Evolutionarily stable strategy, stable state, periodic cycle and chaos in a simple discrete time two-phenotype model.

A simple discrete time two-phenotype matrix game model is investigated. In this model, according to the suggestion of Vincent & Fisher (1988, Evolutionary Ecology 2, 321-337), the fitness of an individual is defined to be an exponential function of its expected pay-off value. The results show that : (i) in our model, the static conditions of ESS are only dependent on the properties of pay-off matrix, but not on the specific form of fitness function. This result implies that the ESS conditions on our model are completely identical with the conditions in the two-phenotype model with linear fitness function. (ii) In our model, the relationship between the static conditions of ESS and the dynamic properties of the pure strategy model is that if the interior fixed point of the pure strategy model is not an ESS-equilibrium, then it must be unstable; conversely, if the interior fixed point of the pure strategy model is an ESS-equilibrium, then it can be stable or unstable, and an unstable ESS-equilibrium must correspond to the cyclic or chaotic behaviour of the population state.