Dynamics of the Logistic Equation with Delay

The logistic equation supplemented with a summand characterizing delay is considered. The local and nonlocal dynamics of this equation are studied. For equations with delay, we use the standard Andronov–Hopf bifurcation methods and the asymptotic method developed by the author and based on the construction of special evolution equations defining the local dynamics of the equations containing delay. In addition, we study the existence and methods of constructing the asymptotics of nonlocal relaxation cycles. A comparison of the results obtained with those for the Hutchinson equation and some of its generalizations is given.