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Reaction-Diffusion Equations, Propagation and Modelling

Final Report Summary - READI (Reaction-Diffusion Equations, Propagation and Modelling)

Reaction and diffusion are fundamental mechanisms ubiquitous in the life, physical and social sciences. Indeed, they are at work for instance in such diverse situations as combustion processes, phase changes, biological invasions, population ecology, or social influence and interactions. The mathematical research is about finding general properties for this kind of equations that can then be relevant to these various settings.

This project combined general mathematical developments with the study of specific models. The advances in the general mathematical theory of reaction-diffusion equations and systems have concerned several areas, notably partial differential equations, non homogeneous problems, infinite dimensional dynamical systems, the theory of principal eigenvalues of elliptic and parabolic operators, non-local diffusion processes, equations with non-local interactions, and non-local free boundary problems.

The mathematical developments allowed us to make progress on some of the underlying mechanisms at work in various modelling issues. We developed a model to study the effect of lines with fast diffusion on overall biological invasion by invasive species such as the tiger mosquito. In particular we quantified the enhancement of invasion speeds due to such lines. In a completely different setting, we proposed a new model to describe the organization of predators into competing packs and how territories emerge. This model allows one to make predictions on territory sizes and shapes and on the role of various ecological parameters. On the social sciences side, we studied the mathematics involved in models for the formation of bubbles in financial markets and developed new models for the onset of rioting activity, how it starts, propagates and eventually subsides.

It is remarkable that all of these questions lend themselves to mathematical modelling that involve reaction and diffusion processes. For some of them, this had been known for some time. For others, we discovered this by proposing new models in the course of the project. Efforts to advance our understanding of these models have led us in turn to new developments in the general mathematical theory of propagation phenomena through reaction-diffusion processes.