Direct parametric analysis of an enzyme-catalyzed reaction model

In this paper, we discuss a substrate-activator system, which exhibits abundant dynamical behaviours as illustrated by numerical simulations. This system depends on a cubic polynomial with such a complicated relation between its coefficients and the original parameters that the coordinates of equilibria or even the number of equilibria can hardly be determined in many cases. All previous results on its qualitative properties and bifurcations are given indirectly for the artificial parameter s(*), a coordinate of a general equilibrium, and the analysis of its dynamics remains far from completion. In this paper, not following the common idea of computing eigenvalues at equilibria, we give a complete analysis of equilibria directly for those original parameters by using continuity, monotonicity and some techniques of inequality. For a global analysis, we also discuss its equilibria at infinity, one of which possesses degeneracy so high sometimes that neither the well-known normal sector method nor the blowing-up method can be used easily. Furthermore, overcoming those difficulties from not solving all coordinates of equilibria, we give a versal unfolding with its original parameters to the degenerate cases and present bifurcation curves of periodic orbits and homoclinic orbits explicitly.