The cancer stem cell hypothesis has evolved to one of the most important paradigms inbiomedical research. During recent years evidence has been accumulating for the existenceof stem cell-like populations in different cancers, especially in leukemias. In thecurrent work we propose a mathematical model of cancer stem cell dynamics in leukemias. Weapply the model to compare cellular properties of leukemic stem cells to those of theirbenign counterparts. Using linear stability analysis we derive conditions necessary andsufficient for expansion of malignant cell clones, based on fundamental cellularproperties. This approach reveals different scenarios of cancer initiation and providesqualitative hints to possible treatment strategies.

A high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.

The aim of this paper is to study the effect of vibrations on convective instability ofreaction fronts in porous media. The model contains reaction-diffusion equations coupledwith the Darcy equation. Linear stability analysis is carried out and the convectiveinstability boundary is found. The results are compared with direct numericalsimulations.

A class of non-linear stochastic models is introduced. Phase transitions, critical points and the domain of attraction are discussed in detail for some concrete examples. For one of the examples the explicit expression for the domain of attraction and the rates of convergence are obtained.