Algebraic structures generating reaction-diffusion models: The activator-substrate system

• The prerequisite of long-distance negative feedback is recognized as a condition on the internal symmetry properties of a system.
• We obtain activator-substrate reaction-diffusion models from integrable towers with algebraic skeleton.
• The appearance of patterns is characterized algebraically: parameters in a given model can be ‘controlled’ already at an algebraic level.
• Our results are of general nature and in principle could be applied to other real world models, in perspective also to models with delay.

We shall construct a class of nonlinear reaction-diffusion equations starting from an infinitesimal algebraic skeleton. Our aim is to explore the possibility of an algebraic foundation of integrability properties and of stability of equilibrium states associated with nonlinear models describing patterns formation.