The flux limited Keller-Segel system;
properties and derivation from kinetic equations
The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyperbolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently.
We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.
Key words and phrases:flux limited Keller-Segel system, chemotaxis, drift-diffusion equation, asymptotic analysis, long time asymptotics
2000 Mathematics Subject Classification:35A01, 35B40, 35B44, 35K57, 35Q92, 92C17
Chemotaxis, the directed movement of an organism in response to a chemical stimulus, is a fundamental cellular process in many important biological processes such as embryonic development , wound healing , blood vessel formation [10, 17], pattern formation [6, 34]) and so on. Well-known examples of biological species experiencing chemotaxis include the slime mold amoebae Dictyostelium discoideum, the flagellated bacteria Escherichia coli and Salmonella typhimurium, and the human endothelial cells . Mathematical models of chemotaxis were derived from either microscopic (individual) or macroscopic (population) perspectives, which have been widely studied in the past four decades. The macroscopic chemotaxis model has been first developed by Keller-Segel in  to describe the aggregation of cellular slime molds Dictyostelium discoideum and in  to describe the wave propagation of bacterial chemotaxis. Because thresholds on the cell number decide when solutions will undergo smooth dispersion or blow-up in finite time, and because of the interest of related functional analysis, this system has attracted an enormous number of studies (cf. ).
In this paper, we are interested in the flux-limited Keller-Segel (FLKS) system in the whole space . Some particular form of such system has already been introduced in [20, 11]. It describes the evolution of cell density and chemical signal concentration at and time , and is based on the physical assumption that the chemotactic flux function is bounded, modeling velocity saturation in large gradient environment. It reads
We denote the cell total number . This system is conservative, that is
Compared to the classical Keller-Segel system, the chemotactic response function depends nonlinearly on the chemical concentration gradient. We assume flux limitation, that means there is a positive constant such that
The motivation to study the FLKS system (1.1) comes from its derivation from mesoscopic kinetic model. The first microscopic/mesoscopic description of chemotaxis model is due to Patlak  whereby the kinetic theory was used to express the chemotactic velocity in term of the average of velocities and run times of individual cells. This approach was essentially boosted by Alt  and developed by Othmer, Dunber and Alt  using a velocity-jump processes which assumes that cells run with some velocity and at random instants of time they changes velocities (directions) according to a Poisson process. The advantage of kinetic models over macroscopic models is that details of the run-and-tumble motion at individual scales can be explicitly incorporated into the tumbling kernel and then passed to macroscopic quantities through bottom-up scaling (cf. [18, 45, 46, 15, 16, 42, 37, 12]), where the rigorous justification of upscaling limits have been studied in many works (see [9, 21, 22, 23, 27] and reference therein). Denoting by the cell number density, at time , position moving with a velocity (compact set of with rotational symmetry), the governing evolution equation of this process is described by a kinetic equation reading as:
The tumbling kernel describes the frequency of changing trajectories from velocity (anterior) to (posterior) depending on the chemical concentration or its gradient. Because cells are able to compare present chemical concentration to previous ones and thus to respond to temporal gradients along their pathways, the tumbling kernel may depend on the pathway (directional derivative) and takes the form ([12, 37])
where denotes a basal meaning tumbling frequency, accounts for the variation of tumble frequency modulation and denotes the signal response (sensing) function which is decreasing to express that cells are less likely to tumble when the chemical concentration increases.
The first goal of the present paper is to derive the FLKS system (1.1) as the parabolic limit of the kinetic equation (1.3)-(1.4) and relate the flux limiting function to . In particular, we introduce a new rescaling, related to the stiffness of signal response, which has been shown to be important to describe the traveling pulses of bacterial chemotaxis observed in the experiment [41, 40, 13] and is related to instabilities both of the FLKS system and the kinetic equation [38, 8]. In particular, we wish to go further than the case proposed in , when the response function is bi-valuated step (stiff) function: , where the parabolic limit equation of (1.3) is
Our second goal is to study the long time behaviour of solutions to the FLKS system (1.1) and the existence of stationary radial solutions. Contrary to the Keller-Segel system for which finite time blow-up of weak solutions is observed, solutions to (1.1) under assumption (1.2) exist globally in time. For a study of the long time convergence towards radially symmetric solutions for the Keller-Segel system, which may be computed explicitely, we refer to . Yet, we do not have an explicit expression of radially symmetric solutions for system (1.1). Then, we prove that when the degradation coefficient is positive (), diffusion takes the advantage over attraction. On the contrary, when the degradation coefficient is disregarded (), the total mass of the system, denoted , appears to be an important parameter. Indeed, when , we observe a threshold phenomenon, with a critical mass in dimension , for the existence of radial stationary solution.
