The Behavior of a Two-Component Population System in Vicinity of the Zero Equilibrium Point

Abstract

A clinical behaviour of the cell-based therapy in recent decades has encouraged a great interest in the culture of cell populations in vitro. One of the directions of cell therapy is transplantation of stem cells. The cell material for transplantation is obtained by culturing the patient’s cells. However, there are often problems because of the genetic mutations of cells in the process of culture, namely, the degeneration of a portion of mutated cells into "immortal" (cancerous) cells, which makes the transplantation of such material unsafe for a patient. To study development dynamics of the cell populations in vitro is quite costly. Such studies are, usually, conducted at the beginning of culture, in the middle of the process, and when completing the process of culture. It is difficult to judge t the cell population development in detail by such data. Here mathematical modelling is of importance.The papers [8-11] propose a cell population system consisting of two types of cells, namely normal (healthy) and abnormal (aneuploid) cells. Interest in such a population system is due to the fact that, although the aneuploid cells have a life time less than the normal ones, a small portion of the aneuploid cells can degenerate into practically "immortal" cancer cells, whose population may, eventually, become dominant.In the qualitative analysis of the nonlinear dynamic systems, a standard component is information on the number of rest points, their nature and location. Earlier, [16] a detailed study of the rest points and their possible nature was performed depending on the biological parameters, such as the proportion of dead cells, the average time of the cell cycle, the proportions of normal cells becoming the population of abnormal ones, etc. However, there is no exhaustive answer, yet, concerning this issueThe paper continues to study the two-component population model considered earlier [9-11, 16]. The study focuses on the zero equilibrium point. The conditions of stability are specified taking into account the fact that the dynamic system, by virtue of its biological content, must be considered in the first quadrant of the plane. In addition, a study of the zero equilibrium point has been conducted in critical cases in which the method to investigate linear approximation stability the does not work.

Keywords

• cell population
• population dynamics
• equilibrium point
• stability