Dynamical effects of loss of cooperation in discrete-time hypercycles

Abstract

Hypercycles' dynamics have been widely investigated in the context of origins of life, especially using time-continuous dynamical models. Different hypercycle architectures jeopardising their stability and persistence have been discussed and investigated, namely the catalytic parasites and the short-circuits. Here we address a different scenario considering RNA-based hypercycles in which cooperation is lost and catalysis shifts to density-dependent degradation processes due to the acquisition of cleaving activity by one hypercycle species. That is, we study the dynamical changes introduced by a functional shift. To do so we use a discrete-time model that can be approached to the time continuous limit by means of a temporal discretisation parameter, labelled $C$. We explore dynamical changes tied to the loss of cooperation in two-, three-, and fourmember hypercycles in this discrete-time setting. With cooperation, the all-species coexistence in two- and three-member hypercycles is governed by an internal stable fixed point. When one species shifts to directed degradation, a transcritical bifurcation takes place and the other hypercycle members go to extinction. The asymptotic dynamics of the four-member system is governed by an invariant curve in its cooperative regime. For this system, we have identified a simultaneous degenerate transcritical-NeimarkSacker bifurcation as cooperation switches to directed degradation. After these bifurcations, as we found for the other systems, all the cooperative species except the one performing degradation become extinct. Finally, we also found that the observed bifurcations and asymptotic dynamical behaviours are independent of $C$. Our results can help in understanding the impact of changes in ecological interactions (i.e., functional shifts) in multi-species systems and to determine the nature of the transitions tied to coextinctions and out-competition processes in both ecosystems and RNA-based systems.