More precisely, our main results may be summarized as follows :
The situation in bounded domain is quite different. Indeed, existence of steady state solutions for on bounded domain with Neumann boundary conditions has been investigated in  in one dimension. Based on a bifurcation analysis, they observe spiky solutions when the chemotactic sensibility is large. See also [28, 31, 43, 44] for spiky steady states in chemotaxis models.
The outline of the paper is as follows. In the next section, we derive the flux-limited Keller-Segel model (1.1) from the kinetic system with the appropriate scaling. Section 3 deals with the existence of radially symmetric stationary states in dimension greater than . The one dimensional case is investigated in section 4. The study of the long time behaviour is performed in section 5 where Theorem 5.1 is proved. Then we summarize briefly our results in a conclusion and provide open questions related to this work. Finally, an appendix is devoted to some technical lemma useful throughout the paper.
2. Derivation of FLKS from kinetic model
Our approach uses the stiffness parameter and a smoothed stiff response function . In other words, we consider the following smooth stiff tumbling kernel
A possible example, as suggested in , is . Other examples include, for instance, . The case corresponds to a stepwise stiff response function mentioned in the introduction. In , it has been measured that . For convenience, we write for some scaling constant and rewrite above tumbling kernel as
In this paper, we shall take as a scaling parameter and derive the parabolic limit of kinetic models of chemotaxis which turns out to be the FLKS model (1.1) as long as the response function is bounded.
2.1. Rescaling of the kinetic equation
We summarize the condition on the response function as follows:
where , and is a constant denoting chemical decay rate.
2.2. Well-posedness, a priori estimates and compactness
The well-posedness and macroscopic limits of kinetic models of chemotaxis where the tumbling kernel depends on the chemical concentration or its spatial derivative have been extensively studied e.g. in [32, 33, 9, 21, 5] either formally or rigorously, based on the advanced functional analytical tools available for kinetic equations. When the tumbling kernel depends on the pathway derivative , the formal limits have been studied in [33, 12] and rigorous justification was given in  for the two species case. The well-posedness of equations (2.3)-(2.5) and the limit as are the direct consequence of the results of . For completeness, we present, without proof, the following result
Theorem 2.1 (Existence, a priori estimates).
Lemma 2.1 (Strong local compactness on ).
The signal function is uniformly bounded and is strongly locally compact in for all .
We use that is bounded in from Theorem 2.1.
From usual elliptic or parabolic regularizing effects (see Lemma A.2 in Appendix), and using only the above bounds for , we conclude that is bounded in , with ( in dimensions , in dimension 4).
Next, the equation on reads
and gives , for any any thanks to the direct estimates on the heat kernel (see (A.1) in Appendix with , , ).
Next, we notice by a similar argument, that is bounded in for . Compactness in for also follows from the convolution formula. Finally, we write
and thus we conclude that , for all , thus providing time compactness. ∎
As a conclusion, we may extract a subsequence such that locally in all spaces , , . Notice that the bounds above tell us that, as ,
2.3. The convergence result
where is a uniform distribution on :
With the symmetry assumption of , it satisfies , and In the limit we infer from (2.9) that
The flux can be identified and we are going to show in the next subsection the following
Theorem 2.2 (Derivation of the FLKS system).
Notice that is well defined by continuity and .
2.4. Asymptotic analysis
In other words, we have identified the flux term
By rotational symmetry of , is proportional to and hence yields the expression of in Theorem 2.2. Due to the hypothesis (H) on , the drift velocity term is uniformly bounded in time and space . This is the main feature of the macroscopic limit model resulting from the stiff response postulated in the kinetic models. ∎
We consider a specific form of signal response function as follows
and derive an explicit flux-limited Keller-Segel system. When , which is a sign function reflecting the stepwise stiff response. However, as , is smooth and .
