Local Dynamics of a Logistic Equation with Delay

We considered a logistic equation with delay and studied its local dynamics. The critical cases have been found in the problem of the equilibrium state stability. We applied standard Andronov–Hopf biffurcation methods for delay differential equations and an asymptotic method, developed by one of the authors, based on the construction of special evolution equations that define the local dynamics equations with delay. It is shown that all solutions of the equation tend to an equilibrium state or result in a single stable cycle. The results of numerical modelling are presented in this paper. The study has proved that analytical and numerical modeling results have a good correlation.