Abstract
This research focuses on stochastic modeling of the evolution over time of biological populations through branching processes. We introduce a new class of discrete time two-sex branching processes with random mating and overlapping generations. Mating and reproduction are considered to be influenced by the numbers of females and males existing in the population. This evolution over time with generational overlap is a novel research in two-sex branching process literature. We study general probabilistic properties and establish some limiting results of biological interest.
Funding statement: This research has been supported by the Ministerio de Ciencia e Innovación of Spain, grant PID2019-108211GB-I00/AEI/10.13039/501100011033.
Acknowledgements
The authors would like to thank the reviewers for providing corrections and comments which have improved this paper.
References
[1] K. L. Chung, Markov Chains with Stationary Transition Probabilities, Grundlehren Math. Wiss. 104, Springer, New York, 1967. Suche in Google Scholar
[2] S. J. Cornell and V. S. Isham, Ultimate extinction of the promiscuous bisexual Galton–Watson metapopulation, Aust. N. Z. J. Statist. 46 (2004), 87–98. 10.1111/j.1467-842X.2004.00315.xSuche in Google Scholar
[3] D. J. Daley, Extinction conditions for certain bisexual Galton–Watson branching processes, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9 (1968), 315–322. 10.1007/BF00531755Suche in Google Scholar
[4] M. González, R. Martínez and M. Mota, Bisexual branching processes to model extinction conditions for Y-linked genes, J. Theoret. Biol. 258 (2009), no. 3, 478–488. 10.1016/j.jtbi.2008.10.034Suche in Google Scholar PubMed
[5] M. González, M. Molina and M. Mota, Limit behaviour for a subcritical bisexual Galton–Watson branching process with immigration, Statist. Probab. Lett. 49 (2000), no. 1, 19–24. 10.1016/S0167-7152(00)00026-2Suche in Google Scholar
[6] M. González, M. Molina and M. Mota, On the limit behavior of a supercritical bisexual Galton–Watson branching process with immigration of mating units, Stoch. Anal. Appl. 19 (2001), no. 6, 933–943. 10.1081/SAP-120000755Suche in Google Scholar
[7] P. Guttorp, Statistical Inference for Branching Processes, Wiley Ser. Probab. Math. Stat., John Wiley & Sons, New York, 1991. Suche in Google Scholar
[8] P. Haccou, P. Jagers and V. A. Vatutin, Branching Processes: Variation, Growth, and Extinction of Populations, Cambridge Studies Adapt. Dyn. 5, Cambridge University, Cambridge, 2005. 10.1017/CBO9780511629136Suche in Google Scholar
[9] D. M. Hull, A survey of the literature associated with the bisexual Galton–Watson branching process, Extracta Math. 18 (2003), no. 3, 321–343. Suche in Google Scholar
[10] C. Jacob, M. Molina and M. Mota, A general class of population-dependent two-sex processes with random mating, Bernoulli 23 (2017), no. 3, 1737–1758. 10.3150/15-BEJ794Suche in Google Scholar
[11] P. Jagers, Branching Processes with Biological Applications, Wiley Ser. Probab. Math. Stat., John Wiley & Sons, London, 1975. Suche in Google Scholar
[12] S. Ma and M. Molina, Two-sex branching processes with offspring and mating in a random environment, J. Appl. Probab. 46 (2009), no. 4, 993–1004. 10.1239/jap/1261670684Suche in Google Scholar
[13] S. Ma and Y. Xing, The asymptotic properties of supercritical bisexual Galton–Watson branching processes with immigration of mating units, Acta Math. Sci. Ser. B (Engl. Ed.) 26 (2006), no. 4, 603–609. 10.1016/S0252-9602(06)60086-6Suche in Google Scholar
[14] S.-X. Ma, Bisexual Galton–Watson branching processes in random environments, Acta Math. Appl. Sin. Engl. Ser. 22 (2006), no. 3, 419–428. 10.1007/s10255-006-0317-4Suche in Google Scholar
[15] C. J. Mode, Multitype Branching Processes. Theory and Applications, American Elsevier, New York, 1971. Suche in Google Scholar
[16] M. Molina, Two-sex branching process literature, Workshop on Branching Processes and Their Applications, Lect. Notes Stat. Proc. 197, Springer, Berlin (2010), 279–293. 10.1007/978-3-642-11156-3_20Suche in Google Scholar
[17] M. Molina, C. Jacob and A. Ramos, Bisexual branching processes with offspring and mating depending on the number of couples in the population, TEST 17 (2008), no. 2, 265–281. 10.1007/s11749-006-0031-9Suche in Google Scholar
[18] M. Molina and M. Mota, Population-dependent two-sex branching processes with random mating: Rates of growth, Stoch. Models 35 (2019), no. 3, 252–268. 10.1080/15326349.2019.1618193Suche in Google Scholar
[19] M. Molina, M. Mota and A. Ramos, Bisexual Galton–Watson branching process in varying environments, Stoch. Anal. Appl. 21 (2003), no. 6, 1353–1367. 10.1081/SAP-120026110Suche in Google Scholar
[20] M. Molina, M. Mota and A. Ramos, Limit behaviour for a supercritical bisexual Galton–Watson branching process with population-size-dependent mating, Stochastic Process. Appl. 112 (2004), no. 2, 309–317. 10.1016/j.spa.2004.02.003Suche in Google Scholar
[21]
M. Molina, M. Mota and A. Ramos,
On
[22] M. Molina, M. Mota and A. Ramos, Stochastic modeling in biological populations with sexual reproduction through branching models: Application to Coho salmon populations, Math. Biosci. 258 (2014), 182–188. 10.1016/j.mbs.2014.10.007Suche in Google Scholar PubMed
[23] M. Molina, M. Mota and A. Ramos, Estimation of parameters in biological species with several mating and reproduction alternatives, Math. Biosci. 329 (2020), Article ID 108471. 10.1016/j.mbs.2020.108471Suche in Google Scholar PubMed
[24] M. Mota, I. del Puerto and A. Ramos, The bisexual branching process with population-size dependent mating as a mathematical model to describe phenomena concerning to inhabit or re-inhabit environments with animal species, Math. Biosci. 206 (2007), no. 1, 120–127. 10.1016/j.mbs.2005.01.007Suche in Google Scholar PubMed
[25] S. Pénisson and C. Jacob, Stochastic methodology for the study of an epidemic decay phase, based on a branching model, Int. J. Stoch. Anal. 2012 (2012), Article ID 598701. 10.1155/2012/598701Suche in Google Scholar
[26] Y. Xing and Y. Wang, On the extinction of a class of population-size-dependent bisexual branching processes, J. Appl. Probab. 42 (2005), no. 1, 175–184. 10.1239/jap/1110381379Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- 80th Jubilee of Professor Nikolay Yanev
- My 55 Years in Stochastics
- On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
- Galton–Watson Theta-Processes in a Varying Environment
- Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
- Critical Multitype Branching Processes with Random Migration
- Branching Processes Under Nonstandard Conditions
Artikel in diesem Heft
- Frontmatter
- 80th Jubilee of Professor Nikolay Yanev
- My 55 Years in Stochastics
- On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
- Galton–Watson Theta-Processes in a Varying Environment
- Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
- Critical Multitype Branching Processes with Random Migration
- Branching Processes Under Nonstandard Conditions