Systems with a distributed memory and applications to population dynamics

本文主要研究连续时滞系统与离散系统之间的相互作用。众所周知，如果具有相同非线性函数的离散映射是稳定的，则具有非线性生产项和线性死亡项的常时滞的非线性方程在任意时滞下是稳定的。最近，我们将这一结果推广到只在产生式中具有分布时滞的标量非自治方程。然而，对于时滞微分方程组，结果不再成立，必须应用更复杂的判据。研究的目的是将这些结果推广到具有分布变时滞的系统。我们计划考虑两种类型的分布式延迟：有限的(但不一定是有界的)和无限的，但具有指数衰减的内存。

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