Some properties of natural selection with single and multiple constraints

The properties of selection restricted by single and multiple constraints are examined by using the Lagrange and Kuhn-Tucker conditions of calculus. We show for a general set of fitness equations containing any number of strategy components and subject to any single differentiable equality constraint that the marginal fitnesses of any two strategy components are equal at the evolutionarily stable strategy (ESS) when expenditures are measured in the same units, those of a binding constraint. Equal marginal advantages are a necessary, though not usually a sufficient, condition for an interior ESS. When selection is operating under more than one constraint, the marginal fitnesses of any two strategy components are equal at the ESS whenever both components are affected by only one, and the same, binding constraint. The equalization of marginal fitnesses allows the positions of constrained fitness maxima to be explored in theoretical models or empirical tests and is a convenient heuristic for understanding selection.