What do phase transitions in physics, propagation of flames, tumours, biological invasions, and the spread of social norms have in common? As an ubiquitous mechanism in nature and in society, diffusion, along with transport and reaction effects, is the main factor explaining changes or transitions in a wide array of systems. It lies at the core of systems where two or several possible states coexist, and accounts for certain states expanding or receding or for giving rise to patterns.
This lecture will describe some mathematical properties of reaction-diffusion equations as an approach to spatial propagation and diffusion. Properties of reaction-diffusion equations and their consequences will be first examined in the context of ecology of populations. Then, I will present a new model for spatial diffusion of illegal behaviour reporting from a joint work with Silvio Franz and Jean-Pierre Nadal. The aim of this theoretical model is to allow one to better understand some of the underlying mechanisms. I will also compare our approach with other existing models and discuss some of its implications.