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

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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<ρ<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=ρ. 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.

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Acknowledgements

W. Hao was supported by the Dunces Chair of the University of Notre Dame and NSF grant DMS-0712910. J.D. Hauenstein was supported by Texas A&M University and NSF grant DMS-0915211. A.J. Sommese was supported by the Dunces Chair of the University of Notre Dame and NSF grant DMS-0712910. Y.-T. Zhang was partially supported by NSF grant DMS-0810413.

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Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Wenrui Hao, Bei Hu, Yuan Liu, Andrew J. Sommese & Yong-Tao Zhang

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Jonathan D. Hauenstein

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  1. Wenrui Hao
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Correspondence to Wenrui Hao.

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Hao, W., Hauenstein, J.D., Hu, B. et al. Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core. J Sci Comput 53, 395–413 (2012). https://doi.org/10.1007/s10915-012-9575-x

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