Journal of Mathematics

Volume 2013 (2013), Article ID 454513, 11 pages

http://dx.doi.org/10.1155/2013/454513

Research Article

Pattern Formation of a Keller-Segel Model with the Source Term

Shengmao Fu1,2 and Fenli Cao¹

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

²School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China

Received 31 January 2013; Accepted 11 August 2013

Academic Editor: Liwei Zhang

Copyright © 2013 Shengmao Fu and Fenli Cao. 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.

Nonlinear dynamics near an unstable constant equilibrium in a Keller-Segel model with the source term is considered. It is proved that nonlinear dynamics of a general perturbation is determined by the finite number of linear growing modes over a time scale of , where is a strength of the initial perturbation.

1. Introduction

Mimura and Tsujikawa in [1] proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis, a dimensionless prototype of which reads

where is the cell density, is the concentration of chemotactic substance, is the amoeboid motility, is the chemotactic sensitivity, is the diffusion rate of cyclic adenosine monophosphate (cAMP), is the rate of cAMP secretion per unit density of amoebae, and is the rate of degradation of cAMP in environment. In [1], the growth term is classified into the three cases: (i) and , for any , (ii) (bistable type) , for some , , for , and , for , and (iii) (Logistic type) and , for .

For model (1), with a Logistic source term , Tello and Winkler [2] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. [3] numerically showed several spatiotemporal patterns in a rectangle. Kuto et al. [4] considered some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints. Banerjee et al. [5] showed that the corresponding dynamics may lead to steady states, to divergences in a finite time as well as to the formation of spatiotemporal irregular patterns. Painter and Hillen [6] demonstrated the capacity of (1) to self-organize into multiple cellular aggregations, which, according to position in parameter space, either form a stationary pattern or undergo a sustained spatiotemporal sequence of merging (two aggregations coalesce) and emerging (a new aggregation appears). Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. They argued that the spatiotemporal irregularity observed here describes a form of spatiotemporal chaos.

For model (1) with a logistic-like growth term , Aida et al. [7] estimated from below the attractor dimension of (1). Efendiev et al. [8] showed that dimension estimates of global attractors for the approximate systems are uniform with respect to the discretization parameter and polynomial order with respect to the chemotactic coefficient in the equation. By using nonnegativity of solutions, Nakaguchi and Efendiev [9] managed significantly to improve dimension estimates with respect to the chemotactic parameter. It is also well-known that the asymptotic behavior of solutions relating to patterns can be described by the dynamical systems of equations and that the degrees of freedom of such processes, which characterize the richness of emerging patterns, correspond to the dimensions of their attractors.

Recently, Guo and Hwang in [10] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model (i.e., (1) with , see [11]). Their result can be interpreted as a rigorous mathematical characterization for pattern formation in the Keller-Segel model.

In the present paper, we consider the nonlinear dynamics near an unstable constant equilibrium for the following chemotaxis-diffusion-growth model:

which satisfies the homogeneous Neumann boundary conditions and initial value conditions for and , that is,

where is a -dimensional box, . By using the bootstrap technique in [10] and higher-order energy estimates, we prove 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 differently from another, which gives rise to the richness of patterns.

2. Local Stability of Positive Constant Equilibrium Solution

The PDE system (2) without chemotaxis is as follows

is the unique positive equilibrium point. We use to denote a column vector. Let , , and . Then

Theorem 1. The positive equilibrium point of (5) is locally asymptotically stable.

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

Let and . The linearization of (5) at is

For each , is invariant under the operator , and is an eigenvalue of on , if and only if it is an eigenvalue of the matrix

Then has two negative eigenvalues and . Hence, is locally asymptotically stable (see [12]).

3. Growing Modes in the System (2)

Let , . Then

The corresponding linearized system takes the form

Let , , and . Then forms a basis of the space of functions in that satisfy Neumann boundary conditions (3). To find a normal mode to the linear system (11) of the following form

where is a vector depending on , we substitute (12) into (11) to get

where . This implies that

Thus, if there exists a such that the linear instability criterion holds, that is,

then (15) has at least one positive root . Obviously,

Therefore, we can denote two distinct real roots for all by

The corresponding (linearly independent) eigenvectors and are given by

It is easy to see from (16) that there exist only finitely many , such that . We therefore denote the largest eigenvalue by and define

Moreover, there is one (possibly two) having , when we regard as a function of . We also denote to be the gap between the and the rest, that is,

Given any initial perturbation , that is,

where

we know that the unique solution of (11) is given by

For any , we denote and and are the inner product of and the scaler product of , respectively. Our main result of this section is the following lemma.

