Logistic Equation with Long Delay Feedback

We study the local dynamics of the delay logistic equation with an additional feedbackcontaining a large delay. Critical cases in the problem of stability of the zero equilibrium state areidentified, and it is shown that they are infinite-dimensional. The well-known methods forstudying local dynamics based on the theory of invariant integral manifolds and normal forms donot apply here. The methods of infinite-dimensional normalization proposed by the author areused and developed. As the main results, special nonlinear boundary value problems of parabolictype are constructed, which play the role of normal forms. They determine the leading terms ofthe asymptotic expansions of solutions of the original equation and are called quasinormal forms.