Fixed point analysis of the dynamics of the NCAX cancer growth model

Abstract

In this work, we explore the properties of the steady state dynamics of the ordinary differential equations governing the population dynamics of the NCAX cancer growth model via fixed-point analysis. In particular, we investigate the resulting fixed-point surfaces for the cancerous (n_C*) and normal (n_N*) cell population against the apoptotic (R) and mutation (P) rates. We have identified four cases of steady-state population curves, two of which are pure states which defines a 3D surface over the parameters R and P. Linear stability analysis via the associated Jacobian matrices of these surfaces show regions of neutral stability and instability (no stable regions were found). These neutrally stable regions are found across the parameter space following intersecting fixed point surfaces such that at each (R,P) pair, there exist only one neutrally stable surface. The resulting final-state values are confirmed using numerical evaluation of the original set of differential equations for each (R,P)-values. It shows that there is a minimal difference between the numerical simulation and the fixed-point analysis. This suggests that our NCAX model is a stable model able to converge to the neutrally stable surfaces in the absence of absolutely stable fixed-points.