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

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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.