Open AccessThe article considers an isolated population system consisting of two types of human stem cells, namely normal cells and cells with chromosomal abnormalities (abnormal ones). The system develops in the laboratory (in vitro). The article analyses possible scenarios of the population system development, which are implemented for different values of its parameters. An investigated model of the cell population system takes into account the limited resources. It is represented as a system of two nonlinear differential equations with continuous right-hand part. The model is considered with non-negative values of the variables; the domain is divided into four sets. The model feature is that in each set the right part of the system of differential equations has a different form.
The article analyses a quality of the rest points of the system in each of four sets. The analytical conditions for determination of the number of rest points and the quality of rest points, with, at least, one zero coordinate, are obtained.
It is shown that the population system under study cannot have more than two points of rest, both coordinates of which are positive (non-zero). It is difficult to determine quality of such rest points depending on the model parameters due to the complexity of the expressions, which define the systems of the first approximation, recorded in a neighborhood of these points of rest. Numerical research results of the stability of these points of rest are obtained, and phase portraits with the specified specific values of the system parameters are demonstrated. The main scenarios for the cell population development are adduced. Analysis of mathematical model shows that a cell population system may remain the system consisting of populations of normal and abnormal cells; it can degenerate into a population of abnormal cells or perish. The scenario, in which there is only the population of normal cells, is not implemented. The numerical simulation allowed us to identify the sets of parameters of the mathematical model, which provide a fairly favorable scenario. In this scenario, the trajectory of the population system starting at the point corresponding to the standard initial cultivating conditions comes in a stable node, with the normal cells to be dominated.