1 Mathematics Department, Université de Liége, Liége, Belgium.
2 INRIA Paris MOKAPLAN, Paris, France.
3 McGill University, Montreal, Canada.
4 IECL Université de Lorraine, Nancy, France.
5 GFM Universidade de Lisboa, Lisboa, Portugal.
In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.