Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core

Abstract

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0<?<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=?. The system depends on a positive parameter μ, which describes tumor aggressiveness, and for a sequence of values μ 2<μ 3<?, there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor μ. By continuously changing μ using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.