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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 854862, 13 pages
Nonlinear Instability for a Leslie-Gower Predator-Prey Model with Cross Diffusion
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Received 18 July 2013; Accepted 29 August 2013
Academic Editor: Benchawan Wiwatanapataphee
Copyright © 2013 Lina Zhang and Shengmao Fu. 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.
A rigorous mathematical characterization for early-stage spatial and temporal patterns formation in a Leslie-Gower predator-prey model with cross diffusion is investigated. Given any general perturbation near an unstable constant equilibrium, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes.
Since Turing proposed the striking idea of “diffusion-driven instability” in 1952 , reaction-diffusion systems are often employed to investigate chemical and biological pattern formations and have received much attention from the scientists [2–7]. However, most of the works concentrate on pattern formation in the case of linear instability, and there is a little discussion about the nonlinear effect of a reaction-diffusion system on the evolution of a nonuniform pattern.
In general, nonlinear instability is treated with great delicacy and difficulty. At first, nonlinear instability was established for nondissipative systems [8–11]. In 2004, Guo et al.  established nonlinear instability for an unstable Kirchhoff ellipse. Based upon a precise linear analysis, they found that the dynamics of general perturbation can be characterized by the linear dynamics of the fastest growing modes. This marks a beginning of a quantitative description of instability. Subsequently, Guo and Hwang dealt with nonlinear stability for a Keller-Segel model in  and described the early-stage pattern formation in that model.
Recently, Guo and Hwang considered the following reaction-diffusion system  in a box with the homogeneous Neumann boundary conditions. In system (1), , denote the densities of two interactive species at time , the functions , are their diffusion rates, and , are the reaction functions. The classical Turing instability and Turing patterns were studied under some suitable conditions. Their result shows that the nonlinear evolution of patterns is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes over a time period.
In this paper, we consider the following Leslie-Gower predator-prey model with cross diffusion: where and represent the densities of the species prey and predator, respectively. The parameters , , , , , , and are all positive constants, where and are the intrinsic growth rates of the prey and predator, is the predation rate, and the term is a modified Leslie-Gower term . The constants , , called diffusion coefficients, represent the natural tendency of each species to diffuse to areas of smaller population concentration, and , called a cross-diffusion coefficient, expresses the population flux of the predator resulting from the presence of the prey species. For more ecological backgrounds about this model, one can refer to [15–17].
System (2) and its variants were studied widely for pattern formation by applying the bifurcation theory and the degree theory [6, 18–20] in the case of linear instability. Inspired by the works [13, 14], in this paper, we attempt to study the nonlinear instability for this system and give a rigorous mathematical characterization for the nonlinear evolution of pattern by using a bootstrap technique. The mathematical approach in this paper is similar in spirit to that of [13, 14]. However, our problem (2) is much more complex. Notice that the diffusion term of the predator equation in the model (2) is In some sense, the coupled degree in (2) is stronger than that in (1). As a result, our analysis here, especially in establishing estimates for nonlinear terms and , is much more difficult and requires some techniques beyond those of [13, 14].
It is obvious that (2) has a unique positive equilibrium if and only if , where Let , be the perturbation around and still denote it by . Then, the perturbation satisfies the following strongly coupled equations: where
This paper is organized as follows. In Section 2, the growing modes in the linearized system are studied, which are important for our later discussions. Section 3 gives some estimates for the perturbation. The key is to control the nonlinear growth of high-order energy. In Section 4, the nonlinear instability is obtained.
2. Growing Modes in the Linearized System
The corresponding linearized system of (5) takes the form of We use to denote a column vector and let , . Then, are eigenvalues of on under the homogeneous Neumann boundary condition, and the corresponding normalized eigenfunctions are given by This set of eigenfunctions forms an orthonormal basis in .
We look for a normal mode to be the linear system (7) of the following form: where is a complex number and is a vector; they depend on . Substituting (9) into (7), we have System (7) possesses a nontrivial normal mode if and only if which is equivalent to Thus, we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring that there exists a , such that the constant term in (12) is In this paper, we always assume that there exists a , such that (13) holds. Then, the discriminant of (12) is and the coefficient of is positive, since (13) implies
For given , we denote the corresponding eigenvalues by and eigenvectors by . We split it into three cases for the linear analysis.(1). Let , and let be two distinct real roots with , being the corresponding (linearly independent) real eigenvectors. It is easy to see that Denote Clearly, for . Note that there are only finitely many in and . Therefore, there are only finitely many linear growing modes, such that the constant equilibrium of (5) is unstable. Furthermore, we define Then, .(2). Let . In this case, (12) possesses repeated real eigenvalues. Consider The corresponding eigenvectors are and we can find another independent vector , satisfying (3). The complex case is where (12) possesses a pair of complex eigenvalues with a negative real part. Denote , and for any , denote Then, where and are linearly independent vectors.
