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

Authors: Lina Zhang; Shengmao Fu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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]
(1)∂U∂t=∇·(D1(U,V)∇U)+f(U,V),∂V∂t=∇·(D2(U,V)∇V)+g(U,V),
in a box 𝕋N=(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:(2)ut-d1Δu=λu-u2-βuv,x∈𝕋N,t>0,vt-d2Δ[(1+d3u)v]=v(μ-vm+u),x∈𝕋N,t>0,∂u∂xi=∂v∂xi=0,xi=0,π,i=1,…,N,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈𝕋N,
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 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
(3)d2Δ[(1+d3u)v]=∇·[d2d3v∇u+d2(1+d3u)∇v].
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 d2d3∇(v∇u) and d2∇[(1+d3u)∇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
(4)u~=λ-βμm1+βμ,v~=μ(m+λ)1+βμ.
Let u^=u(x,t)-u~, 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:
(5)ut-d1Δu=g1(u+u~,v+v~),x∈𝕋N,t>0,vt-d2Δ[(1+d3(u+u~))(v+v~)]=g2(u+u~,v+v~),x∈𝕋N,t>0,∂u∂xi=∂v∂xi=0,xi=0,π,i=1,…,N,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈𝕋N,
where
(6)g1(u,v)=λu-u2-βuv,g2(u,v)=v(μ-vm+u).This paper is organized as follows. In Section2, 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
(7)ut-d1Δu=-u~u-βu~v,vt-d2d3v~Δu-d2(1+d3u~)Δv=μ2u-μv.
We use [·,·] to denote a column vector and let w(x,t)=[u(x,t),v(x,t)], q=(q1,…,qN)∈ℕN. Then, q2=∑i=1N‍qi2 are eigenvalues of -Δ on 𝕋N under the homogeneous Neumann boundary condition, and the corresponding normalized eigenfunctions are given by
(8)eq(x)={(1π)N/2,q=0,(2π)N/2∏i=1N(2π)N/2cos(qixi),q≠0.
This set of eigenfunctions forms an orthonormal basis in L2(𝕋N).We look for a normal mode to be the linear system (7) of the following form:
(9)w(x,t)=rqeλqteq(x),
where λq is a complex number and rq is a vector; they depend on q. Substituting (9) into (7), we have
(10)λqrq=(-u~-d1q2-βu~μ2-d2d3v~q2-μ-d2(1+d3u~)q2)rq:=Lrq.
System (7) possesses a nontrivial normal mode if and only if
(11)det(λq+u~+d1q2βu~-μ2+d2d3v~q2λq+μ+d2(1+d3u~)q2)=0,
which is equivalent to
(12)λq2+{u~+μ+[d1+d2(1+d3u~)]q2}λq+d1d2(1+d3u~)q4λq2+[d1μ+d2(1+d3u~)u~-βd2d3u~v~]q2+μu~+βμ2u~=0.
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
(13)d1d2(1+d3u~)q4+[d1μ+d2(1+d3u~)u~-βd2d3u~v~]q2+μu~+βμ2u~<0.
In this paper, we always assume that there exists a q, such that (13) holds. Then, the discriminant of (12) is
(14)Δ=[d1-d2(1+d3u~)]2q4+{2(u~+μ)[d1+d2(1+d3u~)]-4[d1μ+d2(1+d3u~)u~]+4βd2d3u~v~}q2+(u~-μ)2-4βμ2u~,
and the coefficient of q2 is positive, since (13) implies
(15)d1μ+d2(1+d3u~)u~-βd2d3u~v~<0.For givenq, we denote the corresponding eigenvalues by λq± and eigenvectors by r±(q). We split it into three cases for the linear analysis.(1)
Δ>0. Let Λ1={q∣Δ>0}, and let  λq± be two distinct real roots with λq+>λq-, λq± being the corresponding (linearly independent) real eigenvectors. It is easy to see that
(16)r±(q)=[1,-λq±+u~+d1q2βu~].
Denote
(17)Λ*={q∣d1d2(1+d3u~)q4+[d1μ+d2(1+d3u~)u~-βd2d3u~v~]q2+μu~+βμ2u~<0}.
Clearly, λq+>0 for q∈Λ*. Note that there are only finitely many q in Λ* and Λ*⊂Λ1. Therefore, there are only finitely many linear growing modes, such that the constant equilibrium (0,0) of (5) is unstable. Furthermore, we define
(18)λmax=maxλq+>0λq+,Λmax={q∣λq+=λmax}.
Then, Λmax⊂Λ*⊂Λ1.(2)
Δ=0. Let Λ2={q∣Δ=0}. In this case, (12) possesses repeated real eigenvalues. Consider
(19)λq=λq+=λq-=-12{u~+μ+[d1+d2(1+d3u~)]q2}<0.
The corresponding eigenvectors are
(20)r(q)=r+(q)=r-(q)=[1,-λq+u~+d1q2βu~],
and we can find another independent vector r′(q)=[0,-1/(βu~)], satisfying
(21)(L-λqI)r′(q)=r(q).(3)
Δ<0. The complex case is where (12) possesses a pair of complex eigenvalues with a negative real part. Denote Λ3={qΔ<0}, and for any q∈Λ3, denote
(22)λq+=Reλq+iImλq,r+(q)=Rer(q)+iImr(q).
