Asymptotic properties of molecular mechanism models with functional response

Abstract

In this study, several modified types of the Goodwin model are proposed to demonstrate oscillatory behaviors. The original Goodwind model is the prototype model of a negative feedback loop that exhibits oscillations. But it exhibits oscillations when the cooperativity is very high, n > 8, which is unrealistic in most biological situations. We show two cases of a modified Goodwin model with functional responses of order n = 5 and n = 6 which exhibit periodic oscillations. It is shown that the modified functional responses lead to produce oscillatory solutions in these cases. This result suggests the possibility of constructing models to have oscillatory solutions with lower order functional responses than the requirement n > 8 for the original Goodwin model case.