Flip and Neimark–Sacker bifurcation in a differential equation with piecewise constant arguments model

In this paper, a differential equation with piecewise constant arguments model that describes a population density of a bacteria species in a microcosm is considered. The discretization process of a differential equation with piecewise constant arguments gives us two dimensional discrete dynamical system in the interval . By using the center manifold theorem and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip and Neimark–Sacker bifurcation. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for the discrete model