Global existence and blow-up analysis to a cooperating model with self-diffusion
© Zhu et al.; licensee Springer. 2014
Received: 14 January 2014
Accepted: 27 May 2014
Published: 5 June 2014
In this paper, a two-species cooperating model with free diffusion and self-diffusion is investigated. The existence of the global solution is first proved by using lower and upper solution method. Then the sufficient conditions are given for the solution to blow up in a finite time. Our results show that the solution is global if the intra-specific competition is strong, while if the intra-specific competition is weak and the self-diffusion rate is small, blow-up occurs provided that the initial value is large enough or the free diffusion rate is small. Numerical simulations are also given to illustrate the blow-up results.
where is the Laplace operator, Ω is a bounded domain in with smooth boundary ∂ Ω. , , , , and () are positive constants. System (1.1) is usually referred as the cooperating two-species Lotka-Volterra model describing the interaction of diffusive biological species. The spatial density of the i th species at time t is represented by and its respective free diffusion rate is denoted by . is the self-diffusion rate. The real number is the net birth rate of the i th species and , are the crowding-effect coefficients. The parameters and measure cooperations between the species. The known is a smooth function satisfying the compatibility condition for . The boundary conditions in (1.1) imply that the habitat is surrounded by a totally hostile environment.
here all parameters are positive constants except and which can be chosen positive, zero or negative. He proved that a unique solution of (1.2) exists and is uniformly bounded in if , while if the solution blows up in a finite time for big with any nontrivial nonnegative initial data or for any with big initial data. His results imply that the solution is global if the intra-specific competition is strong, while the solution may blow up if the intra-specific competition is weak. Lou et al.  considered the problem (1.2) with homogeneous Neumann boundary conditions and studied the effect of diffusion on the blow-up. They gave a sufficient condition on the initial data for the solution to blow up in a finite time. Wang et al.  studied a reaction-diffusion system with nonlinear absorption terms and boundary flux. Some sufficient conditions for global existence and finite time blow-up of the solutions are given.
On the other hand, Shigesada et al.  in 1979 proposed a generalization of Lotka-Volterra competing model in order to describe spatial segregation of interacting population species in one space dimension. More and more attention has been given to the SKT model with other types of reaction terms. For example, the two-species prey-predator model is in [13, 14], two-species cooperating model is in , three-species cooperating model is in [16, 17]. These works concentrate on the existence of time-dependent solution or uniform boundedness and stability of global solutions. In this paper we are interested in studying the blow-up properties of the solution and we will consider the effect of self-diffusion coefficients and on the long time behaviors of the solution.
The content of this paper is organized as the follows: In Section 2, the existence and uniqueness of global solution are given by using upper and lower solutions and their associated monotone iterations as in . Section 3 is devoted to a sufficient condition for the solution to blow up. Numerical illustrations are performed in Section 4 and a brief discussion is also given in Section 5.
2 Existence of global solution
This section is devoted to the global existence of (1.1). First we give the definition of ordered upper and lower solutions of (1.1), then a pair of ordered upper and lower solutions is constructed.
and satisfies the above inequalities in reversed order.
Let , , it is easy to see and are ordered upper and lower solutions of (2.3).
Before constructing monotone sequences, we present the following positivity lemma, which will be used in the proof of the monotone property of the sequences.
Lemma 2.1 (Lemma 2.1 of )
then in .
for any m and .
under the same boundary and initial conditions, the existence of the sequence is ensured by  (Chapter V, Section 7). Denote the sequence by if , and by if , and refer to them as maximal and minimal sequences, respectively. The following lemma shows that the sequences are monotone.
, where and .
where , is between and for .
By Lemma 2.1, we have , which leads to and thus . Similarly, we have and . The conclusion of the lemma follows from the induction principle. □
From the proof of Lemma 2.2, we know that the following comparison principle holds.
Lemma 2.3 (Comparison Principle)
exist and satisfy .
Now we show that () and is the unique solution of (1.1).
Moreover, is the unique solution of problem (1.1).
The proof of this theorem is similar to Theorem 3.1 in , so we omit it here.
