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。是 众所周知，在非线性生产项和线性死亡率项中具有常数延迟的非线性方程对于任何 延迟，如果具有相同非线性函数的离散映射是稳定的。我们最近把这个结果推广到标量非自治系统 在生产项中只有分布延迟的方程。然而，对于时滞微分系统，这一结果不再成立 方程，并且必须应用更复杂的标准。提出的研究的目的是将这些结果扩展到系统 具有分布式可变延迟。我们计划考虑两种类型的分布时滞：有限（但不一定有界）和无限 但记忆力呈指数级衰减

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.