The winner takes all: Volume-scavenging populations of networked droplets

Abstract

In this work we present and analyze a fluid-mechanical model of competition (scavenging) amongst $N$ liquid droplets (individual competitors). The eventual outcome of this competition depends sensitively on the average resource (volume) per individual $\overline{V}$. For abundant resource, $\overline{V}>1$, there is one winner only and that winner eventually scavenges all or most of the resource. In the socio-economic realm this is is known as the "winner-take-all" outcome: A disproportionately large reward falls to one or a few winners, even though other competitors start out with comparable (or even slightly more) resource and perform only marginally worse. The losing competitors are not rewarded. For less than abundant resource, $\overline{V}<1$, an outcome with resource that is evenly partitioned amongst the $N$ droplets becomes possible. This is the "all-share-evenly" or egalitarian outcome. For sufficiently scarce resource the egalitarian outcome is the only one that can occur. In addition to predicting what kind and how many winners, our analysis shows that once an individual's resource (droplet volume) falls below a fixed threshold, that individual can neither recover nor emerge as winner. This is the "once down-and-out, always down-and-out" outcome. Selected simulations suggest that the winner depends sensitively on population size and the "trading friction" or inefficiency of resource exchange between individuals (liquid rheology). Of all feasible rest states (equilibria), only certain ones are reachable (stable equilibria). Friction turns out to strongly influence the time to reach an end state (stable equilibrium), in surprising ways. Besides the end states, our analysis reveals an array of rest states, ordered in hierarchies of more versus less costly (energetic) outcomes.