof. □
In order to establish the uniqueness of the solution of (1.1)-(1.4), we need the following comparison principle.
Lemma 2.5 Assume that is the subsolution of problem (1.1)-(1.4), and it has an initial condition , is the supersolution, which has a positive lower bound of problem (1.1)-(1.4) and has an initial condition . If , , then , on .
Then from Gronwall’s lemma, we see that , . The proof is completed. □
Theorem 2.2 Assume that , then initial-boundary value problem (1.1)-(1.4) has a unique weak solution, which is uniformly bounded on .
Namely, is a positive supersolution of problem (1.1)-(1.4). So the weak solution of (1.1)-(1.4) is uniformly bounded. □
3 Periodic solutions
In this section, we will establish the existence of the nontrivial nonnegative periodic solutions by constructing a pair of ordered upper and lower solutions and the method of monotone iteration technique.
Lemma 3.1 Let , then (1.1)-(1.3) has a pair of ordered T-periodic supersolutions and T-periodic subsolutions.
is a subsolution of (1.1)-(1.3) and also is a T-periodic subsolution.
and η, , are chosen as those in Theorem 2.2. Obviously, is a positive T-periodic supersolution of problem (1.1)-(1.3).
The proof is complete. □
for every pair of points .
According to Lemma 2.5, Lemma 3.2 and Theorem 2.1, we can see that the mapping is well defined in and also an ordered preserving and compact map.
for almost every . Since is a compact operator, the limit above also exists in . In addition, are also the fixed points of the Poincare mapping . Using the method similar to that in , we can prove that the even extension of function , which is the solution of problem (1.1)-(1.4) with initial value , is just the nontrivial nonnegative periodic solution of problem (1.1)-(1.3). The existence of can be obtained similarly. In addition, by Lemma 2.5, we can conclude (3.3). The proof is complete. □
Now we consider the asymptotic behavior of the corresponding initial boundary value. Using the similar method as document , we have the following results.
This work is partially supported by the National Science Foundation of China (11271100, 11126222, 11301113), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011006), the Financial Support from Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q11111) and also the Aerospace Supported Fund of China (Contract No. 2011-HT-HGD-06).
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