Traveling wave solutions of spatially periodic nonlocal monostable equations (Q2839825)

The authors study the existence, uniqueness, and stability of traveling wave solutions of an integro-differential equation NEWLINE\[NEWLINE u_t(t,x)=\int\limits_{\mathbb{R}^N} k(x-y)u(t,y)\;dy -u(t,x) +u(t,x)f(x,u(t,x)),\tag{1} NEWLINE\]NEWLINE where \(k\) is a normalized, smooth, compactly supported convolution kernel and \(f\) is \(p\)-periodic for some \(p\in (0,\infty)^N\), i.e., \(f(x+p,u)=f(x,u)\) for all \(x\in\mathbb{R}^N\) and \(u\in\mathbb{R}^+\), and satisfies so-called monostability hypotheses. The latter imply that the equilibrium zero is unstable under the solution semiflow of (1) and that there exists a unique positive equilibrium which is stable. The problem is of great interest in ecology, since it describes the spread of a disease, e.g., under nonlocal dispersal.NEWLINENEWLINEThe authors show under certain assumptions concerning the habitat that given a spreading direction and a speed greater than the spreading speed, (1) has a unique stable traveling wave solution which connects the positive equilibrium and zero. They employ sub- and super-solutions methods and the principal eigenvalue theory for nonlocal dispersal operators which goes back to an earlier work of the authors. The proofs require new sophisticated arguments, which have already found further applications.