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Background In recent years, the average abundance function has attracted much attention as it reflects the degree of cooperation in the population. Then it is significant to analyse how average abundance functions can be increased to promote the proliferation of cooperative behaviour. However, further theoretical analysis for average abundance function with mutation under redistribution mechanism is still lacking. Furthermore, the theoretical basis for the corresponding numerical simulation is not sufficiently understood. Results We have deduced the approximate expressions of average abundance function with mutation under redistribution mechanism on the basis of different levels of selection intensity $$\omega$$ ω (sufficiently small and large enough). In addition, we have analysed the influence of the size of group d, multiplication factor r, cost c, aspiration level $$\alpha$$ α on average abundance function from both quantitative and qualitative aspects. Conclusions (1) The approximate expression will become the linear equation related to selection intensity when $$\omega$$ ω is sufficiently small. (2) On one hand, approximation expression when $$\omega$$ ω is large enough is not available when r is small and m is large. On the other hand, this approximation expression will become more reliable when $$\omega$$ ω is larger. (3) On the basis of the expected payoff function $$\pi \left( \centerdot \right)$$ π ⋅ and function $$h(i,\omega )$$ h ( i , ω ) , the corresponding results for the effects of parameters (d,r,c,$$\alpha$$ α ) on average abundance function $$X_{A}(\omega )$$ X A ( ω ) have been explained.

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