Asymptotic Behavior, Spreading Speeds, and Traveling Waves of Nonmonotone Dynamical Systems

Abstract

In this paper, we study the asymptotic behavior, the spreading speed, and the existence/nonexistence of traveling waves of a class of nonmonotone discrete-time dynamical system. As a byproduct, we also obtain some results on the global attractivity of a nontrivial constant fixed point and on the existence of a nonconstant fixed point. We then apply the main results to three model systems: (i) a spatially nonlocal integro-difference equation; (ii) a reaction-diffusion equation with spatial nonlocality and time delay in the reaction term; and (iii) an equation with nonlocal diffusion and delayed nonmonotone nonlinearity in the reaction term. The obtained results for these three equations improve some existing ones by removing the symmetry of the kernel functions and relaxing the conditions on the nonlinear terms.