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 , 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.
Keller and Segel in their pioneering work  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  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  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.  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.  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 , 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  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 , 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 , 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 , 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