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

Can a population survive in a shifting environment using non-local dispersion

In this article, we analyse the non-local model : $\partial$ t U (t, x) = J $\star$ U (t, x) -- U (t, x) + f (x -- ct, U (t, x)) for t > 0, and x $\in$ R, where J is a positive continuous dispersal kernel and f (x, s) is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population) and shifted through time at a constant speed c. For compactly supported dispersal kernels J, assuming that for c = 0 the population survive, we prove that there exists a critical speeds c * ,$\pm$ and c * * ,$\pm$ such that for all --c * ,-- < c < c * ,+ then the population will survive and will perish when c $\ge$ c * * ,+ or c $\le$ --c * * ,--. To derive this results we first obtain an optimal persistence criteria depending of the speed c for non local problem with a drift term. Namely, we prove that for a positive speed c the population persists if and only if the generalized principal eigenvalue $\lambda$ p of the linear problem cD x [$\Phi$] + J $\star$ $\Phi$ -- $\Phi$ + $\partial$ s f (x, 0)$\Phi$ + $\lambda$ p $\Phi$ = 0 in R, is negative. $\lambda$ p is a spectral quantity that we defined in the spirit of the generalized first eigenvalue of an elliptic operator. The speeds c * ,$\pm$ and c * * ,pm are then obtained through a fine analysis of the properties of $\lambda$ p with respect to c. In particular, we establish its continuity with respect to the speed c. In addition, for any continuous bounded non-negative initial data, we establish the long time behaviour of the solution U (t, x)

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 $\mathcal{L}\_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by :$$\mathcal{L}\_{\Omega}[\varphi] +a\varphi:=\int\_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where $\Omega\subset \mathbb{R}^N$ 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 $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ defined by \lambda\_p(\mathcal{L}\_{\Omega}+a):= \sup\{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such that}\, \mathcal{L}\_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \, \text{in}\;\Omega\}. We establish some new properties of this generalised principal eigenvalue $\lambda\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ 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 $\lambda\_{p} \left(\mathcal{L}\_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}\_{\sigma,m,\Omega}$ is defined by $\displaystyle{\mathcal{L}\_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int\_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. In particular, we prove that $$\lim\_{\sigma\to 0}\lambda\_p\left(\mathcal{L}\_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda\_1\left(\frac{D\_2(J)}{2N}\Delta +a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\varphi\_{p,\sigma}$

On a nonlocal equation arising in population dynamics

We study a one-dimensional nonlocal variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modelled by a convolution operator. We prove that as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits

A counterexample to the Liouville property of some nonlocal problems

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$$\int_{\mathbb{R}^N\setminus K} J(x-y)\,( u(y)-u(x) )\mathrm{d}y+f(u(x))=0, \quad x\in\R^N\setminus K,$$where $K\subset\mathbb{R}^N$ is a bounded compact set, called an ``obstacle", and $f$ is a bistable nonlinearity. When $K$ is convex, it is known that solutions ranging in $[0,1]$ and satisfying $u(x)\to1$ as $|x|\to\infty$ must be identically $1$ in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data $f$ and $J$ for which this property fails