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Boundary dynamics of the replicator equations for neutral models of cyclic dominance

The author is supported by an NSERC Discovery grant

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  • We study the replicator equations, also known as mean-field equations, for a simple model of cyclic dominance with any number $ m $ of strategies, generalizing the rock-paper-scissors model which corresponds to the case $ m = 3 $. Previously the dynamics were solved for $ m\in\{3,4\} $ by consideration of $ m-2 $ conserved quantities. Here we show that for any $ m $, the boundary of the phase space is partitioned into heteroclinic networks for which we give a precise description. A set of $ {\lfloor} m/2{\rfloor} $ conserved quantities plays an important role in the analysis. We also discuss connections to the well-mixed stochastic version of the model.

    Mathematics Subject Classification: Primary: 92D40.


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  • Figure 1.  Simulation output of $ u_0(t) $ from (1) for various $ m $, with initial data $ u(0) = (0.01,0.99/(m-1),\dots,0.99/(m-1)) $, chosen to be close to $ \partial S_m $

    Figure 2.  Simulation output of (1) with initial data $ u(0) = (0,1/(m-1),\dots,1/(m-1)) $

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