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. 2018 Aug 20;8:12476. doi: 10.1038/s41598-018-30945-1

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© The Author(s) 2018

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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Evolutionary trajectories as a function of the probability p of a long-range interaction. Initial conditions are β₀ = 1.0, τ_I0 = 0.5 and L = 100. As in the diffusive case, depending on the value of p there is a transition from the mean-field phase to the ordered phase, albeit with a different value of $R_{0}^{e v}$ . The exact value of p at which the transition occurs is also dependent on the initial conditions.