Given any initial perturbation , we can expand it as follows: where , , , , , and are constants, and The unique solution to (7) is given by
In the sequel, the constant will only depend on the domain and the dimension , and the generic constants , , , , and so forth will depend on , , and the parameters , , , , , , and . Our main result of this section is the following lemma.
Proof. We first consider the case for . For any ,
where is given by (14). Applying Cramer’s rule to (25), we have
It follows from (14) that there exist positive constants and
such that for all . Hence, for any ,
and by (12),
Consequently, there exists a positive constant , such that
for any . Substituting this into (33) yields
for any , where . We thus obtain
there exists a constant , such that
for any .
For any and , as is an increasing function of , we denote With the help of (30) and (31), we get where only depends on , , , , , and . Hence, we conclude that, for any , there exists a positive constant , such that
For all and , by some similar arguments as above we can show that there exist positive constants and , such that
Next, we derive the energy estimate in for . Recall that is an orthonormal basis in . Then, where From (43) and (44), we obtain Thus, where . Finally, for any , we have
For finite time , we multiply the first and second equations of (7) by and , respectively, then add them and use the integration by parts to get Firstly, we claim that the integrand of the second integral in (50) satisfies for some positive constant . Obviously, it suffices to require that This is equivalent to Denote
On the other hand, the term on the right of (50) is Taking , and substituting (51) and (55) into (50), we get Integrating (56) from to leads to If , then it follows from (57) that thus, the Gronwall inequality implies Consequently, there exists a positive constant , such that for all due to the boundedness of
If , in the same way as above, there exists a positive constant , such that The proof is completed by taking .
3. The Estimates for the Solutions of the Full System (5)
Lemma 2. Suppose that is a solution of the full system (5). For and , there exist a and a constant , such that if .
Denote In order to derive the estimate for the solution of (5), we first prove the following energy estimates.
Lemma 3. Suppose that is a solution of the full system (5). Then, for .
Proof. We first notice that system (5) preserves the evenness of the solution; that is, if is a solution to (5), then , , and are also solutions of (5). We can regard system (5) as a special case with the evenness of the periodic problem by a reflective and an even extension. For this reason, we may assume periodicity at the boundary of the extended . Taking the second order partial derivative of the first equation of (5), multiplying , and integrating over the domain to get
are the linear and nonlinear terms, respectively, then, we have
Similarly, taking the second order partial derivative of the second equation of (5), multiplying , and integrating over the domain to get
Substituting (70) (66) into (69) and (73) to get
we apply the Sobolev imbedding to control the norm by
for . From the Hölder inequality, the Poincaré inequality, and the Sobolev imbedding theorem, it follows that
for . Recall the even extension of (5), and the solution satisfies
By (76) and (77), we find that
where is a universal constant. Therefore, when , it follows from (76) and (79) that
Applying the Young inequality to get
which is combined with the interpolation inequality and the -Young inequality to imply
where is a positive constant, in the same way as above, we obtain that the second integral satisfies
Substituting (80)–(83) into (74), take to get
Similar as the proof of Lemma 1, we proceed in the two cases: and . Then, we conclude So, the even extension implies
Next, we control the growth of in terms of its growth.
Lemma 4. Suppose that is a solution of the full system (5), such that Then, where .
Proof. We first consider the second-order derivatives of . From Lemmas 3 and 4 and our assumption for , we have By an integration from 0 to , we deduce that For the first derivations of , we apply the Poincaré inequality to get Applying the even extension, we have It follows from (88)–(93) that where .
4. Nonlinear Instability and Pattern Formation
Let be a fixed constant. For an arbitrary positive number , we define the escape time by or equivalently
Theorem 5. Assume that there exists a , such that the instability criterion (13) holds. Let such that , . Then, there exist , , and , such that for all and , the nonlinear evolution of (5) with the initial perturbation satisfies where the constant , called the gap between and the rest of eigenvalues, is positive, and
Remark 6. First, we notice that the part of the fastest growing modes in (98) is nontrivial. Recall that the initial profile is any function satisfying . Thus, we can choose , such that there is at least one with . Consequently,
Remark 7. Fix to be a positive small number. If is small sufficiently, then is large, and for , . Our estimate (98) implies that the dynamics of a general perturbation can be characterized by such linear dynamics over a long time period , when the initial perturbation is small.
Remark 9. From a view of pattern formation, Theorem 5 shows that if the unique positive equilibrium of (2) is linear unstable, then a general small perturbation near can induce pattern formation. Furthermore, the patterns can be characterized by the fastest growing modes in the corresponding linear dynamics over a long time period .
Proof of Theorem 5. Define