This paper is concerned with entire solutions for bistable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain. Here the entire solutions are defined in the whole space and for all time $t\in \mathbb {R}$ . Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the $x$ -axis and annihilating at a finite time. Furthermore, we show that such an entire solution is unique up to space-time translations and is Liapunov stable. A key idea is to characterize the asymptotic behavior of the solutions as $t\to -\infty$ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are considered.

This paper is concerned with invasion entire solutions of a Lotka-Volterra competition system with nonlocal dispersal, which formulate a new invasion way of the stronger species to the weaker one. We first give the asymptotic behavior of traveling wave solutions at infinity. Then by the comparison principle and sub-super solutions method, we establish the existence of invasion entire solutions which behave as two monotone waves with different speeds and coming from both sides of $x$-axis.

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.