## Residents Have Fitness Zero

For any long-existing community, at the E generated by that community the invasion fitnesses of all composing types equal zero. This is because any type that has invasion fitness different from zero either explodes or dies out, and therefore cannot be a permanent member of the community. For community models having equilibria corresponding to nonfluctuating environments, setting the invasion fitnesses of all types equal to zero produces as many equations for E as there are types. These can be combined with the equilibrium equations derived from the mechanisms that produce the environmental condition from the population outputs to the environment. The latter output can be calculated from the population sizes times the corresponding normalized stationary h-state distributions. The resulting set of equations precisely matches the set of unknowns, so that it is possible in principle through this route to determine the environments that can be generated by the community.

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