Traveling Waves And Spreading Speed In Lattice Differential Equations With Distributed Delay

In our daily life, many problems can be described by the reaction-diffusion equation. For example, in biology, physics, chemistry and other fields. At the same time, lattice differential equations was widely used in chemical reaction, bi-ology, photoshop and materials science. With the rapid development and the uni-versal of the computer science, the spatially discrete models of reaction-diffusion equations are more advantageous of the numerical calculation and numerical anal-ysis. The main purpose of this article is to propose and discuss a single-species population model with infinite distributed maturity delay under the environment of the space patches. In this paper, we first consider the initial problems and established the comparison theorem about the model. We prove that the equilib-rium is stable if there is only a zero equilibrium. We also prove that the positive equilibrium is stable if there is only a positive equilibrium and the birth function b(w) is monotone. Secondly, we consider the existence of traveling wave solutions of the model. Using Schauder's fixed point theorem and the upper-lower solution method, we prove that there exists a traveling wave solution connecting the two equilibriums if c> c*(θ) and the birth function b(w) is monotone, whereθis any fixed direction of propagation. When the birth function b(w) is nonmonotone, we construct two auxiliary functions, and then prove that there exists a non-trivial traveling wave solution. Finally, we consider the existence of the spreading speed in a 1D lattice, and it coincidence with minimum wave.