{"has_accepted_license":"1","ec_funded":1,"publication":"Pnas Nexus","volume":4,"issue":"8","publisher":"Oxford University Press","intvolume":"         4","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"file":[{"creator":"dernst","file_name":"2025_PNASNexus_Brewster.pdf","checksum":"8a5e82c6f842e3220ec96028c9374b69","file_id":"20280","access_level":"open_access","relation":"main_file","date_updated":"2025-09-03T06:20:08Z","file_size":1086419,"content_type":"application/pdf","success":1,"date_created":"2025-09-03T06:20:08Z"}],"abstract":[{"text":"We examine population structures for their ability to maintain diversity in neutral evolution. We use the general framework of evolutionary graph theory and consider birth–death (bd) and death–birth (db) updating. The population is of size N. Initially all individuals represent different types. The basic question is: what is the time TN until one type takes over the population? This time is known as consensus time in computer science and as total coalescent time in evolutionary biology. For the complete graph, it is known that TN is quadratic in N for db and bd. For the cycle, we prove that TN is cubic in N for db and bd. For the star, we prove that TN is cubic for bd and quasilinear (N log N) for db. For the double star, we show that TN is quartic for bd. We derive upper and lower bounds for all undirected graphs for bd and db. We also show the Pareto front of graphs (of size N = 8) that maintain diversity the longest for bd and db. Further, we show that some graphs that quickly homogenize can maintain high levels of diversity longer than graphs that slowly homogenize. For directed graphs, we give simple contracting star-like structures that have superexponential time scales for maintaining diversity.","lang":"eng"}],"PlanS_conform":"1","acknowledgement":"J.S. and K.C. were supported by the European Research Council CoG 863818 (ForM-SMArt) and Austrian Science Fund 10.55776/COE12. J.T. was supported by GAČR grant 25-17377S and by Charles Univ. projects UNCE 24/SCI/008 and PRIMUS 24/SCI/012.","month":"08","arxiv":1,"oa_version":"Published Version","OA_place":"publisher","file_date_updated":"2025-09-03T06:20:08Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2503.09841"]},"title":"Maintaining diversity in structured populations","scopus_import":"1","oa":1,"OA_type":"gold","publication_status":"published","article_type":"original","_id":"20254","article_number":"pgaf252","year":"2025","DOAJ_listed":"1","date_created":"2025-08-31T22:01:32Z","date_updated":"2025-09-03T06:22:24Z","ddc":["000"],"doi":"10.1093/pnasnexus/pgaf252","department":[{"_id":"KrCh"}],"citation":{"mla":"Brewster, David A., et al. “Maintaining Diversity in Structured Populations.” <i>Pnas Nexus</i>, vol. 4, no. 8, pgaf252, Oxford University Press, 2025, doi:<a href=\"https://doi.org/10.1093/pnasnexus/pgaf252\">10.1093/pnasnexus/pgaf252</a>.","apa":"Brewster, D. A., Svoboda, J., Roscow, D., Chatterjee, K., Tkadlec, J., &#38; Nowak, M. A. (2025). Maintaining diversity in structured populations. <i>Pnas Nexus</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/pnasnexus/pgaf252\">https://doi.org/10.1093/pnasnexus/pgaf252</a>","chicago":"Brewster, David A., Jakub Svoboda, Dylan Roscow, Krishnendu Chatterjee, Josef Tkadlec, and Martin A. Nowak. “Maintaining Diversity in Structured Populations.” <i>Pnas Nexus</i>. Oxford University Press, 2025. <a href=\"https://doi.org/10.1093/pnasnexus/pgaf252\">https://doi.org/10.1093/pnasnexus/pgaf252</a>.","ieee":"D. A. Brewster, J. Svoboda, D. Roscow, K. Chatterjee, J. Tkadlec, and M. A. Nowak, “Maintaining diversity in structured populations,” <i>Pnas Nexus</i>, vol. 4, no. 8. Oxford University Press, 2025.","ama":"Brewster DA, Svoboda J, Roscow D, Chatterjee K, Tkadlec J, Nowak MA. Maintaining diversity in structured populations. <i>Pnas Nexus</i>. 2025;4(8). doi:<a href=\"https://doi.org/10.1093/pnasnexus/pgaf252\">10.1093/pnasnexus/pgaf252</a>","ista":"Brewster DA, Svoboda J, Roscow D, Chatterjee K, Tkadlec J, Nowak MA. 2025. Maintaining diversity in structured populations. Pnas Nexus. 4(8), pgaf252.","short":"D.A. Brewster, J. Svoboda, D. Roscow, K. Chatterjee, J. Tkadlec, M.A. Nowak, Pnas Nexus 4 (2025)."},"publication_identifier":{"eissn":["2752-6542"]},"date_published":"2025-08-01T00:00:00Z","article_processing_charge":"Yes","author":[{"first_name":"David A.","full_name":"Brewster, David A.","last_name":"Brewster"},{"orcid":"0000-0002-1419-3267","id":"130759D2-D7DD-11E9-87D2-DE0DE6697425","first_name":"Jakub","full_name":"Svoboda, Jakub","last_name":"Svoboda"},{"first_name":"Dylan","full_name":"Roscow, Dylan","last_name":"Roscow"},{"last_name":"Chatterjee","full_name":"Chatterjee, Krishnendu","first_name":"Krishnendu","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4561-241X"},{"last_name":"Tkadlec","full_name":"Tkadlec, Josef","id":"3F24CCC8-F248-11E8-B48F-1D18A9856A87","first_name":"Josef","orcid":"0000-0002-1097-9684"},{"first_name":"Martin A.","last_name":"Nowak","full_name":"Nowak, Martin A."}],"type":"journal_article","project":[{"call_identifier":"H2020","grant_number":"863818","name":"Formal Methods for Stochastic Models: Algorithms and Applications","_id":"0599E47C-7A3F-11EA-A408-12923DDC885E"}],"day":"01","language":[{"iso":"eng"}],"status":"public","quality_controlled":"1"}