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。 -- 对于离散模型，建议的研究目标将是分析循环而不是稳定的动态以及随机情况下的情况 扰动会显著改变系统的动态。作为一个例子，我们将考虑人口动态系统 Allee效应(对于足够小的种群水平的灭绝)，其中的Allee效应可以通过应用随机的 具有零均值的扰动，或者由于随机的原因，即使对于较大的初始人口值，也存在显著的灭绝概率 微扰。 不稳定离散系统的确定性和随机性控制也将是本研究的主题之一。 最近，在稳定原本不稳定的地图方面取得了重大进展。例如，面向目标的控制 2011年推出的地图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。在这种情况下 如果不是在每一步都施加脉冲控制，则可以稳定一个周期。将这些结果推广到高阶差分方程解 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竞争成功的影响 分布物种，其中模型由偏微分方程组描述。到目前为止，我们已经考虑了两人系统 物种，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比赛中 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)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.