Then,
(23)λq-=Reλq-iImλq,r-(q)=Rer(q)-iImr(q),
where Rer(q) and Imr(q) are linearly independent vectors.Given any initial perturbationw(x,0), we can expand it as follows:
(24)w(x,0)=∑q∈Λ1‍{wq-r-(q)+wq+r+(q)}eq(x)+∑q∈Λ2‍{wqr(q)+wq′r′(q)}eq(x)+∑q∈Λ3‍{wqReRer(q)+wqImImr(q)}eq(x):=∑q∈ℕN‍wqeq(x),
where wq-, wq+, wq, wq′, wqRe, and  wqIm are constants, and
(25)wq=wq-r-(q)+wq+r+(q),q∈Λ1,wq=wqr(q)+wq′r′(q),q∈Λ2,wq=wqReRer(q)+wqImImr(q),q∈Λ3.
The unique solution w(x,t)=[u(x,t),v(x,t)] to (7) is given by
(26)w(x,t)=∑q∈Λ1‍{wq-r-(q)eλq-t+wq+r+(q)eλq+t}eq(x)+∑q∈Λ2‍{wqr(q)+wq′r′(q)+wq′r(q)t}eλqteq(x)+∑q∈Λ3‍{wqRe(Rer(q)cos[(Imλq)t]-Imr(q)sin[(Imλq)t])+wqIm(Rer(q)sin[(Imλq)t]+Imr(q)cos[(Imλq)t])}×e(Reλq)teq(x):=eLtw(x,0).In the sequel, the constantC0 will only depend on the domain 𝕋N and the dimension N, and the generic constants C1, C2, C^1, C^2, and so forth will depend on 𝕋N, N, and the parameters λ, β, m, μ, d1, d2,  and  d3. Our main result of this section is the following lemma.Lemma 1.
Assume that the instability criterion (13) is valid. Suppose that
(27)w(x,t)=[u(x,t),v(x,t)]:=eLtw(x,0)
is a solution to the linearized system (7) with the initial condition w(x,0). Then, there exists a constant C^1≥1, such that
(28)∥w(·,t)∥L2(𝕋N)≤C^1eλmaxt∥w(·,0)∥L2(𝕋N),
for all t≥0.Proof.
We first consider the case fort>1. For any q∈Λ1,
(29)|det[r-(q),r+(q)]|=λq+-λq-βu~=Δβu~,
where Δ is given by (14). Applying  Cramer’s rule to (25), we have
(30)|wq±|=|det[r∓(q),wq]det[r-(q),r+(q)]|≤|r∓(q)|×|wq||det[r-(q),r+(q)]|=βu~|r∓(q)|×|wq|Δ,
where
(31)|r±(q)|=1+(λq±+u~+d1q2)2(βu~)2.
It follows from (14) that there exist positive constants M1 and
(32)C1=(u~+μ)[d1+d2(1+d3u~)]-2[d1μ+d2(1+d3u~)u~]+2βd2d3u~v~,
such that Δ>C1q2 for all |q|>M1. Hence, for any |q|>M1,
(33)|wq±|≤βu~|r∓(q)|×|wq|C1|q|,
and by (12),
(34)lim|q|→∞λq±q2=-d1,-d2(1+d3u~).
Thus,
(35)lim|q|→∞|r±(q)|q2=lim|q|→∞|λq±+u~+d1q2|βu~q2=1βu~(lim|q|→∞|λq±q2+d1|)≤d1+d2(1+d3u~)βu~.
Consequently, there exists a positive constant M2>M1, such that
(36)|r±(q)|≤(d1+d2(1+d3u~)βu~+1)q2
for any |q|>M2. Substituting this into (33) yields
(37)|wq±|≤C2|q||wq|
for any |q|>M2, where C2=(d1+d2(1+d3u~)+βu~)/C1. We thus obtain
(38)|wq±r±(q)eλq±t|≤C2|q||wq||r±(q)|eλq±t.
Since
(39)lim|q|→∞|q||r±(q)|eλq±t≤lim|q|→∞|q|1+(λq±+u~+d1q2)2(βu~)2×e-min{d1/2,d2(1+d3u~)/2}q2t=0,
there exists a constant M3>M2, such that
(40)|wq±r±(q)eλq±t|≤C2|wq|
for any |q|>M3.
For anyq∈Λ1 and |q|≤M3, as Δ is an increasing function of |q|2, we denote
(41)M*=min{|q|Δ(q)>0}.
With the help of (30) and (31), we get
(42)|wq±r±(q)eλq±t|≤C3|wq|eλmaxt,
where C3 only depends on λ, β, μ, m, M*, and M3. Hence, we conclude that, for any q∈Λ1, there exists a positive constant C4=max{C2,C3}, such that
(43)|wq±r±(q)eλq±t|≤C4|wq|eλmaxt.
For allq∈Λ2 and q∈Λ3, by some similar arguments as above we can show that there exist positive constants C5 and C6, such that
(44)|wqr(q)|,|wq′r′(q)|,|wq′r(q)|≤C5|wq|,teλqt≤C5,|wqReRer(q)|,|wqReImr(q)|,|wqImRer(q)|,|wqImImr(q)|≤C6|wq|.
