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

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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 \(\beta \).

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Acknowledgments

The authors Wenrui Hao and Anrew J. Sommese were supported by the Dunces Chair of the University of Notre Dame. We would like to express our thanks to Professor Avner Friedman whose valuable comments and suggestions helped greatly to improve this article. We are grateful to two anonymous referees for their careful reading and valuable suggestions.

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

    Wenrui Hao, Bei Hu & Andrew J. Sommese

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  1. Wenrui Hao
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  2. Bei Hu
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Correspondence to Wenrui Hao.

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Hao, W., Hu, B. & Sommese, A.J. Cell Cycle Control and Bifurcation for a Free Boundary Problem Modeling Tissue Growth. J Sci Comput 56, 350–365 (2013). https://doi.org/10.1007/s10915-012-9678-4

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  • DOI: https://doi.org/10.1007/s10915-012-9678-4

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