Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

Abstract

Abstract. 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.

Stability Analysis for Discrete Biological Models Using Algebraic Methods

“…In [33], several cases of the continuous Cinquin-Demongeot models which describe multistable switches have been studied. By using the Euler discretization, one may obtain a discrete Cinquin-Demongeot model of the form…”

“…A continuous stage-structured model of an Allee effect, derived from [42], has been analyzed by using the algebraic methods described in [33]. Here the Euler discretization method (see, e.g., [24, pp.…”

“…For biological networks modeled as continuous dynamical systems, a general algebraic approach has been proposed in [34,45] for the detection and analysis of stability of real equilibria. This approach has also been applied to the analysis of bifurcations and limit cycles for certain biological systems [33]. In this paper, we adapt, extend, and apply the approach for stability analysis of discrete dynamical systems, which are simply structured and intuitive to biologists.…”