Critical slowing down close to a global bifurcation of a curve of quasineutral equilibria

Abstract

Critical slowing down arises close to bifurcations and involves long transients. Despite slowing down phenomena have been widely studied in local bifurcations i.e., bifurcations of equilibrium points, less is known about transient delay phenomena close to global bifurcations. In this paper, we identify a novel mechanism of slowing down arising in the vicinity of a global bifurcation i.e., zip bifurcation, identified in a mathematical model of the dynamics of an autocatalytic replicator with an obligate parasite. Three different dynamical scenarios are first described, depending on the replication rate of cooperators, $(L)$, and of parasites, $(K)$. If $K<L$ the system is $\underline{\text { bistable }}$ and the dynamics can be either the outcompetition of the parasite or the two-species extinction. When $K>L$ the system is monostable and both species become extinct. In the case $K=L$ coexistence of both species takes place in a Curve of Quasi-Neutral Equilibria (CQNE). The novel slowing down mechanism identified is due to an underlying ghost CQNE for the cases $K \lesssim L$ and $K \gtrsim L$. We show, both analytically and numerically, that the delays caused by the ghost CQNE follow scaling laws of the form $\tau \sim|K-L|^{-1}$ for both $K \lesssim L$ and $K \gtrsim L$. We propose the ghost CQNE as a novel transientgenerator mechanism in ecological systems.