Stability and Hopf bifurcation of a mathematical model describing bacteria–fish interaction in marine environment

In this work, we present and study a model of a host-parasite system in marine environment, which describes the population dynamics of fish (Tilapia) which can be infected by botulinum. The mathematical model is structured by levels of infection. Using the characteristic curves method, we transform the model into a system of distributed delay differential equations. We study the existence of Hopf bifurcation. Following the method presented by Hassard et al. (1981) [5], we prove analytically the stability of limit cycle periodic solutions. We present numerical and computer simulations of the model.