Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth
This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a -dimensional box . It is proved 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 different from another, which gives rise to the richness of patterns.
An important variant of the chemotaxis model was initially proposed by Painter and Hillen in  to model the volume-filling effect. In the volume-filling effect, it is assumed that cells have a certain finite (nonzero) volume and that the occupation of an area limits other cells from penetrating it. A simple version of the volume-filling chemotaxis model (VF) is the following: where , , , , and are given positive constants. is the cell density and denotes the density of the external chemical substance which is secreted by the cells themselves. and denote the cell and chemical diffusion coefficients, respectively. is called chemosensitivity. The term denotes the chemotactic flux under a volume constraint 1 (called crowding capacity), meaning that the chemotactic movement will be inhibited at the aggregation location where the cell density reaches 1.
In recent years, the chemotaxis models with volume-filling effect have been studied extensively. Hillen and Painter  firstly proved the global existence of solutions. Numerical simulations in one and two dimensions show interesting phenomena of pattern formation and formation of stable aggregates. Wrzosek  showed the existence of a compact global attractor in the space , , for some cases. In , the structure of the attractor can be understood using Lyapunov functions. Stationary solutions which are inhomogeneous in space were investigated for a given range of parameters. In , a numerical exploration was conducted to determine the longtime patterning behaviour, revealing formation of multiple plateau type patterns which undergo a coarsening process with increasingly long transient times. Potapov and Hillen  investigated the metastability of steady states. The underlying bifurcation diagram was identified, revealing that the unstable eigenvalues are exponentially small. The plateau interactions were studied using asymptotic methods. In , it was obtained that a priori estimates for the classical chemotaxis model of Patlak, Keller, and Segel when a nonlinear diffusion or a nonlinear chemosensitivity was considered accounting for the finite size of the cells and how entropy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solutions were showed. Burger et al.  discussed the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller-Segel model of chemotaxis with volume-filling effect. Moreover, the global existence of solutions and nontrivial steady states were also studied. Wang and Hillen  established the global existence of classical solutions to a generalized chemotaxis model, which includes the volume-filling effect expressed through a nonlinear squeezing probability. Necessary and sufficient conditions for spatial pattern formation were given and the underlying bifurcations were analyzed. In , the stationary solutions of the volume-filling chemotaxis model without a growth term were obtained by Jiang and Zhang.
Moreover, Wrzosek  considered various assumptions on nonlinear diffusion and chemotactic sensitivity function which lead to the existence of global in time solutions, thus preventing blow-up. In , Winkler proved that if certain conditions were fulfilled, then there were solutions that blow up in either finite or infinite time. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen , his results indicated how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Winkler and Djie  discussed boundedness and finite-time collapse for a chemotaxis system with volume-filling effect. Wang et al.  proved that for a wide range of nonlinear diffusion operators, including singular and degenerate ones, if the taxis force was strong enough with respect to diffusion and the initial data were chosen properly, then there was a classical solution which reaches the threshold at the maximal time of its existence; no matter whether the latter was finite or infinite. Zhang and Zheng  obtained the crucial uniform boundedness of the solution for a quasilinear nonuniform parabolic system modelling chemotaxis with volume-filling effect and the results on convergence to equilibrium and the decay rate using a suitable nonsmooth Simon-Łojasiewicz approach. In , the uniform boundedness, global in time existence and uniqueness of classical solution, were proved. With the help of a suitable nonsmooth Simon-Łojasiewicz approach, the results on convergence of the solution to equilibrium and the convergence rate were obtained. Li and Zhang  classified the existence or nonexistence of steady state solutions of a 1-D chemotaxis model with volume-filling effect. Their results provided insights on how the biological parameters affect pattern formation.
