Systems with a distributed memory and applications to population dynamics

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The main focus of the proposed research is on the interplay of continuous systems with time delay and discrete systems. It is well known that a nonlinear equation with a constant delay in the nonlinear production term and a linear mortality term is stable for any delay, if the discrete map with the same nonlinear function is stable. We have recently generalized this result to a scalar non-autonomous equation with a distributed delay in the production term only. However, the result is no longer true for systems of delay differential equations, and a more complicated criterion must be applied. The purpose of the proposed research is to extend these results to systems with a distributed variable delay. We plan to consider distributed delays of two types: finite (but not necessarily bounded) and infinite but with exponentially decaying memory.