Fast game theory coupled to slow population dynamics: the case of domestic cat populations

We study a deterministic model of a population where individuals alternatively adopt hawk and dove tactics. It is assumed that the hawk and dove individuals compete for some resources at a fast time scale. This fast part of the model is coupled to a slow part that describes the growth of the population. It is shown that, in a constant game matrix, the population grows according to a logistic curve whose r and K parameters are related to the payoff of the tactics. Results show that the highest population density is obtained when all individuals are dove. We also study a density-dependent game matrix for which the gain is a function of the population density. In this case, we show that two stable equilibria can occur, a first one at low density with a high proportion of hawk individuals and a second one at large density with a low proportion of hawk individuals. Our model is applied to domestic cat populations for which the behavior of individuals in competition with one another can be modeled by two tactics: hawk and dove. Such tactics change with density of population. The results of the model agree well with observed data: high-density populations of domestic cats are mainly doves, whereas low-density populations are mainly hawks.