This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion. | Turkish Journal of Mathematics Research Article Turk J Math (2015) 39 ¨ ITAK ˙ c TUB ⃝ doi: Spreading speeds in a lattice differential equation with distributed delay Hui-Ling NIU∗,∗∗ School of Mathematics and Statistics, Lanzhou University, Lanzhou, . China Received: • Accepted: • Published Online: • Printed: Abstract: This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion. Key words: Lattice differential equation, infinite distributed delay, spreading speeds 1. Introduction In biological invasions, the spreading speed (short for the asymptotic speed of spread/propagation) is a very important notion, since it is used to describe the speed at which the geographic range of the species population expands [14, 20, 22, 29, 36, 38]. The concept of the spreading speed was first introduced by Aronson and Weinberger [2, 3] for reaction-diffusion equations and applied by Aronson [1] to an integrodifferential equation. A general theory of spreading speeds has been developed for monotone semiflows [19, 20, 21, 33], for integral and integrodifferential population models [8, 9, 29, 30, 31], for time-delayed reaction-diffusion equations [31, 37], and for lattice differential equations [4, 34]. Recently, Hsu and Zhao [14] and Li et al. [18] extended the theory of spreading speeds in nonmonotone integrodifference equations and Fang et al. [10] in nonmonotone discretedelayed lattice equations. Lattice differential equations arise in many applied subjects, such as chemical reaction, image processing, material science, and biology [7, 15, 17, 28]. In the models of lattice