Next, we derive the energy estimate inL2 for w(x,t). Recall that {eq(x)}q∈ℕN is an orthonormal basis in L2(𝕋N). Then,
(45)∥w(·,t)∥L2(𝕋N)2=A1+A2+A3,
where
(46)A1=∑q∈Λ1‍{wq-r-(q)eλq-t+wq+r+(q)eλq+t}2,A2=∑q∈Λ2‍{wqr(q)+wq′r′(q)+wq′r(q)t}2e2λqt,A3=∑q∈Λ3‍{wqRe(Rer(q)cos[(Imλq)t]-Imr(q)sin[(Imλq)t])+wqIm(Rer(q)sin[(Imλq)t]+Imr(q)cos[(Imλq)t])}2e2(Reλq)t.
From (43) and (44), we obtain
(47)A1≤∑q∈Λ1‍{|wq-r-(q)eλq-t|+|wq+r+(q)eλq+t|}2≤4C42∑q∈Λ1‍|wq|2e2λmaxt,A2≤∑q∈Λ2‍{|wqr(q)+wq′r′(q)|+|wq′r(q)|t}2e2λqt≤8C52∑q∈Λ2‍{|wq|2+|wq′r(q)|2t2}e2λqt≤8C52∑q∈Λ2‍{|wq|2e2λqt+C52|wq′r(q)|2}≤8C52∑q∈Λ2‍{|wq|2e2λqt+C54|wq|2}≤8C52(1+C54)∑q∈Λ2‍|wq|2e2λmaxt,A3≤∑q∈Λ3‍{(|Rer(q)|+|Imr(q)|)(|wqRe|+|wqIm|)}2e2(Reλq)t≤16C62∑q∈Λ3‍|wq|2e2(Reλq)t≤16C62∑q∈Λ3‍|wq|2e2λmaxt.
Thus,
(48)∥w(·,t)∥L2(𝕋N)2≤C72(∑q∈Λ1‍|wq|2+∑q∈Λ2‍|wq|2+∑q∈Λ3‍|wq|2)e2λmaxt=C72∥w(·,0)∥L2(𝕋N)2e2λmaxt,
where C72=max{4C42,8C52(1+C54),16C62}. Finally, for any t>1, we have
(49)∥w(·,t)∥L2(𝕋N)≤C7∥w(·,0)∥L2(𝕋N)eλmaxt.
For finite timet≤1, we multiply the first and second equations of (7) by u and Av, respectively, then add them and use the integration by parts to get
(50)12ddt∫𝕋N{|u|2+A|v|2}dx+∫𝕋N{d1|∇u|2+Ad2(1+d3u~)|∇v|2+Ad2d3v~∇u∇v|∇u|2}dx+∫𝕋N{u~u2+Aμv2}dx=∫𝕋N(Aμ2-βu~)uvdx.
Firstly, we claim that the integrand of the second integral in (50) satisfies
(51)d1|∇u|2+Ad2(1+d3u~)|∇v|2+Ad2d3v~∇u∇v≥12(d1|∇u|2+Ad2(1+d3u~)|∇v|2)
for some positive constant A. Obviously, it suffices to require that
(52)(Ad2d3v~)2-Ad1d2(1+d3u~)<0.
This is equivalent to
(53)A<d1(1+d3u~)d2d32v~2.
Denote
(54)A0=d1(1+d3u~)2d2d32v~2.
On the other hand, the term on the right of (50) is
(55)∫𝕋N‍(Aμ2-βu~)uvdx≤∫𝕋N‍(Aμ2+βu~)|u|·|v|dx≤∫𝕋N‍(Aμ2+βu~)(u2+v22)dx=Aμ2+βu~2∫𝕋N‍(u2+v2)dx.
Taking A=A0, and substituting (51) and (55) into (50), we get
(56)ddt∫𝕋N‍{|u|2+A0|v|2}dx≤(A0μ2+βu~)∫𝕋N‍(u2+v2)dx.
Integrating (56) from 0 to t leads to
(57)∫𝕋N‍{|u(x,t)|2+A0|v(x,t)|2}dx≤∫𝕋N‍{|u(x,0)|2+A0|v(x,0)|2}dx+(A0μ2+βu~)∫0t‍∫𝕋N‍(u2+v2)dxdt.
If A0≥1, then it follows from (57) that
(58)∫𝕋N‍{|u(x,t)|2+|v(x,t)|2}dx≤A0∫𝕋N‍{|u(x,0)|2+|v(x,0)|2}dx+(A0μ2+βu~)∫0t‍∫𝕋N‍(u2+v2)dxdt;
thus, the Gronwall inequality implies
(59)∥w(·,t)∥L2(𝕋N)≤A0∥w(·,0)∥L2(𝕋N)·e((A0μ2+βu~)/2)t.
Consequently, there exists a positive constant C8, such that
(60)∥w(·,t)∥L2(𝕋N)≤C8A0∥w(·,0)∥L2(𝕋N)eλmaxt
for all t∈[0,1] due to the boundedness of
(61)e((A0μ2+βu~)/2-λmax)t.
If0<A0<1, in the same way as above, there exists a positive constant C9, such that
(62)∥w(·,t)∥L2(𝕋N)≤C9∥w(·,0)∥L2(𝕋N)eλmaxt.
The proof is completed by taking C^1=max{C7,C8A0,C9}.


