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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 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 Correspondence should be addressed to Shengmao Fu; Received 18 July 2013; Accepted 29 August 2013 Academic Editor: Benchawan Wiwatanapataphee Copyright © 2013 L. Zhang and S. 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. 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] 𝜕𝑈 = ∇ ⋅ (𝐷1 (𝑈, 𝑉) ∇𝑈) + 𝑓 (𝑈, 𝑉) , 𝜕𝑡 𝜕𝑉 = ∇ ⋅ (𝐷2 (𝑈, 𝑉) ∇𝑉) + 𝑔 (𝑈, 𝑉) , 𝜕𝑡 (1) in a box T 𝑁 = (0, 𝜋)𝑁 ⊂ R𝑁(𝑁 ≤ 3) with the homogeneous Neumann boundary conditions. In system (1), 𝑈(𝑥, 𝑡), 𝑉(𝑥, 𝑡) denote the densities of two interactive species at time 𝑡, the functions 𝐷1 , 𝐷2 are their diffusion rates, and 𝑓, 𝑔 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: 𝑢𝑡 − 𝑑1 Δ𝑢 = 𝜆𝑢 − 𝑢2 − 𝛽𝑢V, V𝑡 − 𝑑2 Δ [(1 + 𝑑3 𝑢) V] = V (𝜇 − 𝜕𝑢 𝜕V = = 0, 𝜕𝑥𝑖 𝜕𝑥𝑖 𝑢 (𝑥, 0) = 𝑢0 (𝑥) , 𝑥 ∈ T 𝑁, 𝑡 > 0, V ), 𝑚+𝑢 𝑥 ∈ T 𝑁, 𝑡 > 0, 𝑥𝑖 = 0, 𝜋, 𝑖 = 1, . . . , 𝑁, 𝑡 > 0, V (𝑥, 0) = V0 (𝑥) , 𝑥 ∈ T 𝑁, (2) where 𝑢(𝑥, 𝑡) and V(𝑥, 𝑡) represent the densities of the species prey and predator, respectively. The parameters 𝜆, 𝛽, 𝜇, 𝑚, 𝑑1 , 𝑑2 , and 𝑑3 are all positive constants, where 𝜆 and 𝜇 are the intrinsic growth rates of the prey and predator, 𝛽 is the predation rate, and the term V/(𝑚 + 𝑢) is a modified Leslie-Gower term [15]. The constants 𝑑1 , 𝑑2 , called diffusion 2 Abstract and Applied Analysis coefficients, represent the natural tendency of each species to diffuse to areas of smaller population concentration, and 𝑑3 , 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 𝑑2 Δ [(1 + 𝑑3 𝑢) V] = ∇ ⋅ [𝑑2 𝑑3 V∇𝑢 + 𝑑2 (1 + 𝑑3 𝑢) ∇V] . (3) In some sense, the coupled degree in (2) is stronger than that in (1). As a result, our analysis here, especially in establishing 𝐻2 estimates for nonlinear terms 𝑑2 𝑑3 ∇(V∇𝑢) and 𝑑2 ∇[(1 + 𝑑3 𝑢)∇V], is much more difficult and requires some techniques beyond those of [13, 14]. It is obvious that (2) has a unique positive equilibrium (̃ 𝑢, ̃V) if and only if 𝜆 > 𝛽𝜇𝑚, where 𝜆 − 𝛽𝜇𝑚 , 𝑢̃ = 1 + 𝛽𝜇 𝜇 (𝑚 + 𝜆) ̃V = . 