In this lecture, the spatio-temporal dynamics of a biological community described by a system of partial differential equations of "diffusion-reaction type". Most of attention will be paid to the problem of "biological invasion" when the initial spatial distribution of some of the species is described by finite functions. Although it is widely believed that in this case the evolution of the system is mainly reduced to a succession of travelling diffusive waves, by now a proper mathematical investigation of this problem has been made only for very few cases. I shall present some recent results obtained in this field.
First, it will be shown that propagation of diffusive waves can lead to rather an unusual phenomenon: locally unstable equilibrium can become stable in the wake of a diffusive front as a result of "dynamical stabilization".
Second, the existence of the diffusive wave of a new type will be shown: moving interface separating two spatial domains with different types of spatio-temporal dynamics inside. Both numerical and analytical results will be presented. Particularly, I shall obtain analytical estimates for the low bound for the spectrum of possible values of the speed of the diffusive waves.