Traveling fronts guided by the environment for reaction-diffusion equations

Traveling fronts guided by the environment for reaction-diffusion equations

Abstract

This paper deals with the existence of traveling fronts for the reaction-diffusion equation:

We first consider the case where is of KPP or bistable type and . This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value and of a nonzero asymptotic profile (a stationary limiting solution) if and only if . When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.

We also study here the case where for and for . This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if is large enough and the non-existence if is too small.

Henri Berestycki

CAMS, UMR 8557, EHESS

190-198 avenue de France, 75244 Paris Cedex 13, France

Guillemette Chapuisat

LATP, UMR 7353, Aix-Marseille Université

39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France

CAMS, UMR 8557, EHESS

190-198 avenue de France, 75244 Paris Cedex 13, France

Dedicated to Hiroshi Matano on the occasion of his Kanreki.


1 Introduction

This paper deals with the existence of bounded traveling fronts for the reaction-diffusion equation

(1)

The function will be of three different forms in this paper. The first two concern non-linear terms where is , and is either of positive type, or of bistable type and is , and . The existence of traveling front depends on the value of . The third case we consider here is when for and for where are given parameters and is of bistable form and for . We study the existence of traveling fronts depending on the value of and .

The problems we study in this paper bear some similarities with the question of traveling fronts in cylinders of [BN]. However there are important differences that have to do with the fact that the cross section in [BN] was bounded and only the Neumann condition was considered there. Whereas here, the problem is posed in the whole space and the solution vanish at infinity in directions orthogonal to the direction of propagation. We follow the same general scheme as in [BN] and in particular make use of the sliding method. But some new ideas are also required. In particular, first, we treat directly the KPP case without the approximation of the KPP non-linearity by a combustion non-linearity as in [BN]. Then in the approach of Berestycki - Nirenberg [BN] to traveling fronts in cylinders for the bistable case, a useful result of H. Matano [HM79] was involved in the proof. Here, we rely on stability ideas but also use energy minimization properties to bound the speed of the solution in the finite domain approximation. In particular, we do not use the precise exponential behavior that was used in [BN]. Actually the developments of this method that we present in this paper can be used to somewhat simplify parts of [BN]. They can also be applied to traveling fronts in cylinder with Robin or Dirichlet boundary conditions. 111The construction of traveling fronts for Neumann and Dirichlet conditions in cylinders given by [Vega93] appears to be incomplete. Indeed, the continuity of the function on page 515 is not established so that using Dini’s Theorem to derive Lemma 3.2 there is not justified.

Equation (1) in the first case comes from a model in population dynamics [DFP03] that we briefly describe now. Let represent the density of individuals at time and position that possess some given quantitative phenotypic trait represented by a continuous variable . For example, the latter could be the size of wings or the height of an individual. We assume that individuals follow a brownian motion (i.e. they diffuse) in space with a constant diffusion coefficient , reproduce identically and disappear with a growth rate that depends on the position and on the trait . Furthermore, they also reproduce with mutation that is represented by a kernel and disappear due to competition with a constant . Thus, one is led to the following equation for :

(2)

We assume moreover that there exists a most adapted trait that may depends on the location . The farther the trait of an individual is from the most adapted trait, the larger the probability of dying and not reproducing. Thus the growth rate can be written for example as with and .

Non-local reaction-diffusion equations of this type raise some new difficulties from a mathematical standpoint as shown in [BNPR09]. There, behaviors that are quite different from those in local equations are brought to light. After this paper was completed, we learned that in [ACR] the existence of traveling front was also derived for equation

This work follows in part the methods of [BNPR09]. As in [BNPR09], the nonzero limiting stationary state is not prescribed. In a forthcoming numerical study [avenir_num], we study the full equation (2) and we discuss the monotonicity of fronts depending on the value of and .

In this paper, we introduce a simplified version of this model that emphasizes propagation guided by the environment. First, we assume that mutations are due to a diffusion process represented by a Brownian motion in the space of trait . Furthermore, we assume that is linear. Then a rotation in the variables allows one to reduce the problem to the case where the most adapted trait is . Therefore we assume and (2) can be rewritten as

(3)

Lastly we assume that competition is only between individuals sharing the same trait which leads us to equation

(4)

Equation (1) is a generalization of this equation. In [DFP03], the authors observe numerically a generalized transition front spreading along the graph of for equation (2) (see [BH06, BH07, BH12, HM12] or [Shen] for the definition of generalized transition fronts). Here we want to prove theoretically (i) that there exists such a front for equation (1) at least for some values of the parameter and (ii) that extinction occurs if is too large. The latter condition can be interpreted as saying that the “area” of adapted traits is too thin compared to the diffusion. To remain consistent with the biological motivation, we only consider here non-negative and bounded solutions of (1).

