An interval polynomial theory approach to fixed-point bifurcation analysis in biological models

A systematic method for fixed point bifurcations analysis in biological models is proposed using interval polynomials theory. Fixed point bifurcation analysis comprises the following stages (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system´s nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To compute the values of the roots of the characteristic polynomial and to study the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov´s stability theorem is proposed. In this approach the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed method for the analysis of fixed points bifurcations in nonlinear models of circadian cells is tested through numerical and simulation experiments.