Linearized stability in case of state-dependent delay: a simple test example

Abstract

In the theory of delay functional (Volterra) equations, as well as in ordinary or delay differential equations, the principle of linearized stability is an effcient tool for studying local asymptotic behaviour of non-linear systems near a steady state. The aim of this thesis is to prove that the principle of linearized stabil- ity for Volterra equations can be used even in some cases when the model does not satisfy assumptions of the standard formulation of the principle (see [5]). We consider a scalar system derived from a Daphnia population model, as an example of a relatively simple model that does not satisfy the as- sumptions of the standard formulation of the principle of linearized sta- bility for Volterra equations and show that the actual principle applies anyway.