Other types of models related to this one have been proposed in the literature. For example, the model developed by Kirkpatrick and Barton in 1997 [KB97] also studies the evolution of a population and of its mean trait. The main difference is that they have a system in and where represents the population and the mean trait is described by a specific equation. This model has been further explored00 [FHB08, HBFF11]. It is worth noting that these models use the same type of non-linearity for the adaptation to the environment and model the mutation with the Laplace operator as well rather than integral operators.

This type of reaction-diffusion process in heterogeneous media also arises in many contexts in medicine. An important class of such models was treated in [C07, PMHC09]. They deal with the propagation of a cortical spreading depression (CSD) in the human brain. These CSD’s are transient depolarizations of the brain that slowly propagate in the cortex of several animal species after a stroke, a head injury, or seizures [somjen]. They also are suspected of being responsible for the aura in migraines with aura. CSD’s are the subject of intensive research in biology since experiments blocking them during strokes in rodents have produced very promising results [DeKeyser99, Nedergaard95]. These observations however have not been confirmed in humans and the existence of CSD’s in the human brain is still a matter of debate [Mayevsky96, Gorgi01, Back00, Strong02]. Since very few experiments and measurements on human brain are available be it for obvious ethical or technical reasons, mathematical models of a CSD is helpful in understanding their existence and conditions for their propagation. In such a problem, the morphology of the brain and thus the geometry of the domain where CSD’s propagate, is believed to play an important role.

The brain is composed of gray matter where neuron’s soma are and of the white matter where only axons are to be found. The rodent brain (on which many of the biological experiments are done) is rather smooth and composed almost entirely of gray matter. On the opposite, the human brain is very tortuous. The gray matter is a thin layer at the periphery of the brain with much thickness variations and convolutions, the rest of the brain being composed of white matter. According to mathematical models of CSDs [Cetal08, somjen, shapiro01, tuckwell80], the depolarization amplitude follows a reaction-diffusion process of bistable type in the gray matter of the brain while it diffuses and is absorbed in the white matter of the brain. The modeling of CSD hence leads one to the study of equations of the following type:

(5)

Here, is of bistable type and corresponds to the transition from gray matter to white matter. This equation is of type (1) and we also study it here in sections 7 and 8 where we extend earlier works on the subject. In [C07], this equation was studied to prove that the thinness of the human gray matter ( small) may prevent the creation or the propagation of CSDs on large distances. It was proved by studying the energy in a traveling referential of the solution of (5) with a specific initial condition. The special case of (5) for was described more completely in [CJ11]. In [PMHC09], a numerical study shows that the convolutions of the brain have also a strong influence on the propagation of CSD. In [CG05], the effect of rapid variations of thickness of the gray matter was studied.

Lastly, let us note that the same kind of equation arises in the modeling of tumor cords but with a slightly more complicate KPP non-linearity. We plan to investigate this model in our forthcoming work [avenir_cancer].

As already mentioned, the study of propagation of fronts and spreading properties in heterogeneous media is of intense current interest. For instance, the existence of fronts propagating in non-homogeneous geometries with obstacles has been established in Berestycki, Hamel and Matano [BHM09]. Definitions of generalized waves have been given by Berestycki and Hamel in [BH07] and [BH12] where they are called generalized transition waves. Somewhat different approaches to generalizing the notions of traveling fronts have been proposed by H. Matano [HM12] and W. Shen [Shen]. The existence of fronts for non-homogeneous equations are established in [NRRZ12] and [Z12].

Let us first introduce some notations before stating the main results.

Notation.

We note where and . Hence is the space variable in , is its first coordinate and is the vector of composed of all the other coordinates of . As usual denotes a ball of radius centered at 0, but here it will always mean the ball in .

First we are interested in solutions of

(6)

with . We will assume that and satisfies either one of the following conditions:

or

The first case will be referred to as the positive case and the second one will be called bistable case. Furthermore, if is in the positive case and if

we will say that is of Fisher-KPP type. Since we are only interested in solutions of (6) in , we will further assume that for . Moreover we assume

(7)

(except in section 3.2 where can vanish) and

(8)

Taking and yields the particular case of equation (4).

This paper is concerned with the long term behavior of (6) and with the existence of curved traveling fronts, i.e. solutions with a constant and such that the limits exist uniformly and are not equal. Regarding these fronts, our main results are the following.

Theorem 1.1.

If is of Fisher-KPP type, there exists such that:

  • For , there exists no traveling front solution of (6),

  • For there exists a threshold such that there exists a traveling front of speed of equation (6) if and only if .

This existence theorem gives us information on the behavior of the solution of the parabolic problem. In this paper we prove the following theorem:

Theorem 1.2.

If is of Fisher-KPP type, for , there exists a unique solution of

  • If , it verifies uniformly with respect to .

  • If and is compactly supported with where is the unique positive asymptotic profile (stationary solution), then

This means that there is a threshold value such that for , there is extinction. On the contrary, when , there is spreading and the state invades the whole space. The asymptotic speed of spreading is then . The property of asymptotic spreading is in the same spirit of the theorem of asymptotic speed of spreading in cylinders established by Mallordy and Roquejoffre in [MR95].

Theorem 1.2 has interesting consequences for the dynamics of the phenotypic diversity in a population. Several studies have tried to understand population migrations through phenotypic diversity [Excoffier09, Hallatschek07, Hallatschek08, Hallatschek10, Roques12, Vlad04]. Our invasion result states that for large times, one expect to see the state at any location (and not the migration process) and it holds whatever the initial distribution of the population is. Note furthermore that the profile is unique. Hence whatever the initial structure of the population is, the phenotypic diversity at large times is completely determined by the profile of the function .

In the slightly more general case of a positive non-linearity, we will prove the following existence theorem.

Theorem 1.3.

If is of positive type, there exists such that for there exists a traveling front of equation (6).

Regarding the case of bistable we have the following result:

Theorem 1.4.

If is of bistable type, there exist such that

  • For , there exists no traveling front solution of (6),

  • For , under condition 43 of Section 7, there exists a traveling front of speed solution of (6).

Lastly, the model for CSD’s leads one to equations of the type

(9)

where verifies

where and are given parameters and is of bistable form.

In this paper we prove the following Theorem.

Theorem 1.5.

There exist critical radii with the following properties:

  • For , there is no traveling front solution of (9).

  • For (independently of ), assuming that there is a unique stable asymptotic profile of (53), there exists a traveling front of speed solution of (9).

The assumption on the uniqueness of the asymptotic profile is proved to be true for the case , and for . This is done in [CJ11] by phase plane method. For want of a uniqueness result for the profile equation in more general cases,

This theorem completes the study in [C07] on the existence of CSD in the human brain. Indeed in [C07] the transition from gray to white matter was instantaneous when biologically there is a smooth transition from gray to white matter. This Theorem confirms the intuition that CSD’s can be found in part of the human brain where the gray matter is sufficiently thick but they can not propagate over large distances due to a thin gray matter in many parts of the human brain.

The paper is organized as follows. In section 2 we state some preliminary results that will be used in the sequel. Section 3 is dedicated to the study of the existence and uniqueness of non-zero asymptotic profiles for a traveling front solution of (6). In section 5 we study the large time behavior. There we prove extinction if and convergence towards the front of minimal speed if . Section 6 extends existence of traveling front results to the case of a positive non-linearity. Then, section 7 is devoted to the study of the asymptotic profiles in the bistable case and section 8 to the existence of traveling front for in the bistable case. Lastly, in section 9 we describe the precise problem arising in the modeling of CSD’s and state our main result in this framework.

2 Preliminary results

In our proofs, we will need several times the exponential decay of the asymptotic profile which can be easily proved from the following theorem established in [BR08].

Theorem 2.1.

Let be a positive function. Assume that there exists and such that

Then, .

This result is established in [BR08], lemma 2.2. In the context of equation (1), we thus have the following corollary.

Corollary 1.

Let be a non-negative and bounded solution of

Then, for any there exists such that

Proof.

The estimate on comes directly from Theorem 2.1 and the estimate on derives from standard global estimates. ∎

3 The case of a Fisher-KPP non-linearity. Asymptotic profiles.

In this section, we are interested in the asymptotic profiles of a traveling front solution of (6) as . Hence, we are looking for solutions of the following equation

(10)

We assume that ,

(11)

and

(12)

Since the constant function 0 is always a solution, the problem is to know when there exist non-zero solutions. As we will see here, the existence of such a positive asymptotic profile is characterized by the sign of the principal eigenvalue of the linearized operator around 0. We now make this notion precise.

3.1 Principal eigenvalue of the linearized operator

To start with, let us define the natural weighted space

and its associated norm. For , we set . The linearized operator about 0 is for . We are interested in the eigenvalues of . Even though the problem is set on all of , the term in yields compactness of the injection . Hence the existence of a principal eigenvalue is obtained as usual.

Theorem 3.1.

Let us define

The operator has a smallest eigenvalue

(13)

Moreover there exists a unique positive eigenfunction associated with of -norm equal to 1, called in the following. The eigenspace associated with is spanned by .

The proof is classical due to the compactness of . We refer for example to [Evans].

Remark 1.

If , the problem can be rescaled and we obtain the harmonic oscillator for which principal eigenvalue and eigenfunction are well known [schwartz]. In that case, and .

Since the existence of a positive solution of (10) will depend on the sign of the principal eigenvalue, the following proposition describes the behavior of as a function of .

Proposition 1.

The function is continuous, increasing and concave for . Moreover and for large enough .

Proof.

Let us fix and . Equation (13) shows that

Similarly, we obtain . From this we derive:

This and similar computation for yields that is increasing and locally Lipschitz on .

Concavity is classical. It suffices to observe that for each fixed ,

is an affine function of and that .

In order to prove that , for any choose a function of compact support with and . Let . From (13) we get

So for any ,

Now we claim that for large enough . Argue by contradiction and assume that for all . Since

we get

and in for all . Furthermore, is bounded in and up to extraction we can assume that converges strongly in , thus converges to in but this is impossible since for all . ∎

Corollary 2.

There exists such that for , and for .

3.2 If vanishes on

The main part of the proof still holds if vanishes on but the result is slightly modified.

In this section, we assume that there exists such that (7) is substituted by the following assumption

(14)

We define the principal eigenvalue of the Laplacian on with Dirichlet boundary conditions, i.e.

In this case, the principal eigenvalue of the linearized operator about 0 is well defined and Proposition 1 becomes

Proposition 2.

The function is continuous, increasing and concave for , and . Now there are two cases:
i) If , then for large enough .
ii) If , then for all .

Proof.

The proof of the first part of the proposition is exactly the same as in Proposition 1. We just have to prove i) and ii).

i) We assume that and argue by contradiction assuming that for all . As in the proof of proposition 1, we have

and this yields in for but now for all only.

As before is bounded in and up to extraction, we have , weak convergence in and strong convergence in of to . The limit verifies , for and

Thus must coincide with in and leading to since . This is a contradiction.

ii) By taking in the Rayleigh quotient (13), where is the principal eigenvalue of the above problem in with Dirichlet boundary conditions, we see that for all . ∎

In the following, we will not state the results specifically for this case (14) and will rather assume (7). However, the proofs and results developed here carry over to this case with the obvious modifications.

3.3 Existence of non-zero asymptotic profile

Theorem 3.2.

For , there is no solution of (10), where is defined in corollary 2. For , there exists a unique positive solution of (10).

Proof.

Let us fix . Then . Assume by contradiction that there exists a solution of (10). Then the strong maximum principle shows that .

Since is an eigenfunction of the linearized operator and is solution of (10), we have

Now from corollary 1, and are rapidly decreasing for and so we can apply Stokes formula . It yields but since is of Fisher-KPP type and thus a contradiction is obtained.


We now turn to the case . For , the eigenvalue is negative. Setting with , we get

if is chosen small enough. Hence is a sub-solution of (10). The constant function 1 is a super-solution and if is small enough. Therefore by the sub- and super-solution method, there exists a solution such that .

Now consider and two non-zero solutions of (10). We argue by contradiction and assume that . Then for example is not empty. Introduce a cutoff function with on , on and on and for all , let us set . Using equation (10), we have

by Lebesgue’s dominated convergence theorem. Owing to corollary 1, , , and have exponential decay and thus Stokes formula can be applied and we obtain

In the term the integrand satisfies

Therefore by Lebesgue’s Theorem of dominated convergence, we infer that . Next the term satisfies . Consequently, we may write:

which is a contradiction in view of (12) as in . Hence and the non-zero solution is unique. ∎

The last point concerns the stability of the asymptotic profiles for . Let us start by studying the energy of . For , we define the energy

(15)

where .

Theorem 3.3.

For , the unique positive solution of (10) is stable in the energy sense, i.e. is the global minimum of and, furthermore .

Proof.

Owing to the maximum principle, solutions of (10) are between 0 and 1. Hence we can modify on such that it becomes odd and as a consequence, can be considered as even. Since the principal eigenvalue of the linearized operator about the zero solution is negative for , 0 cannot be the global minimum of . Now admits a global minimum that will be called for the argument. One can prove that is also a global minimum of and hence is a positive solution of (10). By uniqueness, and thus is a global minimum of . Since is not a global minimum, necessarily . ∎

We now conclude with the linearized stability of .

Theorem 3.4.

For , consider the linearized operator about and denote the principal eigenvalue of this operator. Then .

Proof.

Denote by a positive eigenfunction associated with and assume by contradiction that . If , it is easy to see that for small enough is a sub-solution of (10). From there, it would follow that there exists a solution of (10) between and but this contradicts the uniqueness of .

Now if ,letting be as above, we get

(16)

From the equation and since is unique for every given , it is clear that is differentiable with respect to and that satisfies:

(17)

We know that and from (17) which shows that , we actually see from the maximum principle that in . It is also easily seen that has exponential decay at infinity. From (16) and (17), it then follows that which is a contradiction. Hence .

4 Traveling fronts for a Fisher-KPP non-linearity

This section is devoted to the definition of a speed for which a traveling front of equation (6) exists for . The threshold of existence of the non-zero asymptotic profile is called as in the previous section. For , denotes the unique non-zero asymptotic profile. As shown in the previous section, the energy of the non-zero profile is negative.

A curved traveling front of speed is a function solution of equation (6) and connecting the non-zero asymptotic state to 0. Thus we are looking for a solution of

(18)

where is also an unknown of the problem.

The construction of in Theorem 1.1 uses the sliding method following ideas of [BN]. Note however that there are important differences with [BN]. In that paper, the Fisher-KPP case is derived by first solving the “combustion non-linearity” and then approach the Fisher-KPP non-linearity as a limiting case of truncated functions. Contrary to [BN] here, we derive directly the existence of a solution of the Fisher-KPP case. Actually the method we present here can be applied to somewhat simplify the proof of [BN] in the Fisher-KPP case for cylinder with Neumann conditions.

4.1 Problem on a domain bounded in .

Let us fix and for this subsection and consider the following problem:

(19)

The aim of this subsection is to prove the following theorem:

Theorem 4.1.

There exists a unique solution of (19), denoted in the following. This solution decreases in the -direction, i.e. . Thus on . Moreover is decreasing and continuous from to .

To prove this theorem, we require the following two propositions.

Proposition 3.

Let be a solution of (19). Then for .

Proof.

Let be such that and consider defined on the largest solution of

(20)

Here we think of as having been extended by outside . Since for all , we observe that:

  • by the strong maximum principle, on .

  • since and is a sub-solution of (20), through monotone iterations we have .

  • if , is once again a sub-solution of (20) on and thus on .

  • therefore tends to a function when and through local elliptic estimates, this function is a non-zero solution () of the asymptotic problem (10). By uniqueness, we obtain

Now we consider the problem

(21)

The solution of (19) is a sub-solution of (21) and the constant function is a super-solution. Using monotone iterations starting from the super-solution , we build the same sequence as previously (for problem (20)) since by induction the solutions do not depend on . Hence the sequence converges toward and we have . Now letting yields . ∎

Proposition 4.

Let be such that for where is the Lipschitz norm of on . We set where is an open bounded interval of .

Suppose and are solutions of

(22)

and on . Then on .

Proof.

By contradiction, suppose this is not true. Due to corollary 1 and proposition 3, and converge uniformly to 0 for . Consequently, there exist such that

Since , we have and , and subtracting the equation (22) with from the one with , we obtain

which is impossible since and . ∎

Let us now turn to the proof of Theorem 4.1 using sliding method.
First is a super-solution, 0 is a sub-solution and , so by monotone iterations, there exists a solution of (19).

Lemma 4.2.

Assume and are two solutions of (19). Then

Proof of the lemma.

By proposition 3, we have (resp. ) and using the strong maximum principle, we obtain (resp. ) on .

For , let and for , set .

Let us fix such that for . By compactness and continuity of and , there exists such that on for any