Research Article | Open Access
Pattern Formation of a Keller-Segel Model with the Source Term
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.
Mimura and Tsujikawa in  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 , 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  obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al.  numerically showed several spatiotemporal patterns in a rectangle. Kuto et al.  considered some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints. Banerjee et al.  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  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.  estimated from below the attractor dimension of (1). Efendiev et al.  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  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  investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model (i.e., (1) with , see ). 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  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 ).
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.
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
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