Lemma 2. Assume that the instability criterion (16) holds. Let

be a solution to the linearized system (11) with initial condition . Then there exists a constant depending on and , such that

Proof. We prove the Lemma in the following two cases.

(1) . By analyzing (15), we have

It follows from (18) that

By solving (23), we have

Clearly,

Later on, we will always denote universal constants by . It is not hard to verify that there exist positive constants and , such that

where . It follows from (19) and (28) that

Combining (29)–(32), we can obtain

where . From (31) and (33), we have

For , it follows from (18) that

Therefore, for all , there exists a constant , such that

From (34) and (36), we obtain

where . By (23), we have

Notice that

It follows from (37) and (39) that

Therefore,

(2) . By (11), we have

If , then the integrand of the second integral can be chosen nonnegative as follows:

Combining (42) and Young inequality together, we derive

By Grownwall inequality, we can obtain

where , if ; , if .

4. Bootstrap Lemma

By a standard PDE theory [13], one can establish the existence of local solutions for (10).

Lemma 3 (local existence). For and , there exists a , such that (10) with has a unique solution on , which satisfies

where is a positive constant depending on and .

Lemma 4. Let be a solution of (10). Then

where is the universal constant and .

Proof. Notice that if is a solution of (10) on , then is a solution of (10) on . Let be a solution of (10) on , which is an even extension of on . Then one may assume periodicity at the boundary of the extended .

By (10), one has

where , and

Clearly,

Thus,

By interpolation inequalities, one knows that

for any . If , then one can delete the last term in the above inequality. If , one can choose positive constant such that . Then

where .

One now estimates the second nonlinear term :

Clearly,

Notice that

From (56)–(59), one can obtain

Similarly, one has

Here and later, are universal constants. It follows from (60) and (61) that

where .

Recall that is the even extension of . Combining (49), (53), and (62), one has

where and .

Lemma 5. Let be a solution of (10), such that for ,

Then

where , if and , if .

Proof. By (58), we have

It follows from (67) that

Now, we estimate the second-order derivatives of . By Lemma 4, we have

Integrating on both sides of (69) from to , we get from (65) the following:

We will proceed in the following two cases: and .

(1) If , we have

By (68) and (71), one can obtain

where , .

(2) If , from (70), we have

Therefore,

Combining (68) and (74), we have

where , .

5. Main Result

Let be a small fixed constant, and let be the dominant eigenvalue, which is the maximal growth rate. We also denote the gap between the largest growth rate and the rest by . Then for arbitrary small, we define the escape time by

or equivalently

Our main result of this paper is the following theorem.

Theorem 6. Assume that the set of satisfying instability criterion (16) 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 (15).

Proof. Let be the solutions of (10) with initial data . We define

Choose , such that

Now, we establish estimates for norm of for . For , by Lemma 2, we have

Applying Lemma 5 and the bootstrap argument yields

From this and , we can obtain

Applying Duhamel’s principle, we know that the solution of (10) is

Using Lemma 2, (56), (58), and Lemma 5 yields, for ,

where . Notice that . Thus

For sufficiently small, we claim that

If is the smallest, we can let in (84) and (85) to obtain

for for small, by our choice of in (82) with . This is a contradiction to the definition of . On the other hand, if is the smallest, we let in (88) to get

for for small, by our choice of in (82). This again contradicts the definition of . Hence, the desired assertion follows.

From (24), we obtain

that is,

Using (34), we deduce that

From (18) we know that there is one (or two) satisfying . If there is only one satisfying , we denote it by and if there are and satisfying , we can let . From (94), we obtain

that is,

where . Now, we consider

From (39) and (41), we get

that is,

From (88), (96), and (99), one can obtain

where . This completes the proof.

6. Conclusion

Notice that for , is sufficiently small. As long as for at least one , which is generic for perturbations, the corresponding fastest growing modes

have the dominant leading order of . Theorem 6 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of , for any (see [10]). In particular, choose a fixed and let

Then

Therefore, if , we can obtain from (100)

Note that , , and are fixed constants, and is a fixed vector. Hence, we have

where . Thus,

Let be sufficiently small so that holds. Then , and

This implies nonlinear instability as . Then instability occurs before the possible blow-up time.

Recall that the unique positive equilibrium point of (2) without chemotaxis is locally asymptotically stable. This means that the instability in (2) is called chemotaxis-driven instability since the presence of chemotaxis is essential for the instability mechanism leading to nonuniform patterns in the system (2).

Acknowledgments

The authors would like to thank the referee for their helpful comments. This work is supported by the China National Natural Science Foundation (no. 11061031; 11261053) and the Fundamental Research Funds for the Gansu University.

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