Advances in Mathematical Physics

Volume 2013 (2013), Article ID 934745, 11 pages

http://dx.doi.org/10.1155/2013/934745

Research Article

A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

Shengmao Fu1,2 and Ji Liu¹

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 4 October 2012; Revised 25 December 2012; Accepted 9 January 2013

Academic Editor: M. Lakshmanan

Copyright © 2013 Shengmao Fu and Ji Liu. 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 Keller-Segel model with a cubic source term in a d-dimensional box , which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitude δ, 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 ln(). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

1. Introduction

Keller and Segel in their pioneering work [1] proposed the following model

where is cell density, is chemoattractant concentration, 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. Keller and Segel wanted to model the chemotactic movement of the cellular slime mold Dictyostelium discoideum during its aggregation phase, where population growth does not occur. Therefore, they considered a population in the absence of “death” and “birth.” For some main results on the Keller-Segel model, please see [2–4] and references therein.

Recently, Guo and Hwang in [5] investigated the nonlinear dynamics near an unstable constant equilibrium of the Keller-Segel model satisfying the Neumann boundary conditions for and on a -dimensional box ; that is,

Let be the uniform constant solution of the Keller-Segel model, and . Then satisfies the equivalent Keller-Segel system below:

Guo and Hwang proved that linear fastest growing modes determine unstable patterns for the above system. Their result can be interpreted as a rigorous mathematical characterization for early pattern formation in the Keller-Segel model.

In recent years, more and more attention has been given to the Keller-Segel model with the reaction terms, that is, the following chemotaxis-diffusion-growth model:

For , Painter and Hillen [6] demonstrated the capacity of the above model to self-organize into multiple cellular aggregations which, according to its 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). This spatiotemporal patterning can be further subdivided into either a time-periodic or time-irregular fashion. Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. In particular, they found stationary patterns that bifurcate onto a path of periodic patterns which, prior to the onset of spatiotemporal irregularity, undergo a “periodic-doubling” sequence. Based on these results and comparisons with other systems, they argued that the spatiotemporal irregularity observed here describes a form of spatiotemporal chaos.

For , Banerjee et al. [7] showed that the dynamics of the chemotaxis-diffusion-growth model may lead to steady states, to divergencies in a finite time, and to the formation of spatiotemporal irregular patterns. The latter, in particular, appears to be chaotic in part of the range of bounded solutions, as demonstrated by the analysis of wavelet power spectra. Steady states are achieved with sufficiently large values of the chemotactic coefficient and/or with growth rate below a critical value . For , the solutions of the differential equations of the model diverge in a finite time. They also reported on the pattern formation regime, for different value of , and of the diffusion coefficient . For the same , Kuto et al. [8] considered some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints. They studied the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity and construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, they exhibited several numerical results for the stationary and oscillating patterns.

In [9], Okuda and Osaki studied the chemotaxis-diffusion-growth model with in a rectangular domain by applying the center manifold theory, where constant and either , or . It is observed that the trivial solutions are destabilized due to the chemotaxis term. They obtained the normal form on the center manifold, and it is proved that the locally asymptotically stable hexagonal patterns exist.

Another extended formation of logistic source term is the cubic source term , where is the intrinsic growth rate, the sign of is undetermined, is a positive constant, and is the density restriction term (see [10, 11] for more information and references). Recently, Cao and Fu in [11] studied global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey. In this paper, we investigate dynamics of the chemotaxis-diffusion-growth model with the source term ; that is,

where , , , , and are positive constants and , satisfies the Neumann boundary conditions. We will prove that given any general perturbation of magnitude , its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of . Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model (5).

The organization of this paper is as follows: In Section 2, we prove that the positive constant equilibrium solution of (5) without chemotaxis is globally asymptotically stable if . In Section 3, we investigate the growing modes of (5). In Section 4, we present and prove the Bootstrap lemma. In Section 5, given any 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 .

2. Stability of Positive Equilibrium Point of (5) without Chemotaxis

The corresponding semilinear system of (5) without chemotaxis is as follows:

Obviously, is a positive equilibrium point of (6) if and only if either of the following two cases happens: , . In the following we will discuss the stability of in (6).

Let , and 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 diag and , where

Then the linearization of (6) 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

Notice that is the positive root of and

Thus, has two negative eigenvalues and . It follows from [12, Theorem 5.1.1] that is locally asymptotically stable.

Let be a unique nonnegative global solution of (6). It is not hard to verify by the maximum principle that

Moreover, if , then , , for all .

According to the main result in [13], we have

(i) If , , then . We define the Lyapunov function

where .

Calculating the derivative of along positive solution of (6) by integration by parts and the Cauchy inequality, we have

(ii) If , , then . We define the Lyapunov function

where .

Calculating in the same way as (14), we have

Combining (12)–(16) and Lemma  3.2 in [11], we conclude that

The global asymptotic stability of follows from (17) and the local stability of .

Theorem 1. The positive equilibrium point of (6) is locally asymptotically stable. If either , or , holds, then is globally asymptotically stable.

3. Growing Modes in the System (5)

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. We look for a normal mode to the linearized system (19) of the following form

where is a vector depending on . Plugging (20) into (19) we have the following dispersion formula for

Thus we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring there exists a such that

to ensure that (21) has at least one positive root . This implies that for ,

If , then (21) has two negative roots and . Therefore, the positive equilibrium point of (18) is locally asymptotically stable.

