Oscillations in Monotone Systems with a Negative Feedback

Abstract

We study a finite-dimensional monotone system coupled to a slowly evolving scalar differential equation which provides a negative feedback to the monotone system. We use a theory of multivalued characteristics to show that this system admits a relaxation periodic orbit if a simple model system in $\mathbf{R}^2$ does. Our construction can be used to prove the existence of periodic orbits in slow-fast systems of arbitrary dimension. We apply our theory to a model of a cell cycle in Xenopus embryos. Abrupt changes in signals upon entry to mitosis suggests that the cell cycle is generated by a relaxation oscillation. Our results show that the cell cycle orbit is not a relaxation oscillator. However, we construct a closely related system that exhibits relaxation oscillations and that approximates the cell cycle oscillator for an intermediate range of negative feedback strengths. We show that the cell cycle oscillation disappears if the negative feedback is too weak or too strong.