Stability of non-conservative cross diffusion model and approximation by stochastic particle systems

Abstract

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species move, reproduce and die with rates sensitive to the local densities of the two species. Quantitative estimates are given and convergence is obtained soon as the population per site and the number of sites go to infinity. The proofs rely on the extension of stability estimates via duality approach under a smallness condition and the development of large deviation estimates for structured population models, which are of independent interest. The proofs also involve martingale estimates in H^{-1} and improve the approximation results in the conservative case as well.