The existence and uniqueness of the solution to problem (1.1) can be ensured by constructing a pair of ordered upper and lower solutions of (1.1). In fact, we only need to construct a bounded positive upper solution since that we can take as lower solution. We have the following theorem.
Theorem 2.2 If , then for any nonnegative initial data, problem (1.1) admits a unique global solution , which is uniformly bounded in .
It is easily to check that is the upper solution of problem (1.1). Further, is global and uniformly bounded. The desired results can obtain from Theorem 2.1. □
Remark 2.1 For the problem (1.1) with the Neumann boundary condition instead of the Dirichlet boundary condition, it can be discussed similarly. For the one-dimensional case, we can also obtain the existence and uniform boundedness of the global solution by using Gagliardo-Nirenberg-type inequalities; see  for details.
3 Blow-up of the solution
where k, D, b, and α are positive constants.
has a nontrivial nonnegative solution if , where λ is the first eigenvalue of Laplace operator subject to the homogeneous Dirichlet boundary condition.
The proof of Lemma 3.1 is similar to that of Lemma 11.7.1 of , where , and we omit it here.
then in .
Using the comparison principle and assumptions on , we have in . So in . Then using again the comparison principle yields in for an arbitrarily small . Hence in . □
the initial data is large enough,
then all nontrivial nonnegative solutions of problem (3.1) blow up.
- (i)If and , we get
- (ii)If and the initial data is large enough so that
then there exists such that . Therefore for and . Based on the discussion, we show that if the initial value is large enough the solution of problem (3.1) must blow up in a finite time. Thereby blows up in finite time. □
Our main result in this section can now be stated as follows.
If , then the solution of problem (1.1) with any nontrivial nonnegative initial value blows up.
If the initial value is large enough,
then the solution of (1.1) blows up.
Proof To show this, it suffices to construct a lower solution of (2.3) and prove the lower solution blows up in a finite time with some suitable conditions.
where for .
First we discuss the case when the initial value is large enough. By Lemma 3.1, problem (3.2) has a nontrivial nonnegative solution, denoted by . We define to be the solution of (3.1) with . Then it is easy to see that from Lemma 3.2. So if and , the pair is a lower solution of (1.1). On the other hand, if follows from Lemma 3.3 that the solution must blow up in a finite time provided that . So if and are sufficiently large, then the pair blows up. Therefore the solution of (1.1) blows up in a finite time.
Second, we consider the case that and , are small enough. For an arbitrary nontrivial nonnegative initial data , the solution of (1.1) is positive for by Lemma 3.2. Further, we may assume that and for , otherwise replace the initial function by for some .
Let be the eigenfunction corresponding to the eigenvalue λ, then there exists suitable , such that , . Choosing , then satisfies (3.2) with and and define be the solution of (3.1) with the initial data . Then and is monotone nondecreasing in t by Lemma 3.2. Moreover, it follows from the comparison principle that and in , where is the maximal existent time of , , and . Hence is a lower solution of (1.1).
On the other hand, Lemma 3.3 ensures the existence of a finite such that the solution exists in and is unbounded in Ω as . Thus the solution of (1.1) cannot exist beyond and is nonglobal. This finishes the proof. □
4 Numerical illustrations
In this section, we present some numerical simulations to investigate the blow-up results in Theorem 3.1. Symbolic mathematical software Matlab 7.0 is used to plot numerical graphs. For simplicity, we always take .
Example 4.1 (Case (i))
Example 4.2 (Case (ii))
In this paper, we consider a two-species cooperating model with free diffusion and self-diffusion. Our main purpose is to find sufficient conditions for the solution to blow up in a finite time. The results show that the global solution exists if , i.e. the inter-specific competition is strong.
Comparing Figure 2 with Figure 3, we can see that the solution without self-diffusion blows up fast, which implies the self-diffusion can ‘relax’ the blow-up. We still have no theoretical proof, but we feel it is worth further investigation.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by PRC grant NSFC 61103018, NSFC 11271197 and the NSF of Jiangsu Province BK2012682.
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