## 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 thatw(x,t)=[u,v] is a solution of the full system (5). For s≥1(N=1) and s≥2(N=2,3), there exist a T>0 and a constant C, such that
(63)∥w(·,t)∥Hs(𝕋N)≤C∥w(·,0)∥Hs(𝕋N),0≤t<T
if u0(x),v0(x)∈Hs(𝕋N).Denote(64)∂iju:=∂2u∂xi∂xj,∂iu:=∂u∂xi,i,j=1,…,N.
In order to derive the H2 estimate for the solution of (5), we first prove the following energy estimates.Lemma 3.
Suppose thatw(x,t)=[u,v] is a solution of the full system (5). Then,
(65)12ddt∑i,j=1N‍∫𝕋N‍{|∂iju|2+|∂ijv|2}dx+∑i,j=1N‍∫𝕋N‍{d14|∇∂iju|2dx+A02d2(1+d3u~)|∇∂ijv|2}dx+u~∑i,j=1N‍∫𝕋N‍|∂iju|2dx+A0μ2∑i,j=1N‍∫𝕋N‍|∂ijv|2dx≤C^2(1+η)2∥w(·,t)∥H2(𝕋N)∥∇3w(·,t)∥L2(𝕋N)2+C^2∥u(·,t)∥L2(𝕋N)2
for ∥w∥H2(𝕋N)≤η.Proof.
We first notice that system (5) preserves the evenness of the solution; that is, if w(x1,x2,x3,t) is a solution to (5), then w(-x1,x2,x3,t), w(x1,-x2,x3,t), and w(x1,x2,-x3,t) 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 2𝕋N=(-π,π)N. Taking the second order partial derivative of the first equation of (5), multiplying ∂iju, and integrating over the domain 2𝕋N to get
(66)12ddt∫2𝕋N‍|∂iju|2dx=∫2𝕋N‍∂ij{d1Δu+g1(u+u~,v+v~)}∂ijudx=∫2𝕋N‍∂ij{L1(u,v)+N1(u,v)}∂ijudx,
where
(67)L1(u,v)=d1Δu-u~u-βu~v,(68)N1(u,v)=g1(u+u~,v+v~)+u~u+βu~v=-u2-βuv
are the linear and nonlinear terms, respectively, then, we have
(69)∫2𝕋N‍(∂ijL1(u,v))∂ijudx=-d1∫2𝕋N‍|∇∂iju|2dx-u~∫2𝕋N‍|∂iju|2dx-βu~∫2𝕋N‍∂iju∂ijvdx,∫2𝕋N‍(∂ijN1(u,v))∂ijudx=-∫2𝕋N‍∂ij{u2+βuv}∂ijudx,=-∫2𝕋N‍{2∂iu∂ju+2u∂iju+βv∂iju+β∂iu∂jv+β∂ju∂iv+βu∂ijv}∂ijudx=-∫2𝕋N‍{(2u+βv)∂iju+βu∂ijv+(2∂iu∂ju+β∂iu∂jv+β∂ju∂iv)}∂ijudx≤(2+2β)∥w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)2+(2+2β)∥∇w∥L∞(2𝕋N)∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N).
Similarly, taking the second order partial derivative of the second equation of (5), multiplying ∂ijv, and integrating over the domain 2𝕋N to get
(70)12ddt∫2𝕋N‍|∂ijv|2dx=∫2𝕋N‍∂ij{d2Δ[(1+d3(u+u~))(v+v~)]+g2(u+u~,v+v~)}∂ijvdx=∫2𝕋N‍∂ij{L2(u,v)+N2(u,v)}∂ijvdx,
where
(71)L2(u,v)=d2d3v~Δu+d2(1+d3u~)Δv+μ2u-μv,(72)N2(u,v)=d2Δ[(1+d3(u+u~))(v+v~)]+g2(u+u~,v+v~)-L2(u,v)=d2d3(vΔu+2∇u∇v+uΔv)-(μu-v)2m+u+u~:=N2(1)(u,v)+N2(2)(u,v),
thus,
(73)∫2𝕋N‍(∂ijL2(u,v))∂ijvdx=-d2d3v~∫2𝕋N‍∇∂iju∇∂ijvdx-d2(1+d3u~)∫2𝕋N‍|∇∂ijv|2dx+μ2∫2𝕋N‍∂iju∂ijvdx-μ∫2𝕋N‍|∂ijv|2dx,∫2𝕋N‍(∂ijN2(1)(u,v))∂ijvdx=d2d3∫2𝕋N‍∂ij(vΔu+2∇u∇v+uΔv)∂ijvdx=d2d3∫2𝕋N‍{(∂ijv)Δu+v(Δ∂iju)+(∂iv)(Δ∂ju)+(∂jv)(Δ∂iu)}∂ijvdx+2d2d3∫2𝕋N‍{(∇∂iju)∇v+∇u(∇∂ijv)+(∇∂iu)(∇∂jv)+(∇∂ju)(∇∂iv)}∂ijvdx+d2d3∫2𝕋N‍{(∂iju)Δv+u(Δ∂ijv)+(∂iu)(Δ∂jv)+(∂ju)(Δ∂iv)}∂ijvdx=-d2d3∫2𝕋N‍(∂ijv)(∇∂ijv)∇udx-d2d3∫2𝕋N‍∇v(∂iju)(∇∂ijv)dx-d2d3∫2𝕋N‍v(∇∂ijv)