Asymptotic Behavior of Solutions to Multiple Loop Positive Feedback Systems

Abstract

This paper shows that the asymptotic behavior of solutions to a system of ordinary differential equations which models multiple loop positive feedback in biochemical control circuits is similar to the asymptotic behavior for single loop positive feedback systems. Specifically, the positive orthant is positively invariant and positive-time solutions are bounded. The critical points in the positive orthant are ordered by strict inequality. Each critical point is either asymptotically stable or unstable. Each nondegenerate unstable critical point has two orbits leaving it in opposite directions and each of these orbits is asymptotic to the adjacent critical point. The regions of attraction of the critical points are studied. In particular, for dimensions two and three, the stable manifolds of the unstable critical points separate the positive orthant into regions of attraction. Thus the orbit of each point is asymptotic to some critical point.