Asymptotic limit of the principal eigenvalue of asymmetric nonlocal diffusion operators and propagation dynamics

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{align*} \bf{L}^{J, c}{(-l,l)}φ:=\int{-l}^lJ(x-y)φ(y),dy+cφ’(x), \end{align*} and prove that its principal eigenvalue ${\lambda }_p({\mathbf{L}}_{(-\mathbf{l},\mathbf{l})}^{\mathbf{J},\mathbf{c}})$ has the following asymptotic limit: \begin{equation*}\label{l-to-infty-c} \lim\limits_{l\to \infty}λ_p(\bf {L}^{J, c}{(-l,l)})=\inf\limits{ν\in$\mathbb{R}$}\big[\int_{$\mathbb{R}$}J(x)e^{-νx},dx+cν\big]. \end{equation*} We then demonstrate how this result can be applied to determine the propagation dynamics of the associated Cauchy problem \begin{equation*} \label{cau} \left{ \begin{array}{ll} \displaystyle u_t = d \big[\int_{$\mathbb{R}$} J(x-y) u(t,y) , dy - u(t,x)\big] + f(u), & t > 0, ; x \in $\mathbb{R}$, u(0, x) = u_0(x), & x \in $\mathbb{R}$, \end{array} \right. \end{equation*} 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).