By substitution, we have from (2.4) that
where we have recovered by for brevity and, by rotational symmetry of V,
where is the first component of colinear to : Clearly both and are bounded for all , which implies that the chemotactic (drift) velocity is limited. The system (2.16) with (2.17) gives a specific example of the FLKS system (1.1).
2.6. Global existence for the macroscopic system
Finally, we state the existence result for system (1.1) under the assumption (1.2). A specific example of function satisfying this set of assumptions has been given in (2.17). Under the assumption (1.2), the chemotactic (or drift) velocity term is bounded and hence the global existence of classical solutions of (1.1) can be directly obtained.
Theorem 2.3 (Global existence).
The proof consists of two steps. The first step is the local existence of solutions which can be readily obtained by the standard fixed point theorem (cf. [3, 2]). The second step is to derive the a priori bound of in order to extend local solutions to global ones. This can be achieved by the method of Nash iterations as it is well described in [19, Lemma 1]. Although the procedure therein was shown for bounded domain with Neumann boundary conditions, the estimates directly carry over to the whole space . ∎
3. Radial steady states in dimension
Since it is proved in section 5 that when diffusion takes the advantage over attraction implying the time decay towards zero of the solutions to system (1.1), we are only interested in the case . The stationary problem for system (1.1) is non-trivial due to the nonlinearity. Below we explore a simpler case: existence of radial symmetric stationary solutions. The stationary system of (1.1) when written in radial coordinates for reads
Notice that there is another relation, at infinity, expressing that the mass is given by
We are going to prove the following result.
We use the unknown in order to carry out the analysis. The equation on in (3.1) now reads , and we obtain
From (1.2) and the second equation of (3.3), we get and hence for all . Furthermore from the first equation of (3.3), we know that is non-decreasing and has a limit as which determines the total mass according to (3.2). Hence for finite mass , has a finite limit and thus for large enough, say for some , we have
for some positive constant .
In dimension , we recover a phenomenon similar to the multiple solutions for the critical mass in the Keller-Segel system but explicit solutions are not available. The system (3.3) reduces to
By the boundary conditions in (3.3), we see that positive solutions behave as for and some constant . Then, the proof of Theorem 3.1 is a consequence of Lemma 3.1 and Proposition 3.2 below. Lemma 3.1 states that a necessary condition of existence of radial solution is . Proposition 3.2 shows that for any finite mass larger than the critical mass , there exist radial solutions with mass to system (1.1) with . ∎
Let be a positive solution to (3.5). Then is increasing. If is bounded, then
We split the proof into three steps:
From the behaviour near , we know that for small enough. If we had for some , then the unique solution of (3.5) is which is a contradiction. Therefore for all .
This inequality may be rewritten as
This inequality implies that
because if it were smaller, we would have for some and is not integrable.
As a consequence, we know that as , and thus, for some nonnegative constant , it holds that
We want to prove that for any , there exists such that the solution to (3.5) verifying and . We first simplify the problem by introducing the change of variable . Setting , we deduce, from straightforward computations, that is a solution to the system
We are left to use a shooting method to show there is a number such that (3.8) has a solution satisfying for any .
By definition of and thanks to the above results, we have that and on .
Since , we deduce from (3.8) that
After integration we obtain
Thus, when , we have
Upon integration, we find a positive constant such that for all and , we have
Thus, by continuity and the fact is increasing, we have that .
Let us prove that: For any , there exists a number small enough such that the solution to (3.8) satisfies .
In the vicinity of , we have . Then for small enough, there exists such that . (Indeed, if it is not true, we will have on , which is not possible for since ). The function being bounded on , let us denote . By the same token as above, we deduce from (3.8) that
Integrating above inequality over gives
On one hand, integrating (3.9) from to , we get
On the other hand, integrating (3.9) between and , we obtain
It is clear that as for all . Let small. Then by the continuity of the function , for small enough, we can deduce from the above estimate that . Then, we can redo the same estimate as above, replacing by , we arrive at
which implies by taking ,
Let us prove that .