Motivated by evolutionary biology, we study general infinite-dimensional dynamical systems involving two species - the resident and the invader. Sufficient conditions for competition exclusion phenomena are given when the two species play similar, but distinct, strategies. Those conditions are based on invasibility criteria, for instance, evolutionarily stable strategies in the framework of adaptive dynamics.These types of questions were first proposed and studied by S. Geritz et al. and S. Geritz for a class of ordinary differential equations. We extend and generalize previous work in two directions. Firstly, we consider analytic semiflows in infinite-dimensional spaces. Secondly, we devise an argument based on Hadamard's graph transform method that does not depend on the monotonicity of the two-species system. Our results are applicable to a wide class of reaction-diffusion models as well as models with nonlocal diffusion operators.
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