Zip and velcro bifurcations in competition models in ecology and economics

TL;DR

This paper highlights historical and recent research on bifurcations involving continuous sets of equilibria, specifically zip and velcro bifurcations, in ecology and economics models.

Contribution

It revisits and emphasizes the early work of Miklós Farkas on zip and velcro bifurcations, connecting it to recent studies in the field.

Findings

Identification of zip bifurcations with line segments of equilibria.

Identification of velcro bifurcations with planes of equilibria.

Description of prototypical systems exhibiting these bifurcations.

Abstract

During the last six years or so, a number of interesting papers discussed systems with line segments of equilibria, planes of equilibria, and with more general equilibrium configurations. This note draws attention to the fact that such equilibria were considered previously by Mikl\'os Farkas (1932-2007), in papers published in 1984-2005. He called zip bifurcations those involving line segments of equilibria, and velcro bifurcations those involving planes of equilibria. We briefly describe prototypical situations involving zip and velcro bifurcations.