*Article* **Quasinormal Forms for Chains of Coupled Logistic Equations with Delay**

**Sergey Kashchenko**

Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia; kasch@uniyar.ac.ru

**Abstract:** In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

**Keywords:** logistic equation; delay; quasinormal form; asymptotic behavior; dynamics

**MSC:** 34K11