The chemotaxis models with logistic growth but without a volume-filling effect were studied (see [17–22]). The global attractor and traveling wave solutions of a volume-filling chemotaxis model with logistic growth were obtained in  and , respectively. Ma et al.  studied the existence of stationary solutions of a volume-filling chemotaxis model with logistic cell growth. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, the stability criteria were established and a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue was found. Quite recently, Ma et al. in  studied the nonexistence of nonconstant steady state (i.e., stationary pattern) for a chemotaxis model with the volume-filling effect and logistic cell growth and established the critical value of the chemotactic coefficient between the existence and the nonexistence of stationary pattern.
Guo and Hwang  investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Their results can be interpreted as a rigorous mathematical characterization for the early-stage pattern formation in the Keller-Segel model. Very recently, Fu and Liu in  and  studied instability in the Keller-Segel model with a logistic source and cubic source term, respectively. Their results indicated that chemotaxis-driven nonlinear instability occurs in these models.
In this paper, we mainly consider the nonlinear instability for the following chemotaxis model: which is subject to the Neumann boundary conditions and the nonnegative initial data where is a -dimensional box. The term describes the logistic growth of cells with growth rate and carrying capacity fulfilling . Our main result (see Theorem 6) indicates that the nonlinear dynamics near an unstable constant equilibrium points in the classical Keller-Segel model, the Keller-Segel model with a logistic source and cubic source term, respectively, and a volume-filling chemotaxis model with logistic source term are almost similar.
The organization of this paper is as follows: in Section 2, we show that the unique positive equilibrium point of (2) without chemotaxis is globally asymptotically stable and cross diffusion cannot induce the instability of the positive equilibrium. In Section 3, we consider the growing modes of (2). In Section 4, we present and prove the Bootstrap lemma, which was first introduced in . In Section 5, for any given 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 the order . Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.
2. Stability of Positive Equilibrium Point of (2) without Chemotaxis
We first discuss the following corresponding kinetic equations of (2): We use to denote a column vector. Evidently, (5) has the unique positive equilibrium point . For simplicity, we denote , and a direct calculation yields
The characteristic polynomial of is ; and are the two roots of . Hence is locally stable. Define , where . By the Lyapunov-LaSalle invariance principle , is globally asymptotically stable.
We now consider system (2) without chemotaxis in the following form:
Let be the eigenvalues of the operator on with the homogeneous Neumann boundary condition and the eigenspace corresponding to in . Let , be an orthonormal basis of , and . Then , .
Let and . The linearization of (7) at can be expressed by . For each , is invariant under the operator , and is an eigenvalue of on if and only if it is an eigenvalue of the matrix Thus, has two negative eigenvalues and . It follows from Theorem in  that is locally asymptotically stable.
Let be the unique nonnegative global solution. The maximum principle gives Moreover, by the strong maximum principle , we know that if , , then , on for all .
We define the Lyapunov function where . Then for all . Applying (7) and integration by parts, we have By (9), (11), the basic estimates for parabolic equations , and Lemma 2.1 in  (which is given in  in Chinese), we can conclude that The global asymptotic stability of follows from (12) together with the local stability of .
Next, we consider that the cross diffusion model
Let . Then the linearizing system (13) at can be written as where Then, for each , is invariant under the operator , and is an eigenvalue of on if and only if is an eigenvalue of the matrix Notice that Thus, the two eigenvalues and of have negative real parts.
From this, we see that adding the cross diffusion to the system (7), the positive constant solution is also locally stable, which means that Turing instability does not occur. The information above indicates that the aggregation of individuals does not occur in the absence of chemotactic effect. It is the purpose of the present paper to clarify the effect of the chemotaxis and nonlinear patterns created by chemotaxis for a volume-filling chemotaxis model with logistic growth.
3. Growing Modes in the System (2)
The nonlinear evolution of a perturbation , around satisfies The corresponding linearized system is as follows: Let , , and . Then forms a basis of the space of functions in that satisfies Neumann boundary conditions. We try to find a normal mode to the linear system (19) of the following form: where is a vector depending on . Substituting (20) into (19) yields where . We can obtain the following dispersion formula for : Thus, we deduce the following linear instability criterion by requiring that there exist a such that to ensure that (22) has at least one positive root . This means that There exist two distinct real roots: for all to the quadratic equation (22), which are denoted by . We denote the corresponding (linearly independent) eigenvectors by and , such that Clearly, for large, Thus, there are only finitely many such that . We denote the largest eigenvalue by and define . From (25) we can regard as a function of . Therefore, there is one (possibly two) having . We also denote to be the gap between the and the rest; that is, .
