Delay, impulsive, structured and stochastic systems with applications to population dynamics

本文主要研究变参数时滞微分方程及其与离散、脉冲和混杂系统的联系，以及动力系统在异质或变化环境中的种群动力学中的应用，以及在癫痫发作期间大脑活动的分析。 该提案的目的之一是缩小具有连续和可测量参数的方程的常量之间的差距，连续的为3/2，而具有可测量的系数和可测量的延迟的方程为(1+1/e)。 在前人关于连续时滞方程和离散模型相互作用的研究的基础上，本研究将继续并扩展到系统，并将其应用于具有可变分布时滞的神经网络。首先，这类系统的稳定性与一个向量差分方程解的强吸引性有关。在第二步，一旦内存呈指数衰减，将考虑具有无限分布延迟的系统，而不仅仅是无限分布延迟系统。 噪声的加入可以将不稳定的平衡转变为稳定的平衡，就像卡皮卡钟摆一样，卡皮卡钟摆的基座“颤抖”，可以在顶部(规则的、不稳定的)位置保持稳定。这项研究的目的之一是探索噪声控制，它不仅可以稳定不稳定的平衡点，而且还可以稳定其他不稳定的周期。这项研究将包括当应用噪声确定性控制时的情况，以及仅由噪声控制的情况。为了稳定周期，可以不是在每个步骤而是在选定的步骤上施加控制，即脉冲或脉冲控制。 对于具有变系数和一般分布时滞的方程，我们将研究连续和离散类型的脉冲系统。脉冲控制和时滞微分系统的镇定也是本文研究的一部分。 对各种类型的环境依赖型扩散的影响的探索得出的结论是，一旦一个种群走向更丰富的人均资源，其栖息地就不会受到定期扩散的物种的入侵。调查不同的扩散战略与资源开采的其他差异相结合的情况，或系统结合收获的情况，是拟议研究的一部分。这包括多物种系统和资源消耗专业化的可能性。 该提案中应用最多的部分与癫痫发作前、发作中和发作后人脑神经元连接的特征有关。 这一方案的成功实施将增强我们对系统时滞无关性质的认识，改进离散映射周期的随机镇定方法，分析不同扩散策略对进化成功的影响，推进时滞相关脉冲理论，揭示癫痫发作的数学特征。

My research focuses on delay differential equations with variable parameters, their connection to discrete, impulsive and hybrid systems, and application of dynamical systems to population dynamics in heterogeneous or changing environment, as well as to the analysis of brain activity during epileptic seizures. One of the aims of the proposal is narrowing the gap between the constants for equations with continuous and measurable parameters, 3/2 for continuous compared to (1+1/e) for equations with a measurable coefficient and a measurable delay. Building on previous research on the interplay of continuous delay equations and discrete models, the study will be continued and extended to systems, with applications to neural networks with variable distributed delays. At the first step, stability of such a system will be connected to strong attractivity of a vector difference equation. At the second step, systems with infinite, not just unbounded, distributed delays will be considered, once the memory is exponentially decaying. Addition of noise can turn an unstable equilibrium into a stable one, similarly to the Kapica pendulum, which, with a "trembling" base, can be kept stable in the top (regularly, unstable) position. One of the purposes of the proposed research is to explore noisy controls which can stabilize not only unstable equilibrium points but also otherwise unstable cycles. This study will incorporate the case when a noisy deterministic control is applied, as well as control by noise only. In order to stabilize cycles, a control can be applied not at every, but on chosen steps, i.e. an impulse, or a pulse, control. Impulsive systems of both continuous and discrete type will be studied for equations with variable coefficients and, generally, distributed delays. Impulsive control and stabilization of delay differential systems are also a part of the proposed research. Exploration of the influence of various types of environment-dependent diffusion led to the conclusion that, once a population moves towards more abundant per capita resources, its habitat cannot be invaded by regularly diffusing species. Investigation of the cases when different diffusion strategies are combined with other differences in resources exploitation, or the system incorporates harvesting, is a part of the proposed research. This includes multi-species systems and the possibility of specialization in resource consumption. The most applied part of the proposal is connected with characterization of neuronal connectivity in human brains before, during, and after epileptic seizures. Successful implementation of this proposal will enhance our knowledge of delay-independent properties of systems, improve methods for stochastic stabilization of cycles of discrete maps, analyze the influence of different diffusion strategies on the evolutionary success, advance the theory of delay-dependent impulses and reveal mathematical traits of epileptic seizures.