Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems

Abstract

In this paper, a non-monotone time delayed 2D lattice system with global interaction is studied. The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. The existence of travelling waves for the wave speed $c\ge c_{\ast }\left(\theta \right)$ is established by Schauder’s fixed point theorem and a limiting argument, where $\theta$ is any fixed direction of propagation. The spreading speed is investigated by comparison arguments and a fluctuation method. It is also shown that the spreading speed coincides with the minimal wave speed along every direction. Particularly, letting the direction of propagation $\theta =0$ or $\frac{\pi }{2}$, the wave profile equation of the 2D lattice system can reduce to the wave profile equation of 1D lattice system and therefore, our results can cover the previous works.