Abstract: We introduce a notion of monotonicity for multi-species systems of partial differential equations governed by mass-preserving flow dynamics, extending monotonicity in Banach spaces to the Wasserstein-2 metric space. We show that monotonicity implies the existence of and convergence to a unique steady state, convergence of the velocity fields and second moments, and contraction in the Wasserstein-2 metric. In the special setting where each species follows a Wasserstein-2 gradient flow of its own energy, we prove convergence to the unique Nash equilibrium of the associated energies and delineate the relationship between monotonicity and displacement convexity. This extends known zero-sum results in infinite-dimensional game theory to the general-sum setting. Examples of such multi-species flows include cross-diffusion, gradient flows with potentials, nonlocal interaction, linear and nonlinear diffusion, and min-max systems, and we also draw connections to a class of mean-field games.

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