Hyperbolic traveling waves driven by growth
We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed (), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter : for small the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large the traveling front with minimal speed is discontinuous and travels at the maximal speed . The traveling fronts with minimal speed are linearly stable in weighted spaces. We also prove local nonlinear stability of the traveling front with minimal speed when is smaller than the transition parameter.
Key-words: traveling waves; Fisher-KPP equation; telegraph equation; nonlinear stability.
AMS classification: 35B35; 35B40; 35L70; 35Q92
We consider the problem of traveling fronts driven by growth (e.g. cell division) together with cell dispersal, where the motion process is given by a hyperbolic equation. This is motivated by the occurence of traveling pulses in populations of bacteria swimming inside a narrow channel [1, 38]. It has been demonstrated that kinetic models are well adapted to this problem . We will focus on the following model introduced by Dunbar and Othmer  (see also Hadeler ) and Fedotov [16, 17, 18]
The cell density is denoted by . The parameter is a scaling factor. It accounts for the ratio between the mean free path of cells and the space scale. The growth function is subject to the following assumptions (the so-called monostable nonlinearity)
For the sake of clarity we will sometimes take as an example the logistic growth function .
Equation (1.1) is equivalent to the hyperbolic system
The expression of can be computed explicitly in terms of as follows,
Since the pioneering work by Fisher  and Kolmogorov-Petrovskii-Piskunov , dispersion of biological species has been usually modelled by mean of reaction-diffusion equations. The main drawback of these models is that they allow infinite speed of propagation. This is clearly irrelevant for biological species. Several modifications have been proposed to circumvent this issue. It has been proposed to replace the linear diffusion by a nonlinear diffusion of porous-medium type [40, 31, 35]. This is known to yield propagation of the support at finite speed [32, 33]. The density-dependent diffusion coefficient stems for a pressure effect among individuals which influences the speed of diffusion. Pressure is very low when the population is sparse, whereas it has a strong effect when the population is highly densified. Recently, this approach has been developped for the invasion of glioma cells in the brain . Alternatively, some authors have proposed to impose a limiting flux for which the nonlinearity involves the gradient of the concentration [3, 8, 4].
The diffusion approximation is generally acceptable in ecological problems where space and time scales are large enough. However, kinetic equations have emerged recently to model self-organization in bacterial population at smaller scales [2, 34, 14, 30, 36, 38, 39]. These models are based on velocity-jump processes. It is now standard to perform a drift-diffusion limit to recover classical reaction-diffusion equations [27, 10, 14, 28]. However it is claimed in  that the diffusion approximation is not suitable, and the full kinetic equation has to be handled with. Equation (1.1) can be reformulated as a kinetic equation with two velocities only (see (2.1) below). This provides a clear biological interpretation of equation (1.1) as a simple model for bacteria colonies where bacteria reproduce themselves, and move following a run-and-tumble process.
Hyperbolic models coupled with growth have already been studied in [13, 26, 23, 11]. In  it is required that the nonlinear function in front of the time first derivative is positive (namely here, ). Indeed, this enables to perform a suitable change of variables in order to reduce to the classical Fisher-KPP problem. In our context this is equivalent to since is concave. In  this nonlinear contribution is replaced by 1: the authors study the following equation (damped hyperbolic Fisher-KPP equation),
The long time behaviour of such equation is well understood since the pioneering works by Kolmogorov-Petrovsky-Piskunov  and Aronson-Weinberger . For nonincreasing initial data with sufficient decay at infinity the solution behaves asymptotically as a traveling front moving at the speed . Moreover the traveling front solution with minimal speed is stable in some weighted space .
In this work we prove that analogous results hold true in the parabolic regime . Namely there exists a continuum of speeds for which (1.1) admits smooth traveling fronts. The minimal speed is given by 
Obviously we have . There also exists supersonic traveling fronts, with speed . This appears surprising at first glance since the speed of propagation for the hyperbolic equation (1.1) is (see formulation (1.3) and Section 2). These fronts are essentially driven by growth, since they travel faster than the maximum speed of propagation. The results are summarized in the following Theorem.
Theorem 1 (Parabolic regime).
Assume that . The following alternatives hold:
There exists no smooth or weak traveling front of speed .
For all , there exists a smooth traveling front solution of (1.1) with speed .
For there exists a weak traveling front.
For all there also exists a smooth traveling front of speed .
There is a transition occuring when . In the hyperbolic regime the minimal speed speed becomes:
On the other hand, the front traveling with minimal speed is discontinuous as soon as . In the critical case there exists a continuous but not smooth traveling front with minimal speed .
