Systems with a distributed memory and applications to population dynamics

建议的研究的主要重点是连续系统的时滞和离散系统的相互作用。众所周知，如果具有相同非线性函数的离散映射是稳定的，则非线性生产项具有常数时滞且死亡项具有线性时滞的非线性方程对任意时滞都是稳定的。最近，我们将这个结果推广到一个标量的非线性方程的分布延迟的生产项。然而，对于时滞微分方程组，这一结果不再成立，必须采用更复杂的判据。所提出的研究的目的是将这些结果扩展到具有分布可变延迟的系统。我们计划考虑两种类型的分布延迟：有限（但不一定有界）和无限，但指数衰减的内存。

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.