Symmetry-breaking bifurcation of periodic solutions for a free-boundary tumor model

Abstract

In this paper, we consider a free boundary multi-layer tumor model that incorporates a $T-$periodic provision of external nutrients $\Phi (t)$. The simplified model contains three parameters: the mean of periodic external nutrients $\Phi (t)$, the threshold concentration $\tilde{\sigma }$ for proliferation and the cell to cell adhesiveness coefficient $\gamma$. We first study the flat solution and give a complete classification about $\frac{1}{T}{\int }_0^T\Phi (t)dt$ and $\tilde{\sigma }$ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, (i) a zero flat solution is globally stable under the flat perturbations if and only if $\tilde{\sigma }⩾\frac{1}{T}{\int }_0^T\Phi (t)dt$; (ii) If $\tilde{\sigma }<\frac{1}{T}{\int }_0^T\Phi (t)dt$, then there exists a unique positive flat solution $\left({\sigma }_{\ast }(y,t),p_{\ast }(y,t),{\rho }_{\ast }(t)\right)$ with period $T$ and it is a global attractor of all positive flat solutions for all $\gamma >0$. We further investigate periodic solutions bifurcating from the flat periodic solution $\left({\sigma }_{\ast }(y,t),p_{\ast }(y,t),{\rho }_{\ast }(t)\right)$. By periodicity and symmetry, we not only give symmetry-breaking periodic solutions for all positive parameter ${\gamma }_j$, but also show the existence of a plethora of periodic bifurcations. For the free boundary tumor problem, this is the first result of the existence of periodic bifurcations.

arXiv admin note: text overlap with arXiv:2109.14291