Theorem 2 (Hyperbolic regime).
Assume that . The following alternatives hold:
There exists no smooth or weak traveling front of speed .
There exists a weak traveling front solution of (1.1) with speed . The wave is discontinuous if .
For all there exists a smooth traveling front of speed .
We conclude this introduction by giving the precise definition of traveling fronts (smooth and weak) that will be used throughout the paper.
In the following Section 2 we show some numerical simulations in order to illustrate our results. Section 3 is devoted to the proof of existence of the traveling fronts in the various regimes (resp. parabolic, hyperbolic, and supersonic). Finally, in Section 4 and Section 5 we prove the stability of the traveling fronts having minimal speed . We begin with linear stability (Section 4) since it is technically better tractable, and it let us discuss the case of the hyperbolic regime. We prove the full nonlinear stability in the range (parabolic regime) in Section 5.
2 Numerical simulations
In this Section we perform numerical simulations of (1.1). We choose a logistic reaction term: . We first symmetrize the hyperbolic system (1.3) by introducing and . This results in the following system:
In other words, the population is split into two subpopulations: , where the density denotes particles moving to the right with velocity , whereas denotes particles moving to the left with the opposite velocity.
We discretize the transport part using a finite volume scheme. Since we want to catch discontinuous fronts in the hyperbolic regime , we aim to avoid numerical diffusion. Therefore we use a nonlinear flux-limiter scheme [25, 12]. The reaction part is discretized following the Euler explicit method.
The non-linear reconstruction of the slope is given by
We compute the solution on the interval with the following boundary conditions: and . The discretization of the second equation for (2.1) is similar. The CFL condition reads . It degenerates when , but we are mainly interested in the hyperbolic regime when is large enough. Other strategies should be used in the diffusive regime , e.g. asymptotic-preserving schemes (see [19, 9] and references therein).
Results of the numerical simulations in various regimes (parabolic and hyperbolic) are shown in Figure 1.
3.1 Characteristic equation
We begin with a careful study of the linearization of (1.7) around . We expect an exponential decay as . The characteristic equation reads as follows,
The discriminant is . Hence we expect an oscillatory behaviour in the case , i.e. . We assume henceforth . In the case (subsonic fronts) we have to distinguish between the parabolic regime and the hyperbolic regime . In the former regime equation (3.1) possesses two positive roots, accounting for a damped behaviour. In the latter regime equation (3.1) possesses two negative roots. In the case (supersonic fronts) we get two roots having opposite signs.
Next we investigate the linear behaviour close to . We expect an exponential relaxation as . The characteristic equation reads as follows,
We have . In the case equation (3.2) possesses two roots having opposite signs. In the case it has two positive roots.
We summarize our expectations about the possible existence of nonnegative traveling fronts in Table 1.
|parabolic||if , NO if , YES||YES|
In this section we prove that no traveling front solution exists if the speed is below .
Note that the proof below works in both cases and .
We argue by contradiction. The obstruction comes from the exponential decay at . Assume that there exists such a traveling front . As , one has in the parabolic as well as in the hyperbolic regime. Hence, as is bounded and satisfies the elliptic equation (1.7) in the sense of distributions, classical regularity estimates show that is smooth. It is necessarily decreasing as soon at it is below . Otherwise, it would reach a local minimum at some point , for which , and . It would then follow from (1.7) that and thus . As , the Cauchy-Lipschitz theorem would imply , a contradiction.
Next, we define the exponential rate of decay at :
Consider a sequence such that and define the renormalized shift:
This function is locally bounded by classical Harnack estimates. It satisfies
As , and is concave, the functions and are uniformly bounded, uniformly in . Hence, Schauder elliptic regularity estimates yield that the sequence is locally bounded in the Hölder space for any compact subset and any . The Ascoli theorem and a diagonal extraction process give an extraction, that we still denote , such that converges to some function in for any compact subset and any . The limiting function is a solution in the sense of distributions of
As this equation is linear, one has . If , then as is nonnegative, one would get and thus by uniqueness of the Cauchy problem, which would be a contradiction since . Thus is positive.
3.3 Proof of Theorem 1.(b): Existence of smooth traveling fronts in the parabolic regime
In  the author proves the existence of traveling front, by reducing the problem to the classical Fisher-KPP problem. It is required that the nonlinear function remains positive, which reads exactly in our context. We present below a direct proof based on the method of sub- and supersolutions, following the method developed by Berestycki and Hamel in .
