On a non-autonomous reaction-convection diffusion model to study the bacteria distribution in a river

In this paper, we propose a non-autonomous convection-reaction diffusion system (CDI) with a nonlinear reaction source function. This model refers to the quantification and the distribution of antibiotic resistant bacteria (ARB) in a river. The main contributions of this paper are: (i) the determination of the limit set of the system by applying the semigroups theory, it is shown that it is reduced to the solutions of the associated elliptic system (CDI)(e), (ii) sufficient conditions for the existence of a positive solution of (CDI)(e) based on the Leray-Schauder's degree theory. Numerical simulations which support our theoretical analysis are also given.