A Numerical Method for Analyzing the Stability of Bi-Parametric Biological Systems

For a biological system modeled by a continuous dynamical system defined by rational functions with two parameters, we propose a numerical method to compute the fold and Hopf bifurcation boundaries of the system restricted in a finite region in the parametric space under certain assumptions. The bifurcation boundaries divide their complement in the region into connected subsets, called cells, such that above each of them the number of equilibria is constant and the stability of each equilibrium remains unchanged. The boundaries are generated by first tracing the fold and Hopf bifurcation curves in a higher dimensional space and then projecting them onto the parameter plane. One advantage of this method is that it can exploit global information of real varieties and generate complete boundaries based on homotopy continuation methods and critical point techniques. The bistability properties of several biological systems are successfully analyzed by our method.