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

and Applied Analysis 3 and the coefficient of q is positive, since (13) implies d 1 μ + d 2 (1 + d 3 ?̃?) ?̃? − βd 2 d 3 ?̃?Ṽ < 0. (15) For given q, we denote the corresponding eigenvalues by λ ± q and eigenvectors by r±(q). We split it into three cases for the linear analysis. (1) Δ > 0. LetΛ 1 = {q | Δ > 0}, and let λ±q be two distinct real roots with λ+q > λ − q, λ ± q being the corresponding (linearly independent) real eigenvectors. It is easy to see that r ± (q) = [1, − λ ± q + ?̃? + d1q 2 β?̃? ] . (16) Denote Λ ∗ = {q | d 1 d 2 (1 + d 3 ?̃?) q4 + [d 1 μ + d 2 (1 + d 3 ?̃?) ?̃? − βd 2 d 3 ?̃?Ṽ] q2 +μ?̃? + βμ 2 ?̃? < 0} . (17) Clearly, λ+q > 0 for q ∈ Λ ∗. Note that there are only finitely many q in Λ ∗ and Λ ∗ ⊂ Λ 1 . Therefore, there are only finitely many linear growing modes, such that the constant equilibrium (0, 0) of (5) is unstable. Furthermore, we define λmax = max λ + q>0 λ + q, Λmax = {q | λ + q = λmax} . (18) Then, Λmax ⊂ Λ ∗ ⊂ Λ 1. (2) Δ = 0. Let Λ 2 = {q | Δ = 0}. In this case, (12) possesses repeated real eigenvalues. Consider λq = λ + q = λ − q = − 1 2 {?̃? + μ + [d 1 + d 2 (1 + d 3 ?̃?)] q2} < 0. (19) The corresponding eigenvectors are r (q) = r + (q) = r − (q) = [1, − λq + ?̃? + d1q2 β?̃? ] , (20) and we can find another independent vector r(q) = [0, −1/(β?̃?)], satisfying (L − λqI) r 󸀠 (q) = r (q) . (21) (3) Δ < 0. The complex case is where (12) possesses a pair of complex eigenvalues with a negative real part. Denote Λ 3 = {qΔ < 0}, and for any q ∈ Λ 3 , denote λ + q = Re λq + i Im λq, r+ (q) = Re r (q) + i Im r (q) . (22) Then, λ − q = Re λq − i Im λq, r− (q) = Re r (q) − i Im r (q) , (23) where Re r(q) and Im r(q) are linearly independent vectors. Given any initial perturbation w(x, 0), we can expand it as follows: w (x, 0) = ∑ q∈Λ 1 {w − q r− (q) + w + qr+ (q)} eq (x) + ∑ q∈Λ 2 {wqr (q) + w 󸀠 qr 󸀠 (q)} eq (x) + ∑ q∈Λ 3 {w Re q Re r (q) + w Im q Im r (q)} eq (x) := ∑ q∈NN wqeq (x) , (24) where w q , w + q , wq, w 󸀠 q, w Re q , and w Im q are constants, and wq = w − q r− (q) + w + q r+ (q) , q ∈ Λ 1, wq = wqr (q) + w 󸀠 qr 󸀠 (q) , q ∈ Λ 2 , wq = w Re q Re r (q) + w Im q Im r (q) , q ∈ Λ 3. (25) The unique solution w(x, t) = [u(x, t), V(x, t)] to (7) is given by w (x, t) = ∑ q∈Λ 1 {w − q r− (q) e λ − qt + w + qr+ (q) e λ + qt} eq (x) + ∑ q∈Λ 2 {wqr (q) + w 󸀠 qr 󸀠 (q) + w󸀠 qr (q) t} e λteq (x) + ∑ q∈Λ 3 {w Re q (Re r (q) cos [(Im λq) t] − Im r (q) sin [(Im λq) t]) + w Im q (Re r (q) sin [(Im λq) t] + Im r (q) cos [(Im λq) t])}


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][3][4][5][6][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][9][10][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]   = ∇ ⋅ ( 1 (, ) ∇) +  (, ) ,   = ∇ ⋅ ( 2 (, ) ∇) +  (, ) , in a box T  = (0, )  ⊂ R  ( ≤ 3) with the homogeneous Neumann boundary conditions.In system (1), (, ), (, ) denote the densities of two interactive species at time , the functions  1 ,  2 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: ) ,  ∈ T  ,  > 0,    = V   = 0,   = 0, ,  = 1, . . ., ,  > 0,  (, 0) =  0 () , V (, 0) = V 0 () ,  ∈ T  , where (, ) and V(, ) represent the densities of the species prey and predator, respectively.The parameters , , , ,  1 ,  2 , and  3 are all positive constants, where  and  are the intrinsic growth rates of the prey and predator,  is the predation rate, and the term V/( + ) is a modified Leslie-Gower term [15].The constants  1 ,  2 , called diffusion 2 Abstract and Applied Analysis coefficients, represent the natural tendency of each species to diffuse to areas of smaller population concentration, and  3 , 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][16][17].System (2) and its variants were studied widely for pattern formation by applying the bifurcation theory and the degree theory [6,[18][19][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  2 estimates for nonlinear terms  2  3 ∇(V∇) and  2 ∇[(1 +  3 )∇V], is much more difficult and requires some techniques beyond those of [13,14].
It is obvious that (2) has a unique positive equilibrium (ũ, Ṽ) if and only if  > , where Let û = (, ) − ũ, V = V(, ) − Ṽ be the perturbation around (ũ, Ṽ) and still denote it by (, V).Then, the perturbation (, V) 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.