1 + 𝛽𝜇 (4) Let 𝑢̂ = 𝑢(𝑥, 𝑡) − 𝑢̃, ̂V = V(𝑥, 𝑡) − ̃V be the perturbation around (̃ 𝑢, ̃V) and still denote it by (𝑢, V). Then, the perturbation (𝑢, V) satisfies the following strongly coupled equations: 𝑢𝑡 − 𝑑1 Δ𝑢 = 𝑔1 (𝑢 + 𝑢̃, V + ̃V) , 𝜕𝑢 𝜕V = = 0, 𝜕𝑥𝑖 𝜕𝑥𝑖 𝑢 (𝑥, 0) = 𝑢0 (𝑥) , 𝑥 ∈ T 𝑁, 𝑡 > 0, (5) 𝑥 ∈ T 𝑁, where 𝑔1 (𝑢, V) = 𝜆𝑢 − 𝑢2 − 𝛽𝑢V, 𝑢𝑡 − 𝑑1 Δ𝑢 = −̃ 𝑢𝑢 − 𝛽̃ 𝑢V, V𝑡 − 𝑑2 𝑑3 ̃VΔ𝑢 − 𝑑2 (1 + 𝑑3 𝑢̃) ΔV = 𝜇2 𝑢 − 𝜇V. (7) We use [⋅, ⋅] to denote a column vector and let w(𝑥, 𝑡) = [𝑢(𝑥, 𝑡), V(𝑥, 𝑡)], q = (𝑞1 , . . . , 𝑞𝑁) ∈ N𝑁. Then, q2 = 2 𝑁 ∑𝑁 𝑖=1 𝑞𝑖 are eigenvalues of −Δ on T under the homogeneous Neumann boundary condition, and the corresponding normalized eigenfunctions are given by 1 𝑁/2 { { q = 0, ( ) , { { { 𝜋 (8) 𝑒q (𝑥) = { { 2 𝑁/2 𝑁 { { {( ) ∏ cos (𝑞𝑖 𝑥𝑖 ) , q ≠ 0. 𝑖=1 { 𝜋 This set of eigenfunctions forms an orthonormal basis in 𝐿2 (T 𝑁). We look for a normal mode to be the linear system (7) of the following form: w (𝑥, 𝑡) = rq 𝑒𝜆 q 𝑡 𝑒q (𝑥) , (9) where 𝜆 q is a complex number and rq is a vector; they depend on q. Substituting (9) into (7), we have −𝛽̃ 𝑢 −̃ 𝑢 − 𝑑1 q2 ) r := 𝐿rq . 𝜆 q rq = ( 2 𝜇 − 𝑑2 𝑑3 ̃Vq2 −𝜇 − 𝑑2 (1 + 𝑑3 𝑢̃) q2 q (10) System (7) possesses a nontrivial normal mode if and only if det ( 𝛽̃ 𝑢 𝜆 q + 𝑢̃ + 𝑑1 q2 ) = 0, (11) 2 2 −𝜇 + 𝑑2 𝑑3 ̃Vq 𝜆 q + 𝜇 + 𝑑2 (1 + 𝑑3 𝑢̃) q2 𝑢 + 𝜇 + [𝑑1 + 𝑑2 (1 + 𝑑3 𝑢̃)] q2 } 𝜆 q + 𝑑1 𝑑2 (1 + 𝑑3 𝑢̃) q4 𝜆2q + {̃ 𝑥𝑖 = 0, 𝜋, 𝑖 = 1, . . . , 𝑁, 𝑡 > 0, V (𝑥, 0) = V0 (𝑥) , The corresponding linearized system of (5) takes the form of which is equivalent to 𝑥 ∈ T 𝑁, 𝑡 > 0, V𝑡 − 𝑑2 Δ [(1 + 𝑑3 (𝑢 + 𝑢̃)) (V + ̃V)] = 𝑔2 (𝑢 + 𝑢̃, V + ̃V) , 2. Growing Modes in the Linearized System + [𝑑1 𝜇 + 𝑑2 (1 + 𝑑3 𝑢̃) 𝑢̃ − 𝛽𝑑2 𝑑3 𝑢̃̃V] q2 + 𝜇̃ 𝑢 + 𝛽𝜇2 𝑢̃ = 0. (12) Thus, we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring that there exists a q, such that the co (...truncated)