A first main element in the proof of Theorem 1.5 is the fact that under the assumptions of the theorem, any viscosity solution u to (1.19) is indeed smooth and strictly concave. We prove indeed such result for an unconstrained Hamilton-Jacobi equation where R(z, I)is replaced by R(t, z):
(ut=|∇u|2+R(t, z) (t >0, z∈Rd),
u(0, z) =u0(z). (1.26)
To present our results on (1.26) we provide our assumptions onR(t, z)which are closely related to the ones onR(z, I).
Assumptions on R(t, z):
We choose R to be continuous in t and to have first and second derivatives with respect to z,
Chapter 1. A Hamilton-Jacobi approach for models from evolutionary biology 10 that are continuous both with respect totandz. We suppose that
−B0−B1|z|2≤R(t, z)≤B0−B1|z|2, for t∈R+, (1.27)
−2B1≤D2R(t, z)≤ −2B2<0 as symmetric matrices, (1.28) kD3R(t,·)kL∞(Rd)≤B3, fort∈R+. (1.29) Theorem 1.6([MR16]). Assume (1.20)–(1.22)and (1.27)–(1.29). Equation (1.26)has a unique viscosity solution u that is bounded from above. Moreover, it is a classical solution: u ∈ L∞loc R+;Wloc3,∞(Rd)
The existence and uniqueness of viscosity solution to (1.26) is rather classical. The main ingre-dient to prove the regularity properties above is to use the fact that the viscosity solution to (1.26) is given by the dynamic programming principle:
u(t, z) = sup and the fact thatu0 andRare strictly concave functions.
The second step of the proof is to show that (1.19) is equivalent with the following slightly nonstandard differential system: Note that (1.30) is really a differential system because the assumptions on R imply that I(t) can implicitely be expressed in terms ofz(t). And it is slightly nonstandard because¯ z¯solves an ODE whose nonlinearity depends onu. The precise statement is the following
Theorem 1.7 ([MR16]). Solving the constrained problem (1.19) is equivalent to solving the initial value ODE-PDE problem (1.30)–(1.31).
Finally the last step is to prove that (1.30)–(1.31) has a unique solution. This is obtained using a Banach fixed point argument, by defining an adapted mapping and proving, thanks to some technical computations, that such mapping is a strict contraction.
Chapter 2
Fractional reaction-diffusion equations
2.1 Introduction
In this chapter, we are interested in the study of the following selection-mutation model where the mutations are modeled by a fractional laplacian:
(∂tn+σ(−∆)αn=n R(z, ρ(t)), (t, z)∈R+×Rd n(t= 0,·) =n0(·), ρ(t) =R
Rdn(t, z)dz, (2.1)
with0< α <1and
(−∆)αn(t, z) = p.v.
Z
Rd
n(t, z)−n(t, z+h) dh
|h|d+2α.
Here, as in Chapter 1, z corresponds to a phenotypic trait and n represents the phenotypic density. The total size of the population is denoted byρ. The functionR represents the growth rate of individuals with trait z, and it is a decreasing function with respect to the total size of the populationρ, taking into account in this way competition between the individuals. The difference with (1.1) is that the mutations are modeled by a fractional laplacian instead of a classical laplacian. The choice of a fractional laplacian rather than a classical laplacian or an integral kernel with thin tails, allows to take into account large mutation jumps with a high rate.
Such equation has been derived from stochastic individual based models, in the case where the mutation distribution has algebraic tails [JMW12].
We are interested in an asymptotic analysis of the above equation to understand the effect of large mutation jumps on the dynamics of the phenotypic density. Is there still a rescaling under which one would observe the concentration of the phenotypic density as a Dirac mass? Can the Hamilton-Jacobi approach be extended to study mutation integral kernels with thick tails?
Models closely related to (2.1) but considering regular integral kernels with thin tails or with compact support, instead of the fractional laplacian, have been previously studied (see for in-stance [BMP09, Rao11, GHGR17]). However, not much is known for the models with heavy tailed integral kernels. The presence of large mutations with a high probabilty leads to drasti-cally faster dynamics and the previous scalings are not adapted to the study of such problems.
13
Chapter 2. Fractional reaction-diffusion equations 14 With S. Méléard, in order to obtain a relevant rescaling and to perform an asymptotic analysis for the fractional selection-mutation equation, we first studied a fractional Fisher-KPP equation:
∂tn+ (−∆)αn=n(1−n).
The level sets of the solution to such equation propagate with a speed that is exponential in time [CCR12, CR13]. Note that such propagation is much faster compared to the classical Fisher-KPP equation where the propagation has constant speed [Fis37, Fisher-KPP37, AW78]. In [MM15] we provided an unconventional rescaling that allowed us to provide an asymptotic analysis of this equation and capture the exponential speed of propagation associated to this equation. This work extends previous works on the asymptotic analysis of propagation phenomena, related to large deviation principles [FW98], in classical reaction-diffusion equations to the case of diffusion kernels with fat tails (see for instance [Fre85a, Fre86, ES89]).
Next, being inspired by such analysis we introduced a rescaling for the fractional selection-mutation model which leads to the concentration of the phenotypic density as a single Dirac mass, similarly to the case of classical selection-mutation models [MM15, Mir]. However, signifi-cant difficulties comparing to previous works arise from the fractional laplacian term which lead to considerably different analysis. In particular, the logarithmic transform of the phenotypic density does not converge anymore to a viscosity solution of a Hamilton-Jacobi equation, but to a viscosity supersolution which is minimal in a certain class of functions. Moreover, the limiting Hamilton-Jacobi equation can take infinite values.
The result in [MM15] on the fractional KPP equation has been extended in several directions.
With J.-M. Roquejoffre, we supervise a PhD student, A. Léculier, on reaction-diffusion equations with nonlocal diffusion operators. In a first work [L´19], he has extended the result in [MM15]
on the fractional Fisher-KPP equation to more general stable operators and considering periodic media (see also [CCR12, ST19]). Together with A. Léculier and J.-M. Roquejoffre we also have expanded this approach to study fractional KPP type equations in a periodically hostile environ-ment. Such model describes the nonlocal dispersion of a population in a domain with periodic presence of obstacles.
In [BGHP18], Bouin et al. extended also this method to other dispersion kernels without sin-gularity but with other types of decay. It is expected that the more recent work [Mir] on the fractional selection-mutation model can also be extended to such integral kernels following the same type of adaptation.
In Section 2.2 we present the asymptotic analysis of the fractional Fisher-KPP equation, which is based on the work in [MM15]. In Section 2.3 we go back to the case of the fractional selection-mutation model (2.1) and provide our results on the derivation of a Hamilton-Jacobi equation with constraint and the concentration of nε as a Dirac mass. This part is based on the work in [Mir]. Finally, in Section 2.4 we present a result on the fractional Fisher-KPP equation in a periodically hostile environment [LMR].