We study the stochastic spatial evolutionary game with death-birth updating \to  . The model is a voter model perturbation. Here , we require perturbation strengths w such that 1/N\ll w\ll 1 for typical expanding populations of size N%DIFDELCMD < \to %%% \infty; this range is in contrast to w=\mathcal O(1/N) for the diffusion approximations of the model. The first main result proves that the density processes of strategy converge to the replicator equation in the limit of a large finite population subject to appropriate, but general, conditions. Multiple strategies and mutations are allowed. Moreover, after normalization, the limiting fluctuations are shown to be Gaussian with a covariance function in the replicator equation's solution through the Wright--Fisher noise coefficient. In particular, to the degree of using constants that may depend implicitly on space, we prove a conjecture from the biological literature that the expected density processes approximate the replicator equation on many non-regular graphs.
We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size N\to \infty . The model is a voter model perturbation. For typical populations , we require perturbation strengths satisfying 1/N\ll w\ll %DIFDELCMD < \to %%% 1. Under appropriate conditions on the space, the limiting density processes of strategy are proven to obey the replicator equation , and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the biological literature that the expected density processes approximate the replicator equation on many non-regular graphs.
R
stochastic spatial
multi-strategy stochastic
A
in expanding spatial populations of size N
A
\infty
R
Here
For typical populations
R
w such that
satisfying
D
1 for typical expanding populations of size N
R
\infty; this range is in contrast to w=\mathcal O(1/N) for the diffusion approximations of the model. The first main result proves that the
1. Under appropriate conditions on the space, the limiting
R
converge to
are proven to obey
R
in the limit of a large finite population subject to appropriate, but general, conditions. Multiple strategies and mutations are allowed. Moreover, after normalization, the limiting fluctuations are shown to be Gaussian with a
, and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher
R
replicator equation's solution through the Wright--Fisher noise coefficient. In particular, to the degree of using constants that may depend implicitly on space, we prove
limiting densities. As an application, we resolve in the positive