Abstract and Applied Analysis

Volume 2014 (2014), Article ID 248657, 11 pages

http://dx.doi.org/10.1155/2014/248657

Research Article

Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

Haiyan Gao1,2 and Shengmao Fu³

¹College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

²School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China

³School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China

Received 2 April 2014; Accepted 8 June 2014; Published 3 July 2014

Academic Editor: Yongli Song

Copyright © 2014 Haiyan Gao 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.

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a -dimensional box . It is proved that given any general perturbation of magnitude , its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order . Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

1. Introduction

An important variant of the chemotaxis model was initially proposed by Painter and Hillen in [1] to model the volume-filling effect. In the volume-filling effect, it is assumed that cells have a certain finite (nonzero) volume and that the occupation of an area limits other cells from penetrating it. A simple version of the volume-filling chemotaxis model (VF) is the following:

where , , , , and are given positive constants. is the cell density and denotes the density of the external chemical substance which is secreted by the cells themselves. and denote the cell and chemical diffusion coefficients, respectively. is called chemosensitivity. The term denotes the chemotactic flux under a volume constraint 1 (called crowding capacity), meaning that the chemotactic movement will be inhibited at the aggregation location where the cell density reaches 1.

In recent years, the chemotaxis models with volume-filling effect have been studied extensively. Hillen and Painter [2] firstly proved the global existence of solutions. Numerical simulations in one and two dimensions show interesting phenomena of pattern formation and formation of stable aggregates. Wrzosek [3] showed the existence of a compact global attractor in the space , , for some cases. In [4], the structure of the attractor can be understood using Lyapunov functions. Stationary solutions which are inhomogeneous in space were investigated for a given range of parameters. In [1], a numerical exploration was conducted to determine the longtime patterning behaviour, revealing formation of multiple plateau type patterns which undergo a coarsening process with increasingly long transient times. Potapov and Hillen [5] investigated the metastability of steady states. The underlying bifurcation diagram was identified, revealing that the unstable eigenvalues are exponentially small. The plateau interactions were studied using asymptotic methods. In [6], it was obtained that a priori estimates for the classical chemotaxis model of Patlak, Keller, and Segel when a nonlinear diffusion or a nonlinear chemosensitivity was considered accounting for the finite size of the cells and how entropy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solutions were showed. Burger et al. [7] discussed the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller-Segel model of chemotaxis with volume-filling effect. Moreover, the global existence of solutions and nontrivial steady states were also studied. Wang and Hillen [8] established the global existence of classical solutions to a generalized chemotaxis model, which includes the volume-filling effect expressed through a nonlinear squeezing probability. Necessary and sufficient conditions for spatial pattern formation were given and the underlying bifurcations were analyzed. In [9], the stationary solutions of the volume-filling chemotaxis model without a growth term were obtained by Jiang and Zhang.

Moreover, Wrzosek [10] considered various assumptions on nonlinear diffusion and chemotactic sensitivity function which lead to the existence of global in time solutions, thus preventing blow-up. In [11], Winkler proved that if certain conditions were fulfilled, then there were solutions that blow up in either finite or infinite time. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen [1], his results indicated how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Winkler and Djie [12] discussed boundedness and finite-time collapse for a chemotaxis system with volume-filling effect. Wang et al. [13] proved that for a wide range of nonlinear diffusion operators, including singular and degenerate ones, if the taxis force was strong enough with respect to diffusion and the initial data were chosen properly, then there was a classical solution which reaches the threshold at the maximal time of its existence; no matter whether the latter was finite or infinite. Zhang and Zheng [14] obtained the crucial uniform boundedness of the solution for a quasilinear nonuniform parabolic system modelling chemotaxis with volume-filling effect and the results on convergence to equilibrium and the decay rate using a suitable nonsmooth Simon-Łojasiewicz approach. In [15], the uniform boundedness, global in time existence and uniqueness of classical solution, were proved. With the help of a suitable nonsmooth Simon-Łojasiewicz approach, the results on convergence of the solution to equilibrium and the convergence rate were obtained. Li and Zhang [16] classified the existence or nonexistence of steady state solutions of a 1-D chemotaxis model with volume-filling effect. Their results provided insights on how the biological parameters affect pattern formation.

The chemotaxis models with logistic growth but without a volume-filling effect were studied (see [17–22]). The global attractor and traveling wave solutions of a volume-filling chemotaxis model with logistic growth were obtained in [3] and [23], respectively. Ma et al. [24] studied the existence of stationary solutions of a volume-filling chemotaxis model with logistic cell growth. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, the stability criteria were established and a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue was found. Quite recently, Ma et al. in [25] studied the nonexistence of nonconstant steady state (i.e., stationary pattern) for a chemotaxis model with the volume-filling effect and logistic cell growth and established the critical value of the chemotactic coefficient between the existence and the nonexistence of stationary pattern.

