Spatial dynamics for lattice differential equations with a shifting habitat

Abstract

We study a lattice differential equation model that describes the growth and spread of a species in a shifting habitat. We show that the long term behavior of solutions depends on the speed of the shifting habitat edge c and a number c⁎(∞) that is determined by the maximum linearized growth rate and the diffusion coefficient. We demonstrate that if c>c⁎(∞) then the species will become extinct in the habitat, and that if c<c⁎(∞) then the species will persist and spread along the shifting habitat at the asymptotic spreading speed c⁎(∞). For our purpose the solutions to the model are formulated in the form of integral equations involving modified Bessel functions, for which new asymptotic estimates are provided. To the best of our knowledge, this is the first time that the classical Bessel functions are used to describe the solutions of lattice differential equations, and this approach possesses its own interest in further studying lattice differential equations.