Logistic equation with long delay feedback

Abstract

We study the local dynamics of a logistic equation with delay and with 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 have infinite dimension. Well-known methods for studying local dynamics, based on the application of the theory of invariant integral manifolds and normal forms, are not applicable here. 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 main terms of the asymptotic expansions of solutions to the original equation. They are called quasinormal forms.