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非自治 * 方程的分布延迟的生产项。然而，结果不再是正确的延迟微分方程系统，和一个更复杂的标准必须应用。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 *，但具有指数衰减的记忆。* 对于离散模型，拟议的研究将针对分析循环而不是稳定动态以及随机 * 扰动可以显着改变系统动态的情况。作为一个例子，我们将考虑具有 *Allee效应（在足够小的种群水平下灭绝）的种群动力学系统，其中可以通过应用具有零均值的随机 * 扰动来减轻Allee效应，或者由于随机 * 扰动，即使对于大的初始种群值，也有很大的灭绝概率。不稳定离散系统的确定性和随机控制也将是拟议研究的主题之一。最近，在稳定原本不稳定的地图方面取得了重大进展。例如，2011年引入的目标导向控制 * 允许稳定各种一维映射，甚至可以控制稳定平衡点的选择。在不是每一步都应用脉冲控制的情况下，可以使周期稳定。将这些结果推广到高阶差分方程和 * 系统是本研究的目的之一。研究的另一个重要方面是不同的扩散策略对空间分布物种竞争成功的影响，该模型由偏微分方程系统描述。到目前为止，我们已经考虑了两个种群的系统，它们之间唯一的区别是扩散策略。我们计划将这些结果扩展到Lotka-Volterra竞争 * 模型，并探索以下因素的相互作用：扩散策略，资源开发效率（以较高或较低的承载能力表示）和种内相互作用。

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.**** ***For discrete models, the proposed research will target to analyze cyclic rather than stable dynamics and the cases when stochastic*perturbations can significantly change the dynamics of the system. As an example, we will consider systems of population dynamics with the*Allee effect (extinction for small enough population levels) where either the Allee effect can be alleviated by applying a stochastic*perturbation with zero mean, or there is a significant probability of extinction even for large initial population values, due to stochastic*perturbations.********Deterministic and stochastic control of otherwise unstable discrete systems will also be one of the topics of the proposed research.*Recently significant progress has been achieved on stabilization of otherwise unstable maps. For example, the target oriented control*introduced in 2011 allows to stabilize a wide class of one-dimensional maps, even giving control over the choice of a stable equilibrium point. In the case*of pulse control applied not at every step, a cycle can be stabilized. Extending these results to higher order difference equations and*systems is one of the purposes of the present research.*******Another important aspect of the research is the influence of different diffusion strategies on the success of competing spatially*distributed species, where the model is described by a system of partial differential equations. So far we have considered systems of two*species, where the only difference between them is the diffusion strategy. We plan to extend these results to Lotka-Volterra competition*models and explore the interplay of the following factors: diffusion strategy, efficiency of resource exploitation (expressed in higher or*lower carrying capacity) and intraspecies interactions.***