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

我的研究重点是可变参数的延迟微分方程，它们与离散，脉冲和混合系统的联系，以及动力系统在异质或变化环境中的群体动力学的应用，以及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数之间的差距，与具有可测量系数和可测量延迟的方程的（1+1/e）相比，连续的是3/2。基于以前对连续延迟方程和离散模型相互作用的研究，该研究将继续并扩展到系统，并应用于具有可变分布延迟的神经网络。在第一步，这样的系统的稳定性将连接到一个向量差分方程的强吸引性。在第二步，系统的无限，而不仅仅是无界，分布延迟将被视为，一旦记忆是指数衰减。噪音的加入可以将不稳定的平衡变成稳定的平衡，类似于卡皮卡摆，它有一个“颤抖”的底座，可以保持稳定在顶部（有规律的，不稳定的）位置。所提出的研究的目的之一是探索噪声控制，不仅可以稳定不稳定的平衡点，但也否则不稳定的周期。这项研究将包括的情况下，噪声确定性控制，以及控制噪声。为了稳定循环，可以不是在每一步而是在选定的步骤上施加控制，即脉冲或脉冲控制。连续和离散型的脉冲系统将被研究为具有变系数和一般分布时滞的方程。时滞微分系统的脉冲控制和镇定也是本文研究的一部分。对各种依赖环境的扩散的影响的探索得出的结论是，一旦一个种群向人均资源更丰富的方向迁移，其栖息地就不能被定期扩散的物种入侵。调查的情况下，不同的扩散策略与其他差异相结合的资源开发，或系统采用收获，是拟议的研究的一部分。这包括多鱼种系统和资源消耗专业化的可能性。该提案最适用的部分与癫痫发作之前，期间和之后人类大脑中神经元连接的表征有关。 该方案的成功实施将有助于我们加深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，推进时滞依赖脉冲理论，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.