Guo and Hwang [26] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Their results can be interpreted as a rigorous mathematical characterization for the early-stage pattern formation in the Keller-Segel model. Very recently, Fu and Liu in [22] and [27] studied instability in the Keller-Segel model with a logistic source and cubic source term, respectively. Their results indicated that chemotaxis-driven nonlinear instability occurs in these models.

In this paper, we mainly consider the nonlinear instability for the following chemotaxis model:

which is subject to the Neumann boundary conditions

and the nonnegative initial data

where is a -dimensional box. The term describes the logistic growth of cells with growth rate and carrying capacity fulfilling . Our main result (see Theorem 6) indicates that the nonlinear dynamics near an unstable constant equilibrium points in the classical Keller-Segel model, the Keller-Segel model with a logistic source and cubic source term, respectively, and a volume-filling chemotaxis model with logistic source term are almost similar.

The organization of this paper is as follows: in Section 2, we show that the unique positive equilibrium point of (2) without chemotaxis is globally asymptotically stable and cross diffusion cannot induce the instability of the positive equilibrium. In Section 3, we consider the growing modes of (2). In Section 4, we present and prove the Bootstrap lemma, which was first introduced in [28]. In Section 5, for any given general perturbation of magnitude , we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of the order . Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

2. Stability of Positive Equilibrium Point of (2) without Chemotaxis

We first discuss the following corresponding kinetic equations of (2):

We use to denote a column vector. Evidently, (5) has the unique positive equilibrium point . For simplicity, we denote , and a direct calculation yields

The characteristic polynomial of is ; and are the two roots of . Hence is locally stable. Define , where . By the Lyapunov-LaSalle invariance principle [29], is globally asymptotically stable.

We now consider system (2) without chemotaxis in the following form:

Let be the eigenvalues of the operator on with the homogeneous Neumann boundary condition and the eigenspace corresponding to in . Let , be an orthonormal basis of , and . Then , .

Let and . The linearization of (7) at can be expressed by . For each , is invariant under the operator , and is an eigenvalue of   on if and only if it is an eigenvalue of the matrix

Thus, has two negative eigenvalues and . It follows from Theorem in [30] that is locally asymptotically stable.

Let be the unique nonnegative global solution. The maximum principle gives

Moreover, by the strong maximum principle [31], we know that if , , then , on for all .

We define the Lyapunov function

where . Then for all . Applying (7) and integration by parts, we have

By (9), (11), the basic estimates for parabolic equations [31], and Lemma 2.1 in [22] (which is given in [32] in Chinese), we can conclude that

The global asymptotic stability of follows from (12) together with the local stability of .

Next, we consider that the cross diffusion model

Let . Then the linearizing system (13) at can be written as

where

Then, for each , is invariant under the operator , and is an eigenvalue of on if and only if is an eigenvalue of the matrix

Notice that

Thus, the two eigenvalues and of have negative real parts.

From this, we see that adding the cross diffusion to the system (7), the positive constant solution is also locally stable, which means that Turing instability does not occur. The information above indicates that the aggregation of individuals does not occur in the absence of chemotactic effect. It is the purpose of the present paper to clarify the effect of the chemotaxis and nonlinear patterns created by chemotaxis for a volume-filling chemotaxis model with logistic growth.

3. Growing Modes in the System (2)

The nonlinear evolution of a perturbation , around satisfies

The corresponding linearized system is as follows:

Let , , and . Then forms a basis of the space of functions in that satisfies Neumann boundary conditions. We try to find a normal mode to the linear system (19) of the following form:

where is a vector depending on . Substituting (20) into (19) yields

where . We can obtain the following dispersion formula for :

Thus, we deduce the following linear instability criterion by requiring that there exist a such that

to ensure that (22) has at least one positive root . This means that

There exist two distinct real roots:

for all to the quadratic equation (22), which are denoted by . We denote the corresponding (linearly independent) eigenvectors by and , such that

Clearly, for large,

Thus, there are only finitely many such that . We denote the largest eigenvalue by and define . From (25) we can regard as a function of . Therefore, there is one (possibly two) having . We also denote to be the gap between the and the rest; that is, .

Given any initial perturbation , we can expand it as

In the sequel, denote by and the inner product of and the scalar product of , respectively. For any , we denote . Throughout this paper, we always denote universal constants by .

Clearly,

The unique solution of (19) is given by

Our main result in this section is the following lemma.

Lemma 1. Suppose that the instability criterion (23) holds. Let be a solution to the linearized system (19) with initial condition . Then there exists a constant depending on , , , , , , and , such that

Proof. We divide the proof into the following two cases.

