Pattern Formation of a Keller-Segel Model with the Source Term u p ( 1 − u )

whereU(x, t) is the cell density,V(x, t) is the concentration of chemotactic substance,Du > 0 is the amoeboid motility, χ > 0 is the chemotactic sensitivity,DV > 0 is the diffusion rate of cyclic adenosinemonophosphate (cAMP), α > 0 is the rate of cAMP secretion per unit density of amoebae, and β > 0 is the rate of degradation of cAMP in environment. In [1], the growth termg(U) is classified into the three cases: (i)g(0) = 0 and g(u) < 0, for any u > 0, (ii) (bistable type) g(0) = g(a) = g(1) = 0, for some 0 < a < 1, g(u) < 0, for 0 < u < a, and g(u) > 0, for a < u < 1, and (iii) (Logistic type) g(0) = g(1) = 0 and g(u) > 0, for 0 < u < 1. Formodel (1), with a Logistic source term g(u) = u(1−u), Tello andWinkler [2] obtained infinitelymany 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 andHillen [6] demonstrated the capacity of (1) to selforganize intomultiple 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 g(u) =

For model (1), with a Logistic source term () = (1−), 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 selforganize 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 () =  2 (1 − ), 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 () = 0, 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 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 ln(1/).Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns.

Local Stability of Positive Constant Equilibrium Solution
The PDE system (2) without chemotaxis is as follows Theorem 1.The positive equilibrium point [, ] of (5) is locally asymptotically stable.
Proof.Let 0 =  1 < Let D = diag(1, 1) and L = DΔ + G W (W). The linearization of ( 5) at [, ] is For each  ≥ 1,   is invariant under the operator L, and  is an eigenvalue of L on   , if and only if it is an eigenvalue of the matrix Then −  D + G W (W) has two negative eigenvalues −  − 1 and −  −.Hence, [, ] is locally asymptotically stable (see [12]).

Growing Modes in the System (2)
Let (x, ) = (x, ) − , V(x, ) = (x, ) − .Then The corresponding linearized system takes the form Let w(x, ) ≡ [(x, ), V(x, )], q = ( 1 , . . .,   ) ∈ Ω = N  , and  q (x) = ∏  =1 cos(    ).Then { q (x)} q∈Ω forms a basis of the space of functions in T  that satisfy Neumann boundary conditions (3).To find a normal mode to the linear system (11) of the following form w (x, ) = r q exp ( q )  q (x) , where r q is a vector depending on q, we substitute ( 12) into (11) to get where 2 q + (2 2 +  + 1)  q + ( 2 + 1) ( 2 + ) −  2 = 0. (15) Thus, if there exists a q such that the linear instability criterion holds, that is, then (15) has at least one positive root  q .Obviously, Therefore, we can denote two distinct real roots for all q by The corresponding (linearly independent) eigenvectors r − (q) and r + (q) are given by It is easy to see from ( 16) that there exist only finitely many , such that  + q > 0. We therefore denote the largest eigenvalue by  max > 0 and define Moreover, there is one  2 (possibly two) having  + ( 2 ) =  max , when we regard  + q as a function of  2 .We also denote ] > 0 to be the gap between the  max and the rest, that is, Given any initial perturbation w(x, 0), that is, where we know that the unique solution w(x, ) = [(x, ), V(x, )] of ( 11) is given by For any g(⋅, ) ∈ [ 2 (T  )] 2 , we denote ‖g(⋅, )‖ ≡ ‖g(⋅, )‖  2 and ⟨⋅, ⋅⟩ and (⋅, ⋅) are the inner product of [ 2 (T  )] 2 and the scaler product of R 2 , respectively.Our main result of this section is the following lemma.

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

Main Result
Let  be a small fixed constant, and let  max be the dominant eigenvalue, which is the maximal growth rate.We also denote the gap between the largest growth rate  max and the rest by ] > 0. Then for  > 0 arbitrary small, we define the escape time   by or equivalently Our main result of this paper is the following theorem.
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).