Abstract: This article explores the characteristics of the average abundance function with mutation on the basis of the multi-player snowdrift evolutionary game model by analytical analysis and numerical simulation. The specific field of this research concerns the approximate expressions of the average abundance function with mutation on the basis of different levels of selection intensity and an analysis of the results of numerical simulation on the basis of the intuitive expression of the average abundance function. In addition, the biological background of this research lies in research on the effects of mutation, which is regarded as a biological concept and a disturbance to game behavior on the average abundance function. The mutation will make the evolutionary result get closer to the neutral drift state. It can be deduced that this affection is not only related to mutation, but also related to selection intensity and the gap between payoff and aspiration level. The main research findings contain four aspects. First, we have deduced the concrete expression of the expected payoff function. The asymptotic property and change trend of the expected payoff function has been basically obtained. In addition, the intuitive expression of the average abundance function with mutation has been obtained by taking the detailed balance condition as the point of penetration. It can be deduced that the effect of mutation is to make the average abundance function get close to 1/2. In addition, this affection is related to selection intensity and the gap. Secondly, the first-order Taylor expansion of the average abundance function has been deduced for when selection intensity is sufficiently small. The expression of the average abundance function with mutation can be simplified from a composite function to a linear function because of this Taylor expansion. This finding will play a significant role when analyzing the results of the numerical simulation. Thirdly, we have obtained the approximate expressions of the average abundance function corresponding to small and large selection intensity. The significance of the above approximate analysis lies in that we have grasped the basic characteristics of the effect of mutation. The effect is slight and can be neglected when mutation is very small. In addition, the effect begins to increase when mutation rises, and this effect will become more remarkable with the increase of selection intensity. Fourthly, we have explored the influences of parameters on the average abundance function with mutation through numerical simulation. In addition, the corresponding results have been explained on the basis of the expected payoff function. It can be deduced that the influences of parameters on the average abundance function with mutation will be slim when selection intensity is small. Moreover, the corresponding explanation is related to the first-order Taylor expansion. Furthermore,the influences will become notable when selection intensity is large.

Key words: evolutionary game, average abundance function, mutation, stochastic process