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Relative species abundance of replicator dynamics with sparse interactions

A theory of relative species abundance on sparsely-connected networks is presented by investigating the replicator dynamics with symmetric interactions. Sparseness of a network involves difficulty in analyzing the fixed points of the equation, and we avoid this problem by treating large self interaction $u$, which allows us to construct a perturbative expansion. Based on this perturbation, we find that the nature of the interactions is directly connected to the abundance distribution, and some characteristic behaviors, such as multiple peaks in the abundance distribution and all species coexistence at moderate values of $u$, are discovered in a wide class of the distribution of the interactions. The all species coexistence collapses at a critical value of $u$, $u_c$, and this collapsing is regarded as a phase transition. To get more quantitative information, we also construct a non-perturbative theory on random graphs based on techniques of statistical mechanics. The result shows those characteristic behaviors are sustained well even for not large $u$. For even smaller values of $u$, extinct species start to appear and the abundance distribution becomes rounded and closer to a standard functional form. Another interesting finding is the non-monotonic behavior of diversity, which quantifies the number of coexisting species, when changing the ratio of mutualistic relations $Δ$. These results are examined by numerical simulations, and the multiple peaks in the abundance distribution are confirmed to be robust against a certain level of modifications of the problem. The numerical results also show that our theory is exact for the case without extinct species, but becomes less and less precise as the proportion of extinct species grows.