Fighting for food: a dynamic version of the Hawk-Dove game

The Hawk-Dove game (Maynard Smith, 1982) has been used to analyse conflicts over resources such as food. At the evolutionarily stable strategy (ESS), either a proportion p* of animals always play Hawk, or each animal has a probability p* of playing Hawk. We modify the standard Hawk-Dove game to include a state variable, x, that represents the animal's level of energy reserves. A strategy is now a rule for choosing an action as a function of x and time of day. We consider a non-reproductive period and adopt the criterion of minimizing mortality over this period. We find the ESS, which has the form 'play Hawk if reserves are below c*(t) at time t, otherwise play Dove'. This ESS is very different from the ESS in the standard Hawk-Dove game. It is a pure ESS that depends on the animal's state and on time. Furthermore, it is characterized by the strong condition that any single mutant that does not adopt the ESS suffers a reduction in fitness. The standard Hawk-Dove game assumes pay-offs that are related to fitness; our approach starts from a definition of fitness and derives the pay-offs in the process of finding the ESS. When the environment becomes worse (e. g. food becomes less reliable or energy expenditure increases) the ESS changes in such a way as to increase the proportion of animals that will play Hawk.