Mathematical modeling of cellular regulator in normal and in the tumor process
The method of constructing systems of nonlinear functional-differential equations of cellular regulator in norm and in cancerous formations is considered in the article. A qualitative analysis of the developed equations has shown that taking into account the delay parameter leads to the discovery of a large variety of regimes of behavior of the regulators of living systems. The existence of stable stationary as well as self-oscillatory solutions of the regulator equations, chaotic solutions and the "black hole"effect is established. Mathematical and biological analysis shows that, in the case of cell regulatory, the extinction regime can be identified as a programmed cell death mode - apoptosis, steady state and self-oscillations - as normal cell functioning regimes, dynamic chaos as a mode of irregular, chaotic regulator of tumor cells and the "black hole"effect as a necrosis. The existence in the region of dynamic chaos of "windows"with an oscillatory mode of solutions of the regulator equations (Poincare type limit cycles) is found in the dynamic chaos region, which indicates the possibility of local behavior control in the anomaly zone in order to prepare a global process for moving the system to the region of stable functioning. The presence of small areas with normal behavior in the area of anomalies allows choosing the paths for removing the system from the area of anomalous functioning to a region with a healthy state along the r-windows chains.