Ultradiscretization in discrete limit cycles of tropically discretized and max-plus Sel’kov models

Abstract

Due to phase lock caused by a saddle-node bifurcation, the states of limit cycles for a tropically discretized Sel’kov model become equivalent to those for its ultradiscretized max-plus model. This property is essentially the same as the case of the negative feedback model, and existence of a general mechanism for ultradiscretization of the limit cycles is suggested. Furthermore, we find the logarithmic dependence of the time to pass the bottleneck for phase drift motion in the vicinity of the bifurcation point. This dependency can be understood as a consequence of the piecewise linearization by applying the ultradiscrete limit.