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We provide deterministic approximation results for stochastic processes modeling nite and asymptotically increasing populations. First, we give results concerning continuous time ODE approximations in well-mixed populations. The connection between the stable rest points of resulting differential equations and Nash equilibriums of the game are given. Second, we consider similar approximation results with spatial non-local interactions. We obtain the system of non-local replicator equations as the meso-scopic limit of the patial process and explore the resulting spatio-temporal equation for Prisoners Dilemma game in terms of traveling waves and spreading speeds. As a result, we conclude that cooperation is an evolutionary stable strategy given that the spreading speed is suciently large or small and the initial data are compactly supported. Lastly, we approximate the evolutionary games in Wright-Fisher process by dierence equations. We give an exponential bound on probability of maximal deviations. This result allows us to use the attroctor and repellor sets of the dierence equations to get information about the Morkov chains for large or asymptotically increasing populations.