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 w(, ) = [(, ), V(, )], q = ( 1 , . . .,   ) ∈ N  .Then, q 2 = ∑  =1   2 are eigenvalues of −Δ on T  under the homogeneous Neumann boundary condition, and the corresponding normalized eigenfunctions are given by This set of eigenfunctions forms an orthonormal basis in  2 (T  ).
We look for a normal mode to be the linear system (7) of the following form: where  q is a complex number and r q is a vector; they depend on q.Substituting ( 9) into (7), we have System (7) possesses a nontrivial normal mode if and only if det (  q + ũ +  1 q 2 ũ − 2 +  2  3 Ṽq 2  q +  +  2 (1 +  3 ũ) q 2 ) = 0, (11) which is equivalent to Thus, we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring that there exists a q, such that the constant term in ( 12) is In this paper, we always assume that there exists a q, such that (13) holds.Then, the discriminant of ( 12) is Abstract and Applied Analysis 3 and the coefficient of q 2 is positive, since (13) implies For given q, we denote the corresponding eigenvalues by  ± q and eigenvectors by r ± (q).We split it into three cases for the linear analysis.
Lemma 1. Assume that the instability criterion (13) is valid.Suppose that is a solution to the linearized system (7) with the initial condition w(, 0).Then, there exists a constant Ĉ1 ≥ 1, such that for all  ≥ 0.
Proof.We first consider the case for  > 1.For any q ∈ Λ 1 , where Δ is given by (14).Applying Cramer's rule to (25), we have where It follows from ( 14) that there exist positive constants  1 and such that Δ >  1 q 2 for all |q| >  1 .Hence, for any |q| >  1 , and by (12), Thus, Consequently, there exists a positive constant  2 >  1 , such that for any |q| >  2 .Substituting this into (33) yields for any |q| >  2 , where We thus obtain there exists a constant  3 >  2 , such that for any |q| >  3 .
For any q ∈ Λ 1 and |q| ≤  3 , as Δ is an increasing function of |q| 2 , we denote With the help of ( 30) and (31), we get where  3 only depends on , , , ,  * , and  3 .Hence, we conclude that, for any q ∈ Λ 1 , there exists a positive constant For all q ∈ Λ 2 and q ∈ Λ 3 , by some similar arguments as above we can show that there exist positive constants  5 and  6 , such that       q r (q)      ,        q r  (q)      ,        q r (q)      ≤  5      w q      ,   q  ≤  5 ,       Re q Re r (q)      ,       Re q Im r (q)      ,       Im q Re r (q)      ,       Im q Im r (q)      ≤  6      w q      . (44) Next, we derive the energy estimate in  2 for w(, ).Recall that { q ()} q∈N  is an orthonormal basis in  2 (T  ).Then, where From ( 43) and (44), we obtain Thus, where  2 7 = max{4 2 4 , 8 For finite time  ≤ 1, we multiply the first and second equations of ( 7) by  and V, 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  =  0 , and substituting (51) and ( 55) into (50), we get Integrating (56) from 0 to  leads to If  0 ≥ 1, then it follows from (57) that thus, the Gronwall inequality implies Consequently, there exists a positive constant  8 , such that for all  ∈ [0, 1] due to the boundedness of If 0 <  0 < 1, in the same way as above, there exists a positive constant  9 , such that The proof is completed by taking Ĉ1 = max{ 7 ,  8 √ 0 ,  9 }.

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 w(, ) = [, V] is a solution of the full system (5).For  ≥ 1 ( = 1) and  ≥ 2 ( = 2, 3), there exist a  > 0 and a constant , such that Denote In order to derive the  2 estimate for the solution of (5), we first prove the following energy estimates.
Proof.We first notice that system (5) preserves the evenness of the solution; that is, if w( 1 ,  2 ,  3 , ) is a solution to (5), then w(− 1 ,  2 ,  3 , ), w( 1 , − 2 ,  3 , ), and w( 1 ,  2 , − 3 , ) 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 2T  = (−, )  .Taking the second order partial derivative of the first equation of ( 5), multiplying   , and integrating over the domain 2T  to get where are the linear and nonlinear terms, respectively, then, we have Abstract and Applied Analysis 7 Similarly, taking the second order partial derivative of the second equation of ( 5), multiplying   V, and integrating over the domain 2T  to get where for ∫ 2T   = 0. Recall the even extension of ( 5), and the solution [, V] satisfies By ( 76) and (77), we find that where  0 is a universal constant.Therefore, when  ≤ 3, 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 Similar as the proof of Lemma 1, we proceed in the two cases:  0 ≥ 1 and 0 <  0 < 1.Then, we conclude So, the even extension implies Next, we control the  2 growth of w(, ) in terms of its  2 growth.Lemma 4. Suppose that w(, ) is a solution of the full system (5), such that Then, where Ĉ3 = 2 max{4 Ĉ2 1 , 4( 0 + 1) Ĉ2 1 Ĉ2 / max , ( 0 + 1)/2}.

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Abstract and Applied Analysis
Remark 9. From a view of pattern formation, Theorem 5 shows that if the unique positive equilibrium (ũ, Ṽ) of ( 2) is linear unstable, then a general small perturbation near (ũ, Ṽ) 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 (0,   ].
Proof of Theorem 5. Define Now, we proceed in the following four steps.
From Lemma 1, for any  ≥ 0, we have By the definition of  * , for any 0 ≤  ≤ for any  ≤  * * .