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。它是 众所周知，一个在非线性产生项和线性死亡项中具有常时滞的非线性方程对任意 延迟，如果具有相同非线性函数的离散映射是稳定的。我们最近将这一结果推广到标量非自治 仅在生产项中具有分布延迟的方程。然而，对于时滞差分系统，结果不再成立 方程式，并且必须应用更复杂的标准。拟议研究的目的是将这些结果推广到系统 具有分布式可变延迟。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 但记忆力却呈指数式下降。

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.