Now we investigate nonlinear dynamics near the unstable constant equilibrium solution of (18) in the case . If , the right side of (23) is positive. Therefore, there exist two distinct real roots for all to the quadratic equation (21). We denote the corresponding (linearly independent) eigenvectors by and , such that

Clearly, for large

Hence, there are only finitely many such that . We denote the largest eigenvalue by and define . It is easy to see that there is one (possibly two) having if we regard as a function of . We also denote to be the gap between the and the rest. Given any initial perturbation , we can expand it as

so that

The unique solution to (19) is given by

For any , we denote . Our main result of this section is the following lemma.

Lemma 2. Suppose that the instability criterion (22) 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 first consider the case for . By analyzing (21), for large, we have

respectively. From the quadratic formula of (21), we can see that

It follows from (27) that

Later on we will always denote universal constants by . Note that and . From (24) and (30), for all , there exists a positive constant and , such that , and

By (24), (31), (32), and (33), we deduce that

Thus, it is clear from (33) and (34) that

For , it is not hard to verify that there exists a constant , such that

It follows from (35) and (36) that

Denote by and the inner product of and the scaler product of , respectively. A simple computation shows that

From (28), (37) and (38), we have

On the other hand, for , it is sufficient to derive the standard energy estimate in . By (19), we have

Let

Then the integrand of the second term on the left side of (40) satisfies

By (40), Young's inequality, and , we deduce that

Using Grownwall's inequality and noticing and , we can obtain

If , by (43), , and Grownwall inequality, we have

Let if . Then if and .

4. Bootstrap Lemma

By a standard PDE theory [14], we can establish the existence of local solutions for (18).

Lemma 3 (local existence). For and , there exists , such that (18) with has a unique solution on which satisfies

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

It is not hard to verify the following result.

Lemma 4. 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 5. Let be a solution of (18). Then

where .

Proof. It is known by Lemma 4 that

where is the even extension of on . Taking the second-order derivative of (48) for and making inner product with and , respectively, on both sides then adding the two equations together, we have

Clearly

The nonlinear term is bounded by

We know that

for , and

Applying the Poincaré inequality, we have

It follows from (53) and (54) that . Thus

Furthermore,

This implies that is equivalent to . From (53)–(56), we have

It follows from (56) and (57) that

By (51), (52), (57), and (58), we have

Now we consider . From Gagliardo-Nirenberg inequality and Young inequality, we obtain

Let . Then

It follows from (52), (57), and (58) that

Combining (49), (50), (59), (61), and (62), we have

where , .

Lemma 6. Let be a solution of (18) such that for

Then

where if  if .

Proof. It is clear from (57) that

It follows from (67) that

Now we estimate the second-order derivatives of . By (65) and Lemma 5, we immediately see that

Integrating on both sides of (69) from to and from (65), we have

We will proceed in the following two cases: , .

(1) If , it follows from (70) that

By (68) and (71), we have

where .

(2) If , it follows from (71) that

By (68) and (73), we have

where .

5. Main Result

Let be a small fixed constant, and 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 is as follows.

Theorem 7. Assume that the set of satisfying instability criterion (22) is not empty for given parameters , , , , , , , , . 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 (21).

Proof. Let be the solutions to (18) with initial data . We define

We also define

Choose such that

We now establish a sharper estimate of for . First of all, by the definition of and Lemma 2, for , it is not hard to see that

Applying Lemma 6 and the bootstrap argument, one can prove

From this and , it follows that

Applying Duhamel’s principle, we can obtain

By Lemma 2, (52), (54), and Lemma 6, for , we deduce that

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

We now prove by contradiction that for sufficiently small, . If is the smallest, we can let in (83) and (84). If satisfies (81) with and is sufficiently small such that , we immediately see that

This is a contradiction to the definition of . On the other hand, if is the smallest, we can let in (87). If satisfies (81) and is sufficiently small such that , we also can see that

This again contradicts the definition of . Hence, if is sufficiently small, we have

From (28), we have

that is,

Using (33) and (34), we have

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 let . From (93), we have

where . Now we consider . By (38), we have

From (87), (92), (94), and (95), it follows that

where . 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 .

Our Theorem 7 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of , for any . In particular, choose a fixed and let ; then

Note that , , and are fixed constants and is a fixed vector. From (96) and (98), if , we have

where . Moreover,

Let , and . Then

This implies nonlinear instability as . In particular, instability occurs before the possible blow-up time.

Let us point out that although our proof is based on Guo-Strauss’ bootstrap argument, the adaptation to the procedure to our problem is not trivial at all, since the appearance of a growth restriction of a cubic type, we need more delicate estimates. Notice in our theorem that each initial perturbation can be drastically different from another, which gives rise to the richness of the pattern; on the other hand, the finite number maximal growing modes determines the common characteristics of the pattern, over the time scale of . Therefore, our result indeed provides a mathematical description for the pattern formation in the Keller-Segel model with a cubic source term.

Acknowledgments

The authors would like to thank the referees for their helpful comments. This work is supported by the China National Natural Science Foundation (nos. 11061031; 11261053), the Fundamental Research Funds for the Gansu University.

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