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对任何*时滞都是稳定的。我们最近将这一结果推广到一个仅在生产项中具有分布时滞的标量非自治方程。然而，对于延迟微分方程组，结果不再成立，必须应用更复杂的判据。建议研究的目的是将这些结果推广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时滞：有限的(但不一定是有界的)和无限的，但是具有指数衰减的存储。*对于离散模型，所提出的研究的目标是分析循环而不是稳定的动态，以及当随机扰动可以显著改变系统的动态的情况。作为一个例子，我们将考虑具有*Allee效应的种群动力学系统(对于足够小的种群水平，灭绝)，其中Allee效应可以通过施加具有零均值的随机扰动来缓解，或者由于stochastic*perturbations.********Deterministic而即使对于较大的初始种群值也存在显著的灭绝概率，并且对否则不稳定的离散系统的随机控制也将是拟议研究的主题之一。*最近在其他不稳定映射的稳定化方面取得了重大进展。例如，2011年引入的面向目标的控制*允许稳定一维地图的广泛类别，甚至可以控制稳定平衡点的选择。在不是在每一步都施加脉冲控制的情况下，可以稳定周期。将这些结果推广到高阶差分方程组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竞争的空间分布物种的成功的影响，其中模型由一个偏微分方程组来描述。到目前为止，我们已经考虑了两个物种的系统，它们之间唯一的区别是扩散策略。我们计划将这些结果推广到Lotka-Volterra竞争模型，并探索以下因素之间的相互作用：扩散战略、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.***