Speed of travelling fronts: Two-dimensional and ballistic dispersal probability distributions

– The speed of traveling fronts for a two-dimensional model of a delayed reactiondispersal process is derived analytically and from simulations of molecular dynamics. We show that the one-dimensional (1D) and two-dimensional (2D) versions of a given kernel do not yield always the same speed. It is also shown that the speeds of time-delayed fronts may be higher than those predicted by the corresponding non-delayed models. This result is shown for systems with peaked dispersal kernels which lead to ballistic transport. Introduction. – Wavefronts are spatial profiles of concentration, temperature, etc., which connect two equilibrium states and travel without changing their shapes. Fedotov [1] has considered a kernel in which all particles jump the same distance. This highly idealized case yields very interesting comparisons between alternative evolution equations. Kot and coworkers [2] have derived important results (e.g., accelerating fronts and approximate profiles) for integrodifference evolution equations. In biological invasions, such as animal migrations and the geographic spread of epidemics, it is usual to describe these processes by employing 1D reaction-diffusion models [3]. However, most real processes of biological invasions take place in 2D and, as we show in this work, the modelization of a 2D invasion cannot be analyzed with 1D models if one has to deal with dispersal probability distribution functions (dispersal kernels). We establish how the non-trivial correspondence between a 2D and a 1D model must be performed. In this work we will start from a 2D continuous-time integral equation for the reactiontransport of particles which includes memory (or delay) effects described by a characteristic time T . Our model describes a more realistic situation than the classical 1D models because particles may jump in 2D according to a 2D dispersal kernel and react at the same time.