Logistic Equation with Long Delay Feedback

We study the local dynamics of the delay logistic equation with an additional feedback containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are identified, and it is shown that they are infinite-dimensional. The well-known methods for studying local dynamics based on the theory of invariant integral manifolds and normal forms do not apply here. The methods of infinite-dimensional normalization proposed by the author are used and developed. As the main results, special nonlinear boundary value problems of parabolic type are constructed, which play the role of normal forms. They determine the leading terms of the asymptotic expansions of solutions of the original equation and are called quasinormal forms.