THE SPATIAL STRUGGLE OF TIT-FOR-TAT AND DEFECT

The pioneering work by Trivers (1971), Axelrod (1984) and Axelrod and Hamilton (1981) has stimulated continuing interest in explaining the e volution of cooperation by game theory, in particular, the iterated pr isoner's dilemma and the strategy of tit-for-tat. However these models suffer from a lack of biological reality, most seriously because it i s assumed that players meet opponents at random from the population an d, unless the population is very small, this excludes the repeated enc ounters necessary for tit-for-tat to prosper. To meet some of the obje ctions, we consider a model with two types of players, defectors (D) a nd tit-for-tat players (T), in a spatially homogeneous environment wit h player densities varying continuously in space and time. Players onl y encounter neighbours but move at random in space. The analysis demon strates major new conclusions, the three most important being as follo ws. First, stable coexistence with constant densities of both players is possible. Second, stable coexistence in a pattern (a spatially inho mogeneous stationary state) may be possible when it is impossible for constant distributions (even unstable ones) to exist. Third, invasion by a very small number of T-players is sometimes possible (in contrast with the usual predictions) and so a mutation to tit-for-tat may lead to a population of defectors being displaced by the T-players.