(∇∂iju)dx+d2d3×∫2𝕋N‍{(∂iv)(Δ∂ju)+(∂jv)(Δ∂iu)}∂ijvdx-2d2d3∫2𝕋N‍{(Δ∂iu)∂ijv∂jv+(∇∂iu)(∇∂ijv)∂jv+(Δ∂ju)∂ijv∂iv+(∇∂ju)(∇∂ijv)∂iv}dx-d2d3∫2𝕋N‍u|∇∂ijv|2dx+d2d3×∫2𝕋N‍{(∂iu)(Δ∂jv)+(∂ju)(Δ∂iv)}∂ijvdx≤14d2d3∥∇w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)∥∇3w∥L2(2𝕋N)+2d2d3∥w∥L∞(2𝕋N)∥∇3w∥L2(2𝕋N)2,∫2𝕋N‍(∂ijN2(2)(u,v))∂ijvdx=-∫2𝕋N‍∂ij((μu-v)2m+u+u~)∂ijvdx=∫2𝕋N‍{-(2(μu-v)(μ∂iju-∂ijv)[2∂iu∂ju(m+u+u~)3-∂iju(m+u+u~)2]+2(μ∂iu-∂iv)(μ∂ju-∂jv))×(m+u+u~)-1-(μu-v)2[2∂iu∂ju(m+u+u~)3-∂iju(m+u+u~)2]+(2(μu-v)[(μ∂iu-∂iv)∂ju+(μ∂ju-∂jv)∂iu])×(m+u+u~)-2[2∂iu∂ju(m+u+u~)3-∂iju(m+u+u~)2]}∂ijvdx≤2(μ+1)2m∥w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)2+2(μ+1)2m∥∇w∥L∞(2𝕋N)∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N)+2(μ+1)2m3∥w∥L∞(2𝕋N)2∥∇w∥L∞(2𝕋N)×∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N)+(μ+1)2m2∥w∥L∞(2𝕋N)2∥∂ijw∥L2(2𝕋N)2+4(μ+1)2m2∥w∥L∞(2𝕋N)∥∇w∥L∞(2𝕋N)×∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N)≤(μ+1)2m(2+∥w∥L∞(2𝕋N)m)∥w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)2+2(μ+1)2m(1+∥w∥L∞(2𝕋N)m)2∥∇w∥L∞(2𝕋N)×∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N).
Substituting (70) ×A0+ (66) into (69) and (73) to get
(74)12ddt∫2𝕋N‍{|∂iju|2+A0|∂ijv|2}dx+∫2𝕋N‍{d1|∇∂iju|2+A0d2d3v~∇∂iju∇∂ijv+A0d2(1+d3u~)|∇∂ijv|2}dx+u~∫2𝕋N‍|∂iju|2dx+A0μ∫2𝕋N‍|∂ijv|2dx≤I1+I2,
where
(75)I1=C10(1+η)∥w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)2+C11(1+η)2∥∇w∥L∞(2𝕋N)∥∇w∥L2(2𝕋N)∥∂ijw∥L2(2𝕋N)+C12∥∇w∥L∞(2𝕋N)∥∂ijw∥L2(2𝕋N)∥∇3w∥L2(2𝕋N)+C13∥w∥L∞(2𝕋N)∥∇3w∥L2(2𝕋N)2,I2=(A0μ2-βu~)∫2𝕋N‍∂iju∂ijvdx,
we apply the Sobolev imbedding to control the L∞ norm by
(76)∥g∥L∞(2𝕋N)≤C0∥g∥H2(2𝕋N)
for N≤3. From the Hölder inequality, the Poincaré inequality, and the Sobolev imbedding theorem, it follows that
(77)∥g∥L2(2𝕋N)≤(2π)N/4∥g∥L4(2𝕋N)≤C0∥∇g∥L2(2𝕋N)
for ∫2𝕋N‍gdx=0. Recall the even extension of (5), and the solution [u,v] satisfies
(78)∫2𝕋N‍∇udx=∫2𝕋N‍∇vdx=0,∫2𝕋N‍∂ijudx=∫2𝕋N‍∂ijvdx=0,i,j=1,…,N.
By (76) and (77), we find that
(79)∥∇w∥L∞(2𝕋N)≤C0∥∇w∥H2(2𝕋N)≤C0∥∇3w∥L2(2𝕋N),
where C0 is a universal constant. Therefore, when N≤3, it follows from (76) and (79) that
(80)I1≤C14(1+η)2∥w∥H2(2𝕋N)∥∇3w∥L2(2𝕋N)2.
Applying the Young inequality to get
(81)I2≤(A0μ2+βu~)22A0μ∫2𝕋N‍|∂iju|2dx+A0μ2∫2𝕋N‍|∂ijv|2dx,
which is combined with the interpolation inequality and the ε-Young inequality to imply
(82)∫2𝕋N‍|∂iju|2dx≤C0(a∥∇∂iju∥L2(2𝕋N)2)+14a2∥u∥L2(2𝕋N)2,
where a is a positive constant, in the same way as above, we obtain that the second integral satisfies
(83)∫2𝕋N‍{d1|∇∂iju|2+A0d2d3v~∇∂iju∇∂ijv+A0d2(1+d3u~)|∇∂ijv|2}dx≥{d12|∇∂iju|2+A0d2(1+d3u~)2|∇∂ijv|2}dx.
