Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

Wei Niu (1) , Dongming Wang (1)

1 SALSA - Solvers for Algebraic Systems and Applications

In this paper, we show how to analyze bifurcation and limit cycles for biological systems by using an algebraic approach based on triangular decomposition, Gröbner bases, discriminant varieties, real solution classification, and quantifier elimination by partial CAD. The analysis of bifurcation and limit cycles for a concrete two-dimensional system, the self-assembling micelle system with chemical sinks, is presented in detail. It is proved that this system may have a focus of order 3, from which three limit cycles can be constructed by small perturbation. The applicability of our approach is further illustrated by the construction of limit cycles for a two-dimensional Kolmogorov prey-predator system and a three-dimensional Lotka-Volterra system.

Domaines: Systèmes dynamiques [math.DS]; Calcul formel [cs.SC]

Wei Niu, Dongming Wang. Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems. AB 2008 - 3rd International Conference on Algebraic Biology, Jul 2008, Hagenberg, Austria. pp.156-171, ⟨10.1007/978-3-540-85101-1_12⟩. ⟨hal-00588729⟩