Waves in reaction-transport systems with memory and long-distance dispersal effects

Why do we need to develop the theory of wave propagation into an unstable state for systems of integro-differential and integro-difference equations? Because they provide more realistic models for various phenomena in physics, chemistry, biology. It is well known that a macroscopic transport process comes from the overall effect of many particles performing complex random movements. Classical diffusion is just an approximation to this transport in the long-time, large-distance parabolic limit, but a diffusion approximation is not always appropriate for problems involving propagating fronts. From a practical point of view, this implies that models in physics,mathematical biology, etc. cannot be expected to work properly, in general, if they are based on simple reaction-diffusion ideas. Hence it is desirable to extend these results by considering more suitable, alternative models for transport processes based on integro-differential and integro-difference equations. Very recently there have been some important developments in the theory of wave propagation for a single integro-differential equation. We now wish to extend this groundwork to the solution of systems of integro-differential and integro-difference equations. An advantage is that we will be able to take into account: (i) realistic multi-component cases; (ii) long-range and time-delayed interactions; (iii) long-distance dispersal. Memory effects and long-distance dispersal are a significant feature in many areas of physics, chemistry and biology, but they are often ignored because techniques are not yet readily available for dealing with them. We plan to address this deficiency.