Mathematical Analysis of a Cellular Control Process with Positive Feedback

This paper studies a three-dimensional system of equations proposed by J. S. Griffith to model a cellular process for control of gene expression by positive feedback. The origin is a critical point for all parameter values. For parameter values where the positive octant contains one other critical point, we show that all nonzero solutions in that octant are positively asymptotic to this critical point. For another range of parameter values the positive octant contains two nonzero critical points, lying on a ray from the origin. We show that there is a two-dimensional surface of solutions positively asymptotic to the smaller of these two critical points; all nonzero solutions in the positive octant on one side of this surface are positively asymptotic to the origin and all nonzero solutions on the other side are positively asymptotic to the larger critical point.