arXiv subjects

Jérôme Coville

Publications and source records attributed to Jérôme Coville.

22 records · Page 2Linked to original sources

arXiv2016

Propagation phenomena in monostable integro-differential equations: acceleration or not?

We consider the homogeneous integro-differential equation$\partial \_t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)\textgreater{}0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

math.AP↗

arXiv2016

On the definition and the properties of the principal eigenvalue of some nonlocal operators

In this article we study some spectral properties of the linear operator $\mathcal{L}\_Ω+a$ defined on the space $C(\barΩ)$ by :$$ \mathcal{L}\_Ω[φ] +aφ:=\int\_ΩK(x,y)φ(y)\,dy+a(x)φ(x)$$ where $Ω\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue $λ\_p(\mathcal{L}\_Ω+a)$ defined by $$λ\_p(\mathcal{L}\_Ω+a):= \sup\{λ\in \mathbb{R} \,|\, \exists φ\in C(\bar Ω), φ\textgreater{}0, \textit{such that}\, \mathcal{L}\_Ω[φ] +aφ+λφ\le 0 \, \text{in}\;Ω\}. $$ We establish some new properties of this generalised principal eigenvalue $λ\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $λ\_p(\mathcal{L}\_Ω+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $λ\_{p} \left(\mathcal{L}\_{σ,m,Ω} -\frac{1}{σ^m}+a\right)$ where $\mathcal{L}\_{σ,m,Ω}$ is defined by $\displaystyle{\mathcal{L}\_{σ,m,Ω}[φ]:=\frac{1}{σ^{2+N}}\int\_ΩJ\left(\frac{x-y}σ\right)φ(y)\, dy}$. In particular, we prove that $$\lim\_{σ\to 0}λ\_p\left(\mathcal{L}\_{σ,2,Ω}-\frac{1}{σ^{2}}+a\right)=λ\_1\left(\frac{D\_2(J)}{2N}Δ+a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $λ\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $φ\_{p,σ}$.

math.AP↗

arXiv2015

Concentration Phenomenon in Some Non-Local Equation

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\partial\_t u(t, x) = \left(a(x) -- \int\_Ω k(x, y)u(t, y) dy\right ) u(t, x) + \int\_Ω m(x, y)[u(t, y) -- u(t, x)] dy\quad \text{ for}\quad (t, x) $\in$ \mathbb{R}\_{+} \times Ω,$$ together with the initial condition $u(0, \cdot) = u0 \quad \text{ in }\quad Ω$. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values $Ω$ is a bounded subset of $\mathbb{R}^N$ , the kernels $k$ and $m$ respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on $Ω$. Moreover, when this '' stationary '' measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in $L^{\infty} (Ω) \cap L^1 (Ω)$, the solution of the Cauchy problem converges to this limit measure in $L^2 (Ω)$. We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.

math.AP↗

arXiv2012

Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c

math.AP↗