3.3.1 The linearized problem
3.3.2 Resolution of the problem on a bounded interval
For all and , there exists a solution of
Moreover, this function is nonincreasing over and it is unique in the class of nonincreasing functions.
In order to prove this result, we consider the following sequence of problems:
is solution to
where is defined in Proposition 6 and is large enough so that is increasing.
The sequence is well-defined. The functions are nonincreasing and for all , the sequence is nonincreasing.
We prove this Lemma by induction. Clearly, is nonincreasing. First, one can find a unique weak solution of
using the Lax-Milgram theorem and noticing that the underlying operator is coercive since and .
Let . As is a supersolution of equation (1.7), one has
As , the weak maximum principle gives , that is, .
Define the constant function . It satisfies
in since is increasing and by monotonicity of . The same arguments as above lead to .
Assume that Lemma 8 is true up to rank . The existence and the uniqueness of follow from the same arguments as that of . Let . As is concave and , we know that . As is nonincreasing, we thus get
Hence, and thus . Similarly, one easily proves that in .
Differentiating (3.5) and denoting , one gets
since is increasing and is nonincreasing. As is concave, the zeroth-order term is positive and thus the elliptic maximum principle ensures that reaches its maximum at or at . But as for all , one has
and similarly . Thus , meaning that is nonincreasing. ∎
Proof of Proposition 7..
As the sequence is decreasing and bounded from below, it admits a limit as . It easily follows from the classical regularity estimates that satisfies the properties of Proposition 7.
If and are two nondecreasing solutions of (3.4), then the same arguments as before give that in for all . Hence, and a symmetry argument gives . ∎
For all , there exists such that .
Define . It follows from the classical regularity estimates and from the uniqueness of that is a continuous function. Moreover, as is nonincreasing, one has
where is defined in Proposition 6. As as and as locally uniformly on , one has and . The conclusion follows. ∎
3.3.3 Existence of traveling fronts with speeds
We conclude by giving the proof of Theorem 1 as a combination of the above results.
Proof of Theorem 1..
Consider a sequence such that and define for all . This function is decreasing and satisfies , and
Similar arguments as in the proof of Proposition 4 yield that the sequence converges in as to a function , up to extraction. Then satisfies
it is nonincreasing, and .
Define . Passing to the (weak) limit in the equation satisfied by , one gets . As , the hypotheses on give . On the other hand, as is nonincreasing, one has
We conclude that and . ∎
The following classical inequality satisfied by the traveling profile will be required later.
The traveling profile satisfies: , where is the smallest positive root of (3.1).
We introduce . It is nonnegative, and it satisfies the following first-order ODE with a source term
Since is concave, satisfies the differential inequality
The right-hand-side is the characteristic polynomial of the linearized equation (3.1). Moreover the function verifies . Hence a simple ODE argument shows that . ∎
3.4 Proof of Theorem 1.(c): Existence of weak traveling fronts of speed in the parabolic regime
The aim of this Section is to prove that in the parabolic regime , there still exists traveling fronts in the limit case but in the weak sense.
Assume that . Then there exists a weak traveling front of speed .
Let for all large enough so that . We know from the previous Section that we can associate with the speed a smooth traveling front and that we can assume, up to translation, that . Multiplying equation (1.7) by and integrating by parts over , one gets
Hence, as , the sequence is bounded in and one can assume, up to extraction, that it admits a weak limit in . It follows that the sequence converges locally uniformly to . Passing to the limit in (1.7), we get that this function is a weak solution of
which ends the proof. ∎
3.5 Proof of Theorem 2.(b): Existence of weak traveling fronts of speed in the hyperbolic regime
In this Section we investigate the existence of traveling fronts with critical speed in the hyperbolic regime .
Proof of Theorem 2..
The function is concave, and vanishes when . Furthermore, and . We now distinguish between the two cases and .
First case: . As is decreasing, there exists a unique such that vanishes.
Second case: . The only root of is . In this case we set .
For both cases, we have for all since is strictly concave and . Hence, for all . Set the maximal solution of
Let be the (maximal) interval of definition of , with , and
1- Conclusion of the argument in the first case: .
Since , we have necessarily . From (3.7), is decreasing on . Thus, we have as . Moreover, one easily gets .
We set and we extend by over . We observe that is a weak solution, in the sense of distributions, of
since and .
Up to space shifting , we may assume that the discontinuity arises at .
Example: the case and .
The traveling profile solves
The constant is determined by the condition . Finally the traveling profile satisfies the following implicit relation: