On spatial-temporal dynamics of a Fisher-KPP equation with a shifting environment

Abstract

We consider a generalized Fisher-KPP equation with the growth function being time and space dependent in the form of “shifting with constant speed”. The main concerns are extinction and persistence, as well as spatial-temporal dynamics. By employing a new method relating to semigroup and some subtle estimates, we not only extend the main results in Li et al. [SIAM J. Appl. Math. 74 (2014), pp. 1397-1417] to a scenario when the growth function may have no sign change, but also improve the main results there by dropping some restrictions on the initial functions.