Comet Calendar Event Details

Mathematical Sciences Colloquium by Thomas Hagen
Thursday, Mar. 28
11 a.m. - noon Location: FN 2.104

Thomas Hagen

Department of Mathematical Sciences
The University of Memphis

The Winner Takes All: Volume Scavenging Populations of Networked Droplets

A model of competition (scavenging) over time amongst a population of N individuals (N liquid droplets) for a limited resource (liquid volume) is formulated and solved. The eventual outcome depends sensitively on the average resource per individual V. For abundant resource, V > 1, there is one winner only and that winner eventually scavenges all or most of the resource. This is known as the "winner-take-all" outcome. For less than abundant resource, V < 1, the likely eventual outcome is a resource that is partitioned evenly amongst the N individuals. This is known as the "all-share-evenly" or egalitarian outcome. In addition to predicting what kind and how many winners, the analysis shows that once an individual's resource shrinks below a certain threshold, that individual can never recover. This is the "once down-and-out, always down-and-out" outcome. Simulations can identify winners. Selected simulations suggest that the winner depends sensitively on population size and the trading friction or inefficiency of resource exchange between individuals (liquid theology). Friction turns out to strongly influence the time to reach a rest state (equilibrium), in surprising ways. Of all feasible rest states, only certain ones are reachable (stable equilibria). Finally, the analysis reveals a line-up of rest state "partitionings", ordered in hierarchies of more- versus less-preferential outcomes.

The underlying model we analyze in this presentation is given by a large gradient system of ordinary differential equations on a simple, connected graph. Remarkably, both winner-take-all and all-share-evenly behaviors can be observed within the same model which accounts for scavenging amongst neighbors owing to the action of surface tension. Volume exchange arises by pressure (curvature) differences that drive liquid from one to another droplet along a network of interconnected channels. In this way, certain droplets gain volume at the expense of others. Various networks are considered. We will report about previously unknown equilibrium solutions, a (surprisingly) complete, rigorous classification of their stability in dependence on the relevant bifurcation parameters as well as related results on forward invariant sets and hierarchies of equilibria. The predicted long-term dynamics will be demonstrated with some animations. This is joint work with Paul H. Steen (Cornell).


Coffee to be served outside FN 2.102 30 minutes prior to the talk.



Contact Info:
Viswanath Ramakrishna, 972-883-6873
Questions? Email me.

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