Large time behavior of nonautonomous differential systems modeling antibiotic-resistant bacteria in rivers

In this paper, we study a general nonautonomous model for bacterial dynamics in rivers. The mathematical model is represented by a nonautonomous system of nonlinear ordinary differential equations. We show the existence of a bounded positive invariant and attracting set. By using the Lyapunov function method, we establish global stability of steady-state solutions of the associated autonomous system. Second, the existence of positive periodic solutions of the nonautonomous system is proven using a continuation theorem based on coincidence degree theory.