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

我的研究重点是具有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种群动态的应用，以及癫痫发作期间大脑活动的分析。该提议的目的之一是缩小具有连续参数和可测量参数的方程的常数之间的差距，对于具有可测量系数和可测量延迟的方程，连续方程的常数为3/2，而（1+1/e）。在先前对连续延迟方程和离散模型相互作用的研究基础上，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。首先，将系统的稳定性与矢量差分方程的强吸引性联系起来。在第二步，一旦存储器呈指数衰减，将考虑具有无限（而不仅仅是无界）分布式延迟的系统。噪声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。提出的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，还可以稳定不稳定的循环。本研究将包括应用噪声确定性控制和仅由噪声控制的情况。为了稳定周期，控制可以不应用于每个步骤，而是应用于选定的步骤，即脉冲或脉冲控制。连续型和离散型脉冲系统将研究具有变系数和一般分布延迟的方程。时滞微分系统的脉冲控制和镇定也是拟建研究的一部分。对各种类型的环境依赖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发展，其栖息地就不会被定期扩散的物种入侵。当不同的扩散策略与资源开发中的其他差异相结合时，或者系统包含收获时，对案例的调查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。该提议中应用最多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特征有关。这一建议的成功实施将增强我们对系统延迟无关性质的认识，改进离散映射循环的随机稳定方法，分析不同扩散策略对进化成功的影响，推进延迟依赖脉冲理论，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.