Hopf bifurcation and periodic solutions in a sustainable supply chain defined by a planar system of ordinary differential equations

Abstract

We analyze the stability of the equilibria and bifurcations of a planar system of ordinary differential equations describing the product-resource interaction in a sustainable supply chain. While periodic behavior in supply chain models has often been documented either in systems of higher dimensions or in delayed systems, here a Hopf bifurcation arises in a planar and delay-free system. We show that the interior equilibrium loses stability through a supercritical Hopf bifurcation as the environmental capacity exceeds a critical threshold that depends on the maximum production rate, the resource level at which production reaches half its maximum rate, and the demand and remanufacturing rates. We derive this threshold explicitly and provide estimates for the amplitude and period of the periodic solution close to the Hopf bifurcation, by means of the resulting Hopf normal form. We then further substantiate our analytical results through numerical simulations.