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The Dynamics of a Spatial Cyclic Competition System

We analyze the long term behavior of a system involved in cyclic competition similar to the rock-paper-scissors game. Previous studies have used cellular autonoma simulations to model the stochastic interactions and mean field equations to approximate this stochastic model. However, mean field approximation does not properly account for spatial correlations, leading to loss of spatial significance. We use pair approximations to model the local interactions with a system of differential equations. We then investigate the outcome of various initial conditions of the pair approximation model using numerical integration. Three categories of initial conditions are found that lead to three distinct behaviors: one species is present, all three species oscillate around the fixed point resulting in a heteroclinic cycle, or convergence to an interior fixed point.  We also explore the relative importance of initial conditions and lattice size on fixation probabilities for each species. We finally discuss the implications of these results in a biological context.

  • Poster session award recipient at the 2009 National SACNAS Convention in Dallas, TX
  • Poster session award recipient at the 2010 AMS/MAA Joint Mathematics Meeting in San Francisco, CA

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Jeffery Merckens,
Bryce van de Geijn,
Ioana Hociota,
Don Tadaya,
Benjamin Morin