Score-dependent fertility model for the evolution of cooperation in a lattice.

The evolution of cooperation is studied in a lattice-structured population, in which each individual plays the iterated Prisoner's Dilemma game with its neighbors. The population includes Tit-for-Tat (TFT, a cooperative strategy) and All Defect (AD, a selfish strategy) distributed over the lattice points. An individual dies randomly, and the vacant site is filled immediately by a copy of one of the neighbors in which the probability of colonization success by a particular neighbor is proportional to its score accumulated in the game. This "score-dependent fertility model" (or fertility model) behaves very differently from score-dependent viability model (viability model) studied in a previous paper. The model on a one-dimensional lattice is a analysed by invasion probability analysis, pair-edge method mean-field approximation, pair approximation, and computer simulation. Results are: (1) TFT players come to form tight clusters. When the probability of iteration w is large, initially rare TFT can invade and spread in a population dominated by AD, unlike in the complete mixing model. The condition for the increase of TFT is accurately predicted by all the techniques except mean-field approximation; (2) fertility model is much more favorable for the spread of TFT than the corresponding viability model, because spiteful killing of neighbors is favored in the viability model but not in the fertility model; (3) eight lattice games on two-dimensional lattice with different assumptions are examined. Cooperation and defects can coexist in the models of deterministic state change but not in the models of stochastic state change.