Given any initial perturbation , we can expand it as
In the sequel, denote by and the inner product of and the scalar product of , respectively. For any , we denote . Throughout this paper, we always denote universal constants by .
The unique solution of (19) is given by
Our main result in this section is the following lemma.
Lemma 1. Suppose that the instability criterion (23) 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 divide the proof into the following two cases.(1). It follows from (25) that
respectively. Thus, and there exists a positive constant for all , such that
By the quadratic formula of (22), one can obtain
It follows from (29) that
From (26) and (34), we can conclude that there exists a positive constant such that for all , where . By (34), one can deduce Combining (37) and (38), we find that where . For and large, it is not difficult to verify by (33), (37), and (39) that In view of (30) and (40), we observe that (2). Multiplying the first equation of (19) by and the second by , adding them together, and integrating the result in , we have Let Then the integrand of the second integral can be estimated as follows: Using Young inequality, we deduce that It follows from Gronwall inequality that
If and , then it is clear from (46) that Similarly, if , , then from (46), one has Let if , and let if . Then This completes the proof of Lemma 1.
4. Bootstrap Lemma
Lemma 2 (Local existence). For and , there exists a such that (18) with , has a unique solution on which satisfies where is a positive constant depending on , , , , , and .
It is not difficult to verify the following result.
Lemma 4. Let be a solution of (18). Then where .
Proof. It is known by Lemma 3 that
where is the even extension of on . Taking the second-order derivative of (52) for , and making inner product with and , respectively, and adding them together, we deduce that
We can apply Young inequality and (43) to get
Now we estimate each term on the right-hand sides of (54). By using Hölder inequality, Notice that for . Clearly, Using the Poincaré inequality, we have It follows from (58) and (59) that Furthermore, Combining (56), (57), and (62), we observe that where .
By Hölder inequality, it follows from (57) and (62) that where .
Now we estimate . By interpolation for all , it can be proved that Then it is easy to see that where and .
Finally, from (57) and (61), is bounded by
Combining (54), (55), and (63)–(67), one can obtain where and the proof is completed.
Lemma 5. Let be a solution of (18) such that for , Then where if and if .
Proof. It follows from (61) that
Now we estimate the second-order derivatives of . From Lemma 4 and (69), we can obtain Integrating this from to , we deduce from (70) that
We will proceed in the following two cases: , .(1)If , then it is clear from (75) that Using (73) and (76), we know that where .(2)If , then it is not hard to verify by (70), (73), and (75) that where and thereby completing the proof.
5. Main Result
Let be a small fixed constant, and the dominant eigenvalue which is the maximal growth rate. For arbitrary small we define the escape time by or equivalently
Our main result in this paper reads as follows.
Theorem 6. Assume that the set of satisfying instability criterion (23) 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 (82).
Proof. Let be the solutions of (18) with initial data . Define
We recall (80) and choose such that
We first estimate norm of for . By the definition of and Lemma 1, for , we have From Lemma 5, direct computation gives It follows from (86) and that
Secondly, we establish a sharper estimate for for . Applying Duhamel’s principle, we get By Lemma 1, (57), (59), and Lemma 5, for , we know where . By our choice of , it is further bounded by
Next, We prove by contradiction that for sufficiently small, . If is the smallest, we can let in (86) and (87) to get where satisfies (84) with and is sufficiently small such that . This is a contradiction to the definition of .
If is the smallest, let in (88), we see that for is small such that , by our choice of in (84) and let . This again contradicts the definition of . Hence, is the smallest.
Finally, we can obtain from (31) that By (30), (37), (39), and the definition of , we get From (25) 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 can let . It follows from (30) and (94) that where