Cell Cycle Control and Bifurcation for a Free Boundary Problem Modeling Tissue Growth

Abstract

We consider a free boundary problem for a system of partial differential equations, which arise in a model of cell cycle with a free boundary. For the quasi steady state system, it depends on a positive parameter $$\beta $$ , which describes the signals from the microenvironment. Upon discretizing this model, we obtain a family of polynomial systems parameterized by $$\beta $$ . We numerically find that there exists a radially-symmetric stationary solution with boundary $$r = R$$ for any given positive number $$R$$ by using numerical algebraic geometry method. By homotopy tracking with respect to the parameter $$\beta $$ , there exist branches of symmetry-breaking stationary solutions. Moreover, we proposed a numerical algorithm based on Crandall---Rabinowitz theorem to numerically verify the bifurcation points. By continuously changing $$\beta $$ using a homotopy, we are able to compute non-radially symmetric solutions. We additionally discuss control function $$\beta $$ .