Substituting (80)–(83) into (74), take ((A0μ2+βu~)2/2A0μ)C0a=d1/4 to get
(84)12ddt∫2𝕋N‍{|∂iju|2+A0|∂ijv|2}dx+∫2𝕋N‍{d14|∇∂iju|2+A02d2(1+d3u~)|∇∂ijv|2}dx+u~∫2𝕋N‍|∂iju|2dx+A0μ2∫2𝕋N‍|∂ijv|2dx≤C14(1+η)2∥w∥H2(2𝕋N)∥∇3w∥L2(2𝕋N)2+C15∥u∥L2(2𝕋N)2.
Similar as the proof of Lemma1, we proceed in the two cases: A0≥1 and 0<A0<1. Then, we conclude
(85)12ddt∫2𝕋N‍{|∂iju|2+|∂ijv|2}dx+∫2𝕋N‍{d14|∇∂iju|2+A02d2(1+d3u~)|∇∂ijv|2}dx+u~∫2𝕋N‍|∂iju|2dx+A0μ2∫2𝕋N‍|∂ijv|2dx≤C16(1+η)2∥w∥H2(2𝕋N)∥∇3w∥L2(2𝕋N)2+C17∥u∥L2(2𝕋N)2.
So, the even extension implies
(86)12ddt∑i,j=1N‍∫𝕋N‍{|∂iju|2+|∂ijv|2}dx+∑i,j=1N‍∫𝕋N‍{d14|∇∂iju|2+A02d2(1+d3u~)|∇∂ijv|2}dx+u~∑i,j=1N‍∫𝕋N‍|∂iju|2dx+A0μ2∑i,j=1N‍∫𝕋N‍|∂ijv|2dx≤C^2(1+η)2∥w∥H2(𝕋N)∥∇3w∥L2(𝕋N)2+C^2∥u∥L2(𝕋N)2.Next, we control theH2 growth of w(x,t) in terms of its L2 growth.Lemma 4.
Suppose thatw(x,t) is a solution of the full system (5), such that
(87)∥w(·,t)∥H2(𝕋N)≤min{η,d14C^2(1+η)2,A0d2(1+d3u~)2C^2(1+η)2},(88)∥w(·,t)∥L2(𝕋N)≤2C^1eλmaxt∥w(·,0)∥L2(𝕋N).
Then,
(89)∥w(·,t)∥H2(𝕋N)2≤C^3{∥w(·,0)∥H2(𝕋N)2+e2λmaxt∥w(·,0)∥L2(𝕋N)2},
where C^3=2max{4C^12,4(C0+1)C^12C^2/λmax,(C0+1)/2}.Proof.
We first consider the second-order derivatives ofw(x,t). From Lemmas 3 and 4 and our assumption for w(x,t), we have
(90)12ddt∑i,j=1N‍∫𝕋N‍{|∂iju|2+|∂ijv|2}dx≤C^2∥u∥L2(𝕋N)2≤4C^12C^2e2λmaxt∥w(·,0)∥L2(𝕋N)2.
By an integration from 0 to t, we deduce that
(91)∑i,j=1N‍∫𝕋N‍{|∂iju(x,t)|2+|∂ijv(x,t)|2}dx≤∑i,j=1N‍∫𝕋N‍{|∂iju(x,0)|2+|∂ijv(x,0)|2}dx+4C^12C^2λmaxe2λmaxt∥w(·,0)∥L2(𝕋N)2.
For the first derivations of w(x,t), we apply the Poincaré inequality to get
(92)∥∇w∥L2(2𝕋N)≤C0∥∇2w∥L2(2𝕋N).
Applying the even extension, we have
(93)∥∇w∥L2(𝕋N)≤C0∥∇2w∥L2(𝕋N).
It follows from (88)–(93) that
(94)∥w(·,t)∥H2(𝕋N)2=∥w(·,t)∥L2(𝕋N)2+∥∇w∥L2(𝕋N)2+∥∇2w∥L2(𝕋N)2≤∥w(·,t)∥L2(𝕋N)2+(C0+1)∥∇2w∥L2(𝕋N)2≤4C^12e2λmaxt∥w(·,0)∥L2(𝕋N)2+(C0+1)∥w(·,0)∥H2(𝕋N)2+4(C0+1)C^12C^2λmaxe2λmaxt∥w(·,0)∥L2(𝕋N)2≤C^3{∥w(·,0)∥H2(𝕋N)2+e2λmaxt∥w(·,0)∥L2(𝕋N)2},
where C^3=2max{4C^12,4(C0+1)C^12C^2/λmax,(C0+1)/2}.


## 4. Nonlinear Instability and Pattern Formation

Letθ be a fixed constant. For an arbitrary positive number δ, we define the escape time Tδ by
(95)θ=δeλmaxTδ
or equivalently
(96)Tδ=1λmaxlnθδ.Theorem 5.
