Abstract and Applied Analysis

Volume 2013 (2013), Article ID 854862, 13 pages

http://dx.doi.org/10.1155/2013/854862

Research Article

Nonlinear Instability for a Leslie-Gower Predator-Prey Model with Cross Diffusion

Lina Zhang and Shengmao Fu

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.

Abstract.

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.

1. Introduction

Since Turing proposed the striking idea of “diffusion-driven instability” in 1952 [1], 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. [12] 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 [13] and described the early-stage pattern formation in that model.

Recently, Guo and Hwang considered the following reaction-diffusion system [14]

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 [15]. 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.

Lemma 1. Assume that the instability criterion (13) is valid. Suppose that

is a solution to the linearized system (7) with the initial condition . Then, there exists a constant , such that

for all .

Proof. We first consider the case for . For any ,

where is given by (14). Applying Cramer’s rule to (25), we have

where

It follows from (14) that there exist positive constants and

such that for all . Hence, for any ,

and by (12),

Thus,

Consequently, there exists a positive constant , such that

for any . Substituting this into (33) yields

for any , where . We thus obtain

Since

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)

The general theory in [21] guarantees that (5) has a unique nonnegative local solution. The results can be summarized as follows.

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

where

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

where

thus,

Substituting (70) (66) into (69) and (73) to get

where

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 8. In particular, if we take

in Remark 6, then, at the time , the estimate (98) gives

Therefore,

For sufficiently small, as , which implies that nonlinear instability occurs.

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

Now, we proceed in the following four steps.

Step  1. We establish estimate for the solution of .

From Lemma 1, for any , we have

By the definition of , for any , it follows that

Substituting (107) into (108), we obtain

Furthermore, it follows from (109) and Lemma 4 that

for any .

Step  2. We establish estimate for of . Applying Duhamel’s principle, we have

where , , and are given by (68) and (72). Using Lemma 1, it follows that

where . Therefore,

Substituting (110) into (114), we obtain

Step  3. We prove by contradiction that

for sufficiently small and

If , we can let in (111) to obtain

By choosing sufficiently small, such that

and the choice of in (117), we have

This is a contradiction to the definition of .

On the other hand, if , we can let in (115) to get

By choosing sufficiently small, such that

and the choice of in (117), we have

This again contradicts the definition of . From these arguments, (116) holds.

Step  4. Rewriting the left-hand term in (115) as the form of (26), and separating and moving to the right-hand side, it follows that

Next, we process the first term on the right side of (124) to get

Recall that there are only finite elements in and for any , there exists a constant , such that . Therefore,

Similar to the arguments in the proof of Lemma 1, we can treat the second, third, and fourth terms to obtain some similar estimates as (126), and our theorem follows.

Acknowledgments

This work was supported by the National Natural Science Foundation of China as received Grant nos. 10871160 and 11361055, the Fundamental Research Funds for the Gansu University, and NWNU-LKQN-11-21 Foundations.

References

A. M. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society B, vol. 237, no. 641, pp. 37–72, 1952.

T. K. Callahan and E. Knobloch, “Pattern formation in three-dimensional reaction-diffusion systems,” Physica D, vol. 132, no. 3, pp. 339–362, 1999.

M. Wang, “Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion,” Mathematical Biosciences, vol. 212, no. 2, pp. 149–160, 2008.

Y. Du and Y. Lou, “S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model,” Journal of Differential Equations, vol. 144, no. 2, pp. 390–440, 1998.

F. Yi, J. Wei, and J. Shi, “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,” Journal of Differential Equations, vol. 246, no. 5, pp. 1944–1977, 2009.

Y. Du and S. B. Hsu, “A diffusive predator-prey model in heterogeneous environment,” Journal of Differential Equations, vol. 203, no. 2, pp. 331–364, 2004.

Y. Du and M. Wang, “Asymptotic behaviour of positive steady states to a predator-prey model,” Proceedings of the Royal Society of Edinburgh A, vol. 136, no. 4, pp. 759–778, 2006.

Y. Guo and W. A. Strauss, “Instability of periodic BGK equilibria,” Communications on Pure and Applied Mathematics, vol. 48, no. 8, pp. 861–894, 1995.

Y. Guo, “Instability of symmetric vortices with large charge and coupling constant,” Communications on Pure and Applied Mathematics, vol. 49, no. 10, pp. 1051–1080, 1996.

C. Bardos, Y. Guo, and W. Strauss, “Stable and unstable ideal plane flows,” Chinese Annals of Mathematics B, vol. 23, no. 2, pp. 149–164, 2002.

H. J. Hwang and Y. Guo, “On the dynamical Rayleigh-Taylor instability,” Archive for Rational Mechanics and Analysis, vol. 167, no. 3, pp. 235–253, 2003.

Y. Guo, C. Hallstrom, and D. Spirn, “Dynamics near an unstable Kirchhoff ellipse,” Communications in Mathematical Physics, vol. 245, no. 2, pp. 297–354, 2004.

Y. Guo and H. J. Hwang, “Pattern formation (I): the Keller-Segel model,” Journal of Differential Equations, vol. 249, no. 7, pp. 1519–1530, 2010.

Y. Guo and H. J. Hwang, “Pattern formation. II: the turing instability,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2855–2866, 2007.

M. A. Aziz-Alaoui and M. Daher Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1069–1075, 2003.

B. Dubey, B. Das, and J. Hussain, “A predator-prey interaction model with self and cross-diffusion,” Ecological Modelling, vol. 141, no. 1–3, pp. 67–76, 2001.

W. Wang, Y. Lin, L. Zhang, F. Rao, and Y. Tan, “Complex patterns in a predator-prey model with self and cross-diffusion,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2006–2015, 2011.

Y. Du, R. Peng, and M. Wang, “Effect of a protection zone in the diffusive Leslie predator-prey model,” Journal of Differential Equations, vol. 246, no. 10, pp. 3932–3956, 2009.

R. Peng and M. Wang, “Positive steady states of the Holling-Tanner prey-predator model with diffusion,” Proceedings of the Royal Society of Edinburgh A, vol. 135, no. 1, pp. 149–164, 2005.

R. Peng, M. Wang, and G. Yang, “Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 570–577, 2008.

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralcera, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monograph, AMS, Providence, RI, USA, 1968.