Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats

Abstract

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution u * (t, x). In this paper, we show that such an equation has a spatial spreading speed c * (ξ) in the direction of any given unit vector ξ. A variational characterization of c * (ξ) is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution 0 has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting u * (·, ·) and 0 propagating in any given direction of ξ with speed c > c * (ξ).

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

“…The existence of transition waves of (1.1) with non-periodic time and/or space dependence has been studied in [35,41,42,43,52,53,54,63]. Applying Theorem 2.2 or the arguments in the proof of Theorem 2.2, we can establish the asymptotic stability of the transition waves proved in [35,41,42,43,48,52,53,54,63]. For convenience, we introduce the following assumptions.…”

“…A huge amount of research has been carried out toward the transition waves or generalized traveling waves of various time and/or space dependent monostable equations. See, for example, [7,8,9,10,11,20,22,25,27,28,32,33,34,36,39,41,42,43,44,45,46,51,52,53,58,60,61,62,63], and references therein for space and/or time dependent Fisher-KPP type equations with random dispersal, and see, for example, [15,16,17,35,48,54,55,56,57], and references therein for space and/or time dependent Fisher-KPP type equations with nonlocal dispersal.…”

“…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.…”

The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries

“…Problem (1.3) and its many variations have been extensively studied in recent years; see, for example, [2,4,8,10,25,26,32,[35][36][37][38]41] and the references therein. In particular, several classical results on (1.2) as a model for spreading of species have been successfully extended to (1.3) and its variations.…”

Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment

“…These two fundamental issues along with some new types of entire solutions have been widely used for the description of species invasion and disease transmission. Regarding the nonlocal dispersal equation with time and/or space periodic dependence, we refer the readers to [25,29,31] for spreading speeds, and [1,7,25,30] for traveling wave solutions, and [18] for new types of entire solutions.…”