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. 2019 Apr 3;9:5580. doi: 10.1038/s41598-019-41614-2

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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The results of the mean field predictions. (A) Species interaction network for 7 species where each species i has one mutualistic partner j, i.e. M_ij = 1, $ϵ_{1}$ = 1, $ϵ_{2}$ = 1. (B) Time evolution of the populations of the 7 species as predicted by the mean field dynamics Eq. (2). (C) Species interaction network for 7 species where one species is not helped by any species and the iterative pruning process, as described in the main text, leads to a cascade of extinctions. (D) As the time evolution of the mean field Eq. (2), only one species dominates the community. (E) Nested structure for fruit eating birds community in Mexico³⁹. (F) All species coexist, as predicted by our theoretical framework. In the ordinate axis use the notation $\bar{η}$ and not η.