A weakly nonlinear analysis of a model of avascular solid tumour growth

In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations.