Propagation dynamics for a periodic delayed lattice differential equation without quasi-monotonicity

Abstract

This paper is concerned with the propagation dynamics for a periodic delayed lattice differential equation with age structures. In the quasi-monotone case, the existence of the spreading speed and its coincidence with the minimal wave speed of periodic traveling fronts has been established in Wang and Weng (2014). In this paper, we consider the spatial dynamics of the equation without quasi-monotonicity. We first establish the existence of the spreading speed c∗ and the non-existence of the periodic traveling waves with speed c∈(0,c∗). Then, we prove the existence of the periodic traveling waves by using the asymptotic fixed point theorem. Our result indicates that the spreading speed still equals to the minimal wave speed in the non-quasi-monotone case. It may be the first work to study the spatial dynamics of the time-periodic and delayed lattice differential systems without quasi-monotonicity.