Evolutionary Game Dynamics Of The Mutants Under Random Payoff Matrices

Evolutionary game theory originates from mathematical biology. It becomes a multi-disciplinarytheory which involves mathematics, biology, economics and sociology after decades of development.Classical evolutionary game theory is based upon infinite populations. Replicator dynamics equationprovides effective approaches to the study on evolutionary dynamics in infinite populations. However,the real populations in nature are often finite, and the evolution of the populations is affected by anumber of random factors. Therefore, it is meaningful to consider randomness in the study based onfinite populations.Based on two kinds of classical evolutionary models for finite populations, namely Moranprocess and Wright-Fisher process, we introduce random variables into the payoff matrices of thegames and focus on the fixation probabilities and fixation times of the mutants. In chapter2, we studythe fixation probabilities of the mutants in Moran process. According to the updating mechanisms ofMoran process, we first simulate the distributions of the fixation probabilities of the mutants. Theresults indicate that the fixation probabilities of the mutants represent evidently random distributionsunder random payoff matrices, and the distributions change with the selection intensities. Then wecalculate the expressions for the expectations of the fixation probabilities under weak selection, anddiscuss their relations with the distributions of the random variables in payoff matrices. At last wepresent the changes of the fixation probabilities of the mutants under strong selection. Fixation timesare as important as the fixation probabilities in describing the process of biological evolution. Thus inchapter3, based on Moran process again we derive the expressions of the expectations of bothunconditional and conditional fixation probabilities of the mutants under weak selection, and discusstheir relations with the payoff and selection intensity. We then demonstrate our results by numericalsimulations. Based on an important synchronous updating process, namely Wright-Fisher process, inchapter4we also introduce random variables into the payoff matrices and derive the expressions ofthe fixation probabilities of the mutants by using the weak selection approximations and the totalprobability formula.