Assume that there exists aq, such that the instability criterion (13) holds. Let
(97)w0(x)=∑q∈Λ1‍{wq-r-(q)+wq+r+(q)}eq(x)+∑q∈Λ2‍{wqr(q)+wq′r′(q)}eq(x)+∑q∈Λ3‍{wqReRer(q)+wqImImr(q)}eq(x),
such that w0(x)∈H2(𝕋N), ∥w0∥L2(𝕋N)=1. Then, there exist θ0>0, δ0>0, and C>0, such that for all θ∈(0,θ0) and δ∈(0,δ0], the nonlinear evolution wδ(x,t) of (5) with the initial perturbation wδ(x,0)=δw0(x) satisfies
(98)∥wδ(·,t)-δeλmaxt∑q∈Λmax‍wq+r+(q)eq(x)∥L2(𝕋N)≤C{e-σt+δ∥w0∥H2(𝕋N)2+δeλmaxt}δeλmaxt,∀t∈[0,Tδ],
where the constant σ, called the gap between λmax and the rest of eigenvalues, is positive, and
(99)δ0=min{λmax4C^3C^4∥w0∥H2(TN)212C^3∥w0∥H2(TN),min{η,d14C^2(1+η)2,A0d2(1+d3u~)2C^1(1+η)2},λmax4C^3C^4∥w0∥H2(TN)2},θ0=min{λmax4C^3C^4,η2C^3,d18C^2C^3(1+η)2,A0d2(1+d3u~)4C^2C^3(1+η)2}.Remark 6.
First, we notice that the part of the fastest growing modes(100)δeλmaxt∑q∈Λmax‍wq+r+(q)eq(x)
in (98) is nontrivial. Recall that the initial profile w0(x) is any H2 function satisfying ∥w0∥L2(𝕋N)=1. Thus, we can choose w0(x), such that there is at least one q0∈Λmax with wq0+≠0. Consequently,
(101)∥δeλmaxt∑q∈Λmax‍wq+r+(q)eq(x)∥L2(𝕋N)≥δeλmaxt|wq0+||r+(q0)|>0.Remark 7.
Fixθ to be a positive small number. If δ is small sufficiently, then Tδ is large, and for 0≤t≤Tδ, δeλmaxt≤θ. Our estimate (98) implies that the dynamics of a general perturbation can be characterized by such linear dynamics over a long time period (0,Tδ], when the initial perturbation is small.Remark 8.
In particular, if we take(102)w0(x)=r+(q0)|r+(q0)|eq0(x)
in Remark 6, then, at the time t=Tδ, the estimate (98) gives
(103)∥wδ(·,Tδ)-δeλmaxTδr+(q0)|r+(q0)|eq0(·)∥L2(𝕋N)≤C{e-σTδ+δ∥w0∥H2(𝕋N)2+θ}θ,=C{θ1-σ/λmaxδσ/λmax+δθ∥w0∥H2(𝕋N)2+θ2}.
Therefore,
(104)∥wδ(·,Tδ)∥≥θ-C{∥w0∥H2(𝕋N)2+θ2θ1-σ/λmaxδσ/λmax+δθ∥w0∥H2(𝕋N)2+θ2}.
For θ sufficiently small, ∥wδ(.,Tδ)∥≥θ/2 as δ→0, which implies that nonlinear instability occurs.Remark 9.
From a view of pattern formation, Theorem5 shows that if the unique positive equilibrium (u~,v~) of (2) is linear unstable, then a general small perturbation near (u~,v~) 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,Tδ].Proof of Theorem5.
Define(105)T*=sup{t∣∥wδ(·,t)-δeLtw0∥L2(𝕋N)≤C^12δeλmaxt},(106)T**=sup{A0d2(1+d3u~)2C^2(1+η)2t∣∥wδ(·,t)∥H2(𝕋N)≤min{η,d14C^2(1+η)2,A0d2(1+d3u~)2C^2(1+η)2}}.
Now, we proceed in the following four steps.
Step  1. We establish H2 estimate for the solution wδ(x,t) of 0≤t≤min{Tδ,T*,T**}.
From Lemma1, for any t≥0, we have
(107)∥δeLtw0∥L2(𝕋N)≤δC^1eλmaxt.
By the definition of T*, for any 0≤t≤T*, it follows that
(108)∥wδ(·,t)∥L2(𝕋N)-∥δeLtw0∥L2(𝕋N)≤C^12δeλmaxt.
Substituting (107) into (108), we obtain
(109)∥wδ(·,t)∥L2(𝕋N)≤3C^12δeλmaxt.
Furthermore, it follows from (109) and Lemma 4 that
(110)∥wδ(·,t)∥H2(𝕋N)≤C^3·δ2∥w0∥H2(𝕋N)2+δ2e2λmaxt(111)≤C^3·(δ∥w0∥H2(𝕋N)+δeλmaxt)
for any t≤T**.
