A generalization of Pontryagin's maximum principle for dynamic evolutionary games among relatives

We present two theorems that generalize Pontryagin's maximum principle to the setting of dynamic evolutionary games between genetically related individuals. The two theorems correspond to two types of interactions among individuals: patch-structured populations in which individuals locally "play the field" and pairwise interactions. These generalizations can be used in the same way that Pontryagin's maximum principle is used and they are valid for diploid organisms under a single locus, diallelic genetic model. These generalizations involve an interesting, dynamic version of Hamilton's Rule from inclusive fitness theory. We illustrate how these theoretical results can be applied by modeling the evolution of lifetime resource allocation to growth and reproduction in an annual plant when there is competition for resources among related individuals.