Traveling waves in lattice differential equations with distributed maturation delay

In this paper we derive a lattice model with infinite distributed delay to describe the growth of a single-species population in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction. We consider the existence of traveling wave solutions when the birth rate is large enough that each patch can sustain a positive equilibrium. When the birth function is monotone, we prove that there exists a traveling wave solution connecting two equilibria with wave speed c > c∗(θ) by using the monotone iterative method and super and subsolution technique, where θ ∈ [0, 2π] is any fixed direction of propagation. When the birth function is non-monotone, we prove the existence of non-trivial traveling wave solutions by constructing two auxiliary systems satisfying quasi-monotonicity.