On the principal eigenvalue of asymmetric nonlocal diffusion operators and associated propagation dynamics

Abstract

For fixed $c\in \mathbb{R}$, $l>0$ and a general non-symmetric kernel function $J(x)$ satisfying a standard assumption, we consider the nonlocal diffusion operator $$\begin{array}{r}{\mathcal{L}}_{(-l,l)}^{J,c}[\varphi ](x):={\int }_{-l}^{l}J(x-y)\varphi (y)dy+c{\varphi }^{\prime }(x),\end{array}$$

and prove that its principal eigenvalue ${\lambda }_p({\mathcal{L}}_{(-l,l)}^{J,c})$ has the following asymptotic limit: $$\underset{l\to \mathrm{\infty }}{lim}{\lambda }_p\left({\mathcal{L}}_{(-l,l)}^{J,c}\right)=\underset{\nu \in \mathbb{R}}{inf}[{\int }_{\mathbb{R}}J(x)e^{-\nu x}dx+c\nu ].$$ We then demonstrate how this result can be applied to determine the propagation dynamics of the associated Cauchy problem $$\{\begin{array}{ll}{\displaystyle u_t=d[{\int }_{\mathbb{R}}J(x-y)u(t,y)dy-u(t,x)]+f(u),}& t>0,x\in \mathbb{R},\\ u(0,x)=u_0(x),& x\in \mathbb{R},\end{array}$$ with a KPP nonlinear term $f(u)$. This provides a new approach to understand the propagation dynamics of KPP type models, very different from those based on traveling wave solutions or on the dynamical systems method of Weinberger (1982).

This talk is based on joint work with Dr Xiangdong Fang (Dalian Univ Tech) and Dr Wenjie Ni (Univ New England).