Volume 30, Issue 1

A method for determination of a nonnegative solution to a system of linear equations and linear programming problems is proposed in the paper. The method is based on the algorithmic selection of extreme vectors of a convex sharpened polyhedral cone corresponding to those problems.

Optimal control problems for mathematical models describing HIV infection dynamics in a human body are considered in the paper. An overview of current approaches to solution of control problems for models of HIV dynamics is presented for techniques related to construction of optimal programme (open loop) or positional (feedback) controls for various criteria of control process quality and is based on the Pontryagin’s maximum principle and the Bellman’s theory of dynamic programming, respectively. In the framework of the theory of positional differential games of Krasovskii, there are constructive and efficient methods of synthesis of controls for different classes of dynamical systems. In this paper we present a formalization of a control problem for HIV infection considered as a corresponding positional differential game. The control algorithm based on the method of extremal shift is applied to the ODEs model of HIV infection. The numerical implementation of the extremal shift method is developed to construct the control taking the system to a neighbourhood of a given trajectory using the information on the dynamics of the guides or leaders.

The article presents the developments of the kinetic consistent approach of solution of the magneto gas dynamics problems on the high performance computing systems with massive parallelism. The main idea is to derive the magneto gas dynamic equations from the Boltzmann equation using a complex distribution function, including the electromagnetic terms. The derived equations were used for the formulation of the explicit numerical method of solution for the high performance parallel computing systems. The numerical experiments were performed for the verification of the chosen method.

We investigate three tensor product numerical data compression techniques in solution of the chemical master equation for the enzymatic futile cycle and compare them with the previously reported results, obtained by the stochastic simulation algorithm. On this particular example from systems biology, we show the history how the newly proposed tensor product methods reduced the computational complexity of the futile cycle modelling from days on a HPC cluster to hours and even minutes on a workstation.

New iterative solution methods are proposed for the finite element approximation of a class of variational inequalities with nonlinear diffusion-convection operator and constraints to the gradient of solution. Implementation of every iteration of these methods reduces to the solution of a system of linear equations and a set of two-dimensional minimization problems. Convergence is proved by the application of a general result on the convergence of the iterative methods for a nonlinear constrained saddle point problem.

We present a mathematical model of an aquatic community, which includes zooplankton and fish reproduction, and age-weight-structured trophic relationships. We show that interactions between separate components of the aquatic community can give rise to long-period oscillations in fish population size. The period of these oscillations is on the order of decades.We show that predatory fish can be an element, which gives rise to the long-period oscillations. With this model we also show that an increase in the zooplankton growth rate may entail a sequence of bifurcations in the fish population dynamics: steady states → regular oscillations → quasicycles → dynamic chaos. Since aquatic, and in particular, lake ecosystems are spatially heterogeneous, they are often considered as consisting of separate habitats, which are distinguished by their hydrophysical and ecological characteristics. We show that interhabitat fish migration can lead to dramatic changes in the fish population dynamics. In particular, the fish migration can destabilize both stationary states and chaotic regimes giving rise to regular and quasiregular oscillations in the fish population size.