(1). It follows from (25) that

respectively. Thus, and there exists a positive constant for all , such that

By the quadratic formula of (22), one can obtain

It follows from (29) that

From (26) and (34), we can conclude that there exists a positive constant such that

for all , where . By (34), one can deduce

Combining (37) and (38), we find that

where . For and large, it is not difficult to verify by (33), (37), and (39) that

In view of (30) and (40), we observe that

(2). Multiplying the first equation of (19) by and the second by , adding them together, and integrating the result in , we have

Let

Then the integrand of the second integral can be estimated as follows:

Using Young inequality, we deduce that

It follows from Gronwall inequality that

If and , then it is clear from (46) that

Similarly, if , , then from (46), one has

Let if , and let if . Then

This completes the proof of Lemma 1.

4. Bootstrap Lemma

By a standard PDE theory [31], we can establish the existence of local solutions for (18).

Lemma 2 (Local existence). For and , there exists a such that (18) with , has a unique solution on which satisfies

where is a positive constant depending on , , , , , and .

It is not difficult to verify the following result.

Lemma 3. Let be a solution of (18). Then the even extension of on is also the solution of (18) which satisfies homogeneous Neumann boundary conditions and periodical boundary conditions on .

Lemma 4. Let be a solution of (18). Then

where .

Proof. It is known by Lemma 3 that

where is the even extension of on . Taking the second-order derivative of (52) for , and making inner product with and , respectively, and adding them together, we deduce that

We can apply Young inequality and (43) to get

Now we estimate each term on the right-hand sides of (54). By using Hölder inequality,

Notice that

for . Clearly,

Using the Poincaré inequality, we have

It follows from (58) and (59) that

Furthermore,

Combining (56), (57), and (62), we observe that

where .

By Hölder inequality, it follows from (57) and (62) that

where .

Now we estimate . By interpolation for all , it can be proved that

Then it is easy to see that

where and .

Finally, from (57) and (61), is bounded by

Combining (54), (55), and (63)–(67), one can obtain

where and the proof is completed.

Lemma 5. Let be a solution of (18) such that for ,

Then

where if and if .

Proof. It follows from (61) that

Thus,

Now we estimate the second-order derivatives of . From Lemma 4 and (69), we can obtain

Integrating this from to , we deduce from (70) that

We will proceed in the following two cases: , .

(1)If , then it is clear from (75) that

Using (73) and (76), we know that

where .

(2)If , then it is not hard to verify by (70), (73), and (75) that

where and thereby completing the proof.

5. Main Result

Let be a small fixed constant, and the dominant eigenvalue which is the maximal growth rate. For arbitrary small we define the escape time by

or equivalently

Our main result in this paper reads as follows.

Theorem 6. Assume that the set of satisfying instability criterion (23) is not empty for given parameters , , , , , , and . Let

such that . Then there exist constants , , and , depending on , , , , , , and , such that for all , if the initial perturbation of the steady state is , then its nonlinear evolution satisfies

for , and is the gap between and the rest of in (82).

Proof. Let be the solutions of (18) with initial data . Define

We recall (80) and choose such that

We first estimate norm of for . By the definition of and Lemma 1, for , we have

From Lemma 5, direct computation gives

It follows from (86) and that

Secondly, we establish a sharper estimate for for . Applying Duhamel’s principle, we get

By Lemma 1, (57), (59), and Lemma 5, for , we know

where . By our choice of , it is further bounded by

Next, We prove by contradiction that for sufficiently small, . If is the smallest, we can let in (86) and (87) to get

where satisfies (84) with and is sufficiently small such that . This is a contradiction to the definition of .

If is the smallest, let in (88), we see that

for is small such that , by our choice of in (84) and let . This again contradicts the definition of . Hence, is the smallest.

Finally, we can obtain from (31) that

By (30), (37), (39), and the definition of , we get

From (25) we know that there is one (or two) satisfying . If there is only one satisfying , we denote it by ; if there are and satisfying , we can let . It follows from (30) and (94) that

where . Furthermore, by (29), (30), and the definition of , we know that

Substituting (90), (95), and (96) into (93), one can obtain

where with and thereby concluding the proof.

Theorem 6 implies that the dynamics of a general perturbation are characterized by such linear dynamics over a long time period of , for any . In particular, choose a fixed and let , where so that . Then

Therefore, if , we can obtain from (97) and (98)

where . If and , then

This means nonlinear instability as .

Respecting the above-mentioned facts, the models (7) and (13) have no nonconstant positive steady state no matter what the self-diffusion coefficients , and cross diffusion are; in other words, self-diffusion and cross diffusion (without chemotaxis) cannot drive instability and cannot generate patterns for the corresponding model. This means that chemotaxis-driven instability occurs. We prove that for any given general perturbation of magnitude , linear fastest growing modes determine the nonlinear evolution for the model (2), over a time period of the order . Therefore, our results indeed provide a rigorous mathematical description for the nonlinear pattern formation in a volume-filling chemotaxis model with logistic growth.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the China National Natural Science Foundation (nos. 11361055 and 11261053) and the Fundamental Research Funds for the Gansu University.

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