Recent developments on spatial propagation for diffusion equations in shifting environments

Abstract

<p style='text-indent:20px;'>In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented.</p>

Spatial propagation of a lattice predation–competition system with one predator and two preys in shifting habitats

“…8, 9, 13, 14 for predator–prey models. For more researches in this area, we refer the readers to a recent review article by the second author of this paper and his collaborators 24 . In recent years, the propagation of lattice equations in shifting habitats has also attracted many scholars' interest.…”

“…For more researches in this area, we refer the readers to a recent review article by the second author of this paper and his collaborators. 24 In recent years, the propagation of lattice equations in shifting habitats has also attracted many scholars' interest. Hu and Li 15 first studied the persistence and spreading speed for the lattice Kolmogorov-Petrovski-Piskuno (KPP) equation in a shifting environment.…”

Nonlocal dispersal equations in domains becoming unbounded

“…This consideration, in the absence of external sources, leads immediately to that u satisfies (1). For recent references on nonlocal dispersal equations, see [32,25,30] and references therein.…”