Supervision : Dongming WANG

Qualitative Analysis of Biological Systems Using Algebraic Methods

This thesis is dedicated to qualitative analysis of biological systems, modeled as systems of differential or difference equations, using algebraic methods. We study the problems of detecting steady states, analyzing stability and different kinds of bifurcations, and constructing limit cycles for both continuous and discrete biological models, show how to reduce these problems to those of solving polynomial or semi-algebraic systems according to stability criteria and techniques from the qualitative theory of dynamical systems, and explain how the latter problems can be solved by using an algebraic approach based on the methods of triangular sets, Gröbner bases, quantifier elimination, and real solution isolation and classification. Experiments with various biological systems show the effectiveness of our algebraic approach. In particular, the stability, three kinds of bifurcations, and limit cycles for the self-assembling micelle system with chemical sinks are successfully analyzed. Exact algebraic conditions on the parameters of this system are derived to describe the kinds of bifurcations and the stability and types of the bifurcation points, and three limit cycles are constructed from a steady state by small perturbation.

2008-2012 Publications

• 2012
  • W. Niu, D. Wang : “Algebraic Analysis of Stability and Bifurcation of a Self-assembling Micelle System”, Applied Mathematics and Computation, vol. 219 (1), pp. 108-121, (Elsevier) (2012)
• 2011
  • W. Niu : “Qualitative Analysis of Biological Systems Using Algebraic Methods”, thesis, defence 06/17/2011, supervision Wang, Dongming (2011)
  • X. Li, Ch. Mou, W. Niu, D. Wang : “Stability Analysis for Discrete Biological Models Using Algebraic Methods”, Mathematics in Computer Science, vol. 5 (3), pp. 247-262, (Springer) (2011)
• 2009
  • X. Li, Ch. Mou, W. Niu, D. Wang : “Stability Analysis for Discrete Biological Models Using Algebraic Methods”, Proceedings of the Joint Conference of ASCM 2009 and MACIS 2009, vol. 22, COE Lecture Note Series, Fukuoka, Japan, pp. 382-385, (Kyushu University) (2009)
• 2008
  • W. Niu, D. Wang : “Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems”, Proceedings of the 3^rd International Conference on Algebraic Biology (Hagenberg, Austria, July 31-August 2, 2008), vol. 5147, Lecture Notes in Computer Science, Hagenberg, Austria, pp. 156-171, (Springer-Verlag) (2008)
  • W. Niu, D. Wang : “Algebraic approaches to stability analysis of biological systems”, Mathematics in Computer Science, vol. 1 (3), pp. 507-539, (Springer) (2008)