Step  2. We establish L2 estimate for wδ(x,t) of 0≤t≤min{Tδ,T*,T**}. Applying Duhamel’s principle, we have
(112)wδ(x,t)=δeLtw0-∫0t‍eL(t-τ)N(wδ(x,τ))dτ,
where N[wδ(x,τ)]=(N1(wδ(x,τ)),N2(wδ(x,τ))), N1(wδ(x,τ)), and N2(wδ(x,τ)) are given by (68) and (72). Using Lemma 1, it follows that
(113)∥wδ(·,t)-δeLtw0∥L2(𝕋N)=∥∫0t‍eL(t-τ)N(wδ(x,τ))dτ∥L2(𝕋N)≤∫0t‍∥eL(t-τ)N(wδ(x,τ))∥L2(𝕋N)dτ≤C^1∫0t‍eλmax(t-τ)∥N(wδ(x,τ))∥L2(𝕋N)dτ,∥N(wδ(x,τ))∥L2(𝕋N)≤∥N1(wδ(x,τ))∥L2(𝕋N)+∥N2(wδ(x,τ))∥L2(𝕋N)=∥-(uδ)2-βuδvδ∥L2(𝕋N)+∥(μuδ-vδ)2m+uδ+u~d2d3vδΔuδ+2d2d3∇uδ∇vδ+d2d3uδΔvδ-(μuδ-vδ)2m+uδ+u~∥L2(𝕋N)≤(β+1)∥wδ∥L∞(𝕋N)∥wδ∥L2(𝕋N)+2d2d3∥wδ∥L∞(𝕋N)×∥wδ∥H2(𝕋N)+2d2d3∥∇wδ∥L4(𝕋N)2+(μ+1)2m∥wδ∥L∞(𝕋N)∥wδ∥L2(𝕋N)≤C^4∥wδ∥H2(𝕋N)2,
where C^4=β+1+4d2d3+(μ+1)2/m. Therefore,
(114)∥wδ(·,t)-δeLtw0∥L2(𝕋N)≤C^1C^4∫0t‍eλmax(t-τ)∥wδ(τ)∥H2(𝕋N)2dτ.
Substituting (110) into (114), we obtain
(115)∥wδ(·,t)-δeLtw0∥L2(𝕋N)≤C^1C^3C^4∫0t‍eλmax(t-τ)×(δ2∥w0∥H2(𝕋N)2+δ2e2λmaxτ)dτ≤C^1C^3C^4{δ∥w0∥H2(𝕋N)2λmax+δeλmaxtλmax}δeλmaxt.
Step  3. We prove by contradiction that
(116)Tδ=min{Tδ,T*,T**}
for δ sufficiently small and
(117)θ<min{λmax4C^3C^4,η2C^3,d18C^2C^3(1+η)2,A0d2(1+d3u~)4C^2C^3(1+η)2}.
IfT**=min{Tδ,T*,T**}, we can let t=T**≤Tδ in (111) to obtain
(118)∥wδ(·,T**)∥H2(𝕋N)≤C^3(δ∥w0∥H2(𝕋N)+δeλmaxT**)≤C^3(δ∥w0∥H2(𝕋N)+θ).
By choosing δ sufficiently small, such that
(119)C^3δ∥w0∥H2(𝕋N)≤12min{η,d14C^2(1+η)2,A0d2(1+d3u~)2C^2(1+η)2},
and the choice of θ in (117), we have
(120)∥wδ(·,T**)∥H2(𝕋N)<min{η,d14C^2(1+η)2,A0d2(1+d3u~)2C^2(1+η)2}.
This is a contradiction to the definition of T**.
On the other hand, ifT*=min{Tδ,T*,T**}, we can let t=T* in (115) to get
(121)∥wδ(·,T*)-δeLT*w0∥L2(𝕋N)≤C^1C^3C^4{δ∥w0∥H2(𝕋N)2λmax+θλmax}δeλmaxT*.
By choosing δ sufficiently small, such that
(122)C^3C^4δ∥w0∥H2(𝕋N)2λmax≤14,
and the choice of θ in (117), we have
(123)∥wδ(·,T*)-δeLT*w0∥L2(𝕋N)<C^12δeλmaxT*.
This again contradicts the definition of T*. From these arguments, (116) holds.
Step  4. Rewriting the left-hand term in (115) as the form of (26), and separating q∈Λmax and moving q∉Λmax to the right-hand side, it follows that
(124)∥wδ(·,t)-δeλmaxt∑q∈Λmaxwq+r+(q)eq(x)∥L2(𝕋N)≤δ∥∑q∈Λmaxwq-r-(q)eλq-teq(x)∥L2(𝕋N)+δ∥∑q∈Λ1∖Λmax{wq-r-(q)eλq-t+wq+r+(q)eλq+t}eq(x)∑q∈Λ1/Λmax∥L2(𝕋N)+δ∥∑q∈Λ2{wqr(q)+wq′r′(q)+wq′r(q)t}eλqteq(x)∑q∈Λ2∥L2(𝕋N)+δ∥∑q∈Λ3{wqRe(Rer(q)cos[(Imλq)t]-Imr(q)sin[(Imλq)t])+wqIm(Rer(q)sin[(Imλq)t]+Imr(q)cos[(Imλq)t])}×e(Reλq)teq(x)∑q∈Λ3∥L2(𝕋N)+C^1C^3C^4{δ∥w0∥H2(𝕋N)2λmax+δeλmaxtλmax}δeλmaxt.
Next, we process the first term on the right side of (124) to get
(125)δ∥∑q∈Λmaxwq-r-(q)eλq-teq(x)∥L2(𝕋N)=δ(∑q∈Λmax(wq-)2|r-(q)|2e2λq-tdx)1/2.
Recall that there are only finite elements in Λmax and for any q∈Λmax, there exists a constant M0, such that 0<|q|<M0. Therefore,
(126)δ∥∑q∈Λmaxwq-r-(q)eλq-teq(x)∥L2(𝕋N)≤Cδe(λmax-σ)t.
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.



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*Source: 854862-2013-11-07.xml*