Hamilton's rule meets the Hamiltonian: kin selection on dynamic characters

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Abstract
Many biological characters of interest are temporal sequences of decisions. The evolution of such characters is often modelled using dynamic optimization methods such as the maximum principle. A quantity central to these analyses is the 'Hamiltonian' function, named after the mathematician William R. Hamilton. On the other hand, evolutionary models in which individuals interact with relatives are usually based on Hamilton's rule, named after the evolutionary biologist William D. Hamilton. In this article we present a generalized maximum principle that includes the effects of interactions among relatives and we show that a time-dependent (dynamic) version of Hamilton's rule holds involving the Hamiltonian. This result brings together the power and generality of both the maximum principle and Hamilton's rule thereby providing a natural framework for understanding the evolution of 'dynamic' characters under kin selection.

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