article published yes Jakub Svoboda author 130759D2-D7DD-11E9-87D2-DE0DE66974250000-0002-1419-3267 Josef Tkadlec author 3F24CCC8-F248-11E8-B48F-1D18A9856A870000-0002-1097-9684 Kamran Kaveh author Krishnendu Chatterjee author 2E5DCA20-F248-11E8-B48F-1D18A9856A870000-0002-4561-241X KrCh department Formal Methods for Stochastic Models: Algorithms and Applications project Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size N, each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate μ between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when μ is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in N. By contrast, when μ is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in N. Moreover, we precisely identify the threshold migration rate μ⋆ that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results. https://research-explorer.ista.ac.at/download/12787/12796/2023_ProceedingsRoyalSocietyA_Svoboda.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ The Royal Society2023 eng 1364-5021 1471-2946 00095712550000210.1098/rspa.2022.0685 4792271 https://research-explorer.ista.ac.at/record/20138 https://doi.org/10.6084/m9.figshare.21261771.v1 J. Svoboda, J. Tkadlec, K. Kaveh, K. Chatterjee, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 479 (2023). Svoboda J, Tkadlec J, Kaveh K, Chatterjee K. 2023. Coexistence times in the Moran process with environmental heterogeneity. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 479(2271), 20220685. J. Svoboda, J. Tkadlec, K. Kaveh, and K. Chatterjee, “Coexistence times in the Moran process with environmental heterogeneity,” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, vol. 479, no. 2271. The Royal Society, 2023. Svoboda, Jakub, et al. “Coexistence Times in the Moran Process with Environmental Heterogeneity.” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, vol. 479, no. 2271, 20220685, The Royal Society, 2023, doi:<a href="https://doi.org/10.1098/rspa.2022.0685">10.1098/rspa.2022.0685</a>. Svoboda J, Tkadlec J, Kaveh K, Chatterjee K. Coexistence times in the Moran process with environmental heterogeneity. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. 2023;479(2271). doi:<a href="https://doi.org/10.1098/rspa.2022.0685">10.1098/rspa.2022.0685</a> Svoboda, Jakub, Josef Tkadlec, Kamran Kaveh, and Krishnendu Chatterjee. “Coexistence Times in the Moran Process with Environmental Heterogeneity.” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. The Royal Society, 2023. <a href="https://doi.org/10.1098/rspa.2022.0685">https://doi.org/10.1098/rspa.2022.0685</a>. Svoboda, J., Tkadlec, J., Kaveh, K., &#38; Chatterjee, K. (2023). Coexistence times in the Moran process with environmental heterogeneity. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. The Royal Society. <a href="https://doi.org/10.1098/rspa.2022.0685">https://doi.org/10.1098/rspa.2022.0685</a> 127872023-04-02T22:01:09Z2026-09-16T06:59:32Z