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

This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a d-dimensional box (d = 1, 2, 3), 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 growingmodes, over a time period of the order ln(1/δ). 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.


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][3][4] and references therein.
Then [, ] satisfies the equivalent Keller-Segel system below: =   ∇ 2  +  − . ( 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:   = ∇ (  ∇ − ∇) +  () ,   =   ∇ 2  +  − . (4)

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For () = (1 − /), 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 () = (1 − ), 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  = (1−) + (1−)(−) in a rectangular domain by applying the center manifold theory, where constant  ∈ (0, 1) and either  > 0,  = 0, or  = 0,  > 0. 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  = ( 1 +  2  −  3  2 ), where  1 ≥ 0 is the intrinsic growth rate, the sign of  2 is undetermined,  3 > 0 is a positive constant, and  2  −  3  2 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 predatorprey system with stage structure for the prey.In this paper, we investigate dynamics of the chemotaxis-diffusion-growth model with the source term  = ( 1 +  2  −  3  2 ); 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 ln(1/).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  1 > 0. 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 ln(1/).

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: (i)  1 > 0,  2 ∈ R, (ii)  1 = 0,  2 > 0. In the following we will discuss the stability of [, ] in (6).
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 Notice that Moreover, if  0 ,  0 ≥ ( ̸ ≡ )0, then (, ) > 0, (, ) > 0, for all  > 0.
Calculating the derivative of () along positive solution of (6) by integration by parts and the Cauchy inequality, we have where  = 2 2 / 3 .
Calculating in the same way as ( 14), we have Combining ( 12)-( 16) and Lemma 3.2 in [11], we conclude that lim The global asymptotic stability of [, ] follows from (17) and the local stability of [, ].

Growing Modes in the System
The corresponding linearized system takes the form Let w(x, ) ≡ [(x, ), (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.We look for a normal mode to the linearized system (19) of the following form where r q is a vector depending on q.Plugging (20) into (19) we have the following dispersion formula for  q Thus we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring there exists a q such that to ensure that (21) has at least one positive root  q .This implies that for q, If q = 0, then (21) has two negative roots − and −3 3  2 + 2 2  +  1 .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 q ̸ = 0.If q ̸ = 0, the right side of (23) is positive.Therefore, there exist two distinct real roots  ± q for all q ̸ = 0 to the quadratic equation (21).We denote the corresponding (linearly independent) eigenvectors by r − (q) and r + (q), such that Clearly, for  large Hence, there are only finitely many q such that  + q > 0. We denote the largest eigenvalue by  max > 0 and define It is easy to see that there is one  2 (possibly two) having  + ( 2 ) =  max if we regard  + q as a function of  2 .We also denote  > 0 to be the gap between the  max and the rest.Given any initial perturbation w(x, 0), we can expand it as so that The unique solution w(x, ) = [(x, ), (x, )] to (19) is given by For any g(⋅, ) ∈ [ 2 (T  )] 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   ( = 1, 2, . ..).Note that q ∈ N  and q ̸ = 0. From ( 24) and ( 30), for all  > 0, there exists a positive constant  1 and  2 , such that By ( 24), ( 31), (32), and (33), we deduce that Thus, it is clear from (33) and (34) that For  ≥ 1, it is not hard to verify that there exists a constant  4 > 0, such that It follows from ( 35) and (36) that       ± q r ± (q) exp ( ± q )      ≤  5      w q      .

Bootstrap Lemma
By a standard PDE theory [14], we can establish the existence of local solutions for (18).
It is not hard to verify the following result.

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Now we estimate the second-order derivatives of w(⋅, ).By (65) and Lemma 5, we immediately see that Integrating on both sides of (69) from 0 to  and from (65), we have We will proceed in the following two cases: By ( 68) and (71), we have where By ( 68) and (73), we have where

Main Result
Let  be a small fixed constant, and  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 is as follows.
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 ln(1/).Therefore, our result indeed provides a mathematical description for the pattern formation in the Keller-Segel model with a cubic source term.