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We analyze the stability of a nonlinear dynamical model describing the noncooperative strategic interactions among the agents of a finite collection of populations. Each agent selects one strategy at a time and revises it repeatedly according to a protocol that typically prioritizes strategies whose payoffs are either higher than that of the current strategy or exceed the population average. The model is predicated on well-established research in population and evolutionary games, and has two sub-components. The first is the payoff dynamics model (PDM), which ascribes the payoff to each strategy according to the proportions of every population adopting the available strategies. The second sub-component is the evolutionary dynamics model (EDM) that accounts for the revision process. In our model, the social state at equilibrium is a best response to the payoff, and can be viewed as a Nash-like solution that has predictive value when it is globally asymptotically stable (GAS). We present a systematic methodology that ascertains GAS by checking separately whether the EDM and PDM satisfy appropriately defined system-theoretic dissipativity properties. Our work generalizes pioneering methods based on notions of contractivity applicable to memoryless PDMs, and more general system-theoretic passivity conditions. As demonstrated with examples, the added flexibility afforded by our approach is particularly useful when the contraction properties of the PDM are unequal across populations.