{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:39:49Z","timestamp":1760060389731,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T00:00:00Z","timestamp":1756080000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["ORF-2025-1068"],"award-info":[{"award-number":["ORF-2025-1068"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce and analyze a novel two-dimensional Moran random walk model, where each component evolves by either increasing by one unit or resetting to zero at each time step, with transition probabilities dependent on time. The primary contribution of this work is the derivation of an explicit closed-form expression for the probability distribution of the final maximum altitude, Un=max(Un(1),Un(2)). Additionally, we provide a detailed analysis of the height statistics and cumulative distribution associated with the model. Using probabilistic techniques, we establish the exact distribution of the final altitude and its moments. Furthermore, we conduct numerical simulations to illustrate the behavior of the model for various parameter values. These results offer new insights into the statistical properties of two-dimensional Moran processes and have potential applications in population genetics, nuclear physics, and related fields.<\/jats:p>","DOI":"10.3390\/sym17091387","type":"journal-article","created":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T06:25:57Z","timestamp":1756189557000},"page":"1387","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Time-Dependent Probability Distribution of Two-Dimensional Moran Walk"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5660-555X","authenticated-orcid":false,"given":"Mohamed","family":"Abdelkader","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1017\/S0305004100033193","article-title":"Random processes in genetics","volume":"54","author":"Moran","year":"1958","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_2","unstructured":"Moran, P.A.P. (1962). The Statistical Processes of Evolutionary Theory, Oxford University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1038\/nature03204","article-title":"Evolutionary dynamics on graphs","volume":"433","author":"Lieberman","year":"2005","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0304-3975(02)00007-5","article-title":"Basic analytic combinatorics of directed lattice paths","volume":"281","author":"Banderier","year":"2002","journal-title":"Theor. Comput. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Flajolet, P., and Sedgewick, R. (2009). Analytic Combinatorics, Cambridge University Press.","DOI":"10.1017\/CBO9780511801655"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"750","DOI":"10.1239\/aap\/1013540243","article-title":"The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions","volume":"32","author":"Flajolet","year":"2000","journal-title":"Adv. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability, Springer.","DOI":"10.1007\/978-3-211-75357-6"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Abdelkader, M. (2023). On the Height of One-Dimensional Random Walk. Mathematics, 11.","DOI":"10.3390\/math11214513"},{"key":"ref_9","first-page":"35","article-title":"Bounded discrete walks","volume":"AM","author":"Banderier","year":"2010","journal-title":"Discret. Math. Theor. Comput. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"de Bruijn, N.G., Knuth, D.E., and Rice, S.O. (1972). The average height of planted plane trees. Graph Theory and Computing, Academic Press.","DOI":"10.1016\/B978-1-4832-3187-7.50007-6"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Aguech, R., and Abdelkader, M. (2023). Two-Dimensional Moran Model: Final Altitude and Number of Resets. Mathematics, 11.","DOI":"10.3390\/math11173774"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Althagafi, A., and Abdelkader, M. (2023). Two-Dimensional Moran Model. Symmetry, 15.","DOI":"10.3390\/sym15051046"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1080\/08898480.2015.1087775","article-title":"Random walk Green kernels in the neutral Moran modelconditioned on survivors at a random time to origin","volume":"23","author":"Huillet","year":"2016","journal-title":"Math. Popul. Stud."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1239\/jap\/1253279856","article-title":"Duality and asymptotics for a class of nonneutral discrete Moran models","volume":"46","author":"Huillet","year":"2009","journal-title":"J. Appl. Probab."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1387\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:32:23Z","timestamp":1760034743000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1387"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,25]]},"references-count":14,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091387"],"URL":"https:\/\/doi.org\/10.3390\/sym17091387","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,8,25]]}}}