Dynamical properties of discrete negative feedback models

TL;DR

This paper investigates the dynamical behaviors of discrete negative feedback models, including bifurcations and limit cycles, using tropical discretization and max-plus algebra, revealing new insights into their stability and ultradiscrete states.

Contribution

It rederives bifurcation conditions with a novel approach and uncovers ultradiscrete states and multiple limit cycles in discretized and max-plus negative feedback models.

Findings

Neimark-Sacker bifurcation conditions are rederived.

Ultradiscrete states emerge for large time intervals.

Two limit cycles are identified in the max-plus model.

Abstract

Dynamical properties of tropically discretized and max-plus negative feedback models are investigated. Reviewing the previous study [S. Gibo and H. Ito, J. Theor. Biol. 378, 89 (2015)], the conditions under which the Neimark-Sacker bifurcation occurs are rederived with a different approach from their previous one. Furthermore, for limit cycles of the tropically discretized model, it is found that ultradiscrete state emerges when the time interval in the model becomes large. For the max-plus model, we find the two limit cycles; one is stable and the other is unstable. The dynamical properties of these limit cycles can be characterized by using the Poincar\'e map method. Relationship between ultradiscrete limit cycle states for the tropically discretized and the max-plus models is also discussed.