Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

Abstract

Abstract. This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in [34,39,45,46,61] for random dispersal Fisher-KPP equations with time and space periodic dependence, in [41,42,43,51,52,53,58,63] for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in [17,48,56] for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation. Up to the author's knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with general time and space dependence is studied.

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media

“…In particular, Rawal and Shen [39] showed that the properties of positive time periodic solutions are determined by the sign of the principal spectrum point of the corresponding linearized equation of ( 8)-( 10) at the null state. We refer the reader to [40] for the study of spreading properties and traveling waves of (8) in time and space periodic case and to [42] for the study of properties of transition waves and positive entire solutions of (8) in general time and space dependence. For the study of other aspects of nonlocal dispersal models, the reader is referred to [43,48] and the references therein.…”

Non-local dispersal equations with almost periodic dependence. II. Asymptotic dynamics of Fisher-KPP equations

“…(2) In [26], results similar to theorems 1.1 and 1.2 for the case D = R were obtained for general time dependence under the condition lim inf t−s→∞ [26] in the case when D = R and f (t, x, u) is almost periodic in t.…”

On the mean speed of bistable transition fronts in unbounded domains

“…Lastly, even in the homogeneous space R N , non-standard transition fronts which are not invariant in any moving frame were also constructed in [24] under assumptions (1.2)-(1.5). More generally speaking, there is now a large literature devoted to transition fronts for bistable reactions in homogeneous or heterogeneous settings [6,13,18,22,48,53,60], as well as for other types of homogeneous or space/time dependent reactions in dimension 1 [16,29,30,34,35,37,40,42,43,51,52,57,58] and in higher dimensions [1,9,38,39,49,50,59,61].…”