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We investigate the replicator dynamics with "sparse" symmetric interactionswhich represent specialist-specialist interactions in ecological communities.By considering a large self interaction $u$, we conduct a perturbativeexpansion which manifests that the nature of the interactions has a directimpact on the species abundance distribution. The central results are allspecies coexistence in a realistic range of the model parameters and that acertain discrete nature of the interactions induces multiple peaks in thespecies abundance distribution, providing the possibility of theoreticallyexplaining multiple peaks observed in various field studies. To get morequantitative information, we also construct a non-perturbative theory whichbecomes exact on tree-like networks if all the species coexist, providing exactcritical values of $u$ below which extinct species emerge. Numericalsimulations in various different situations are conducted and they clarify therobustness of the presented mechanism of all species coexistence and multiplepeaks in the species abundance distributions.

Multiple peaks of species abundance distributions induced by sparse interactions