Two-sex branching processes with offspring and mating in a random environment

Two-sex branching processes with offspring and mating in a random environment

S. MA and M. MOLINA

Source: J. Appl. Probab. Volume 46, Number 4 (2009), 993-1004.

Abstract

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

First Page: Show

Primary Subjects: 60J80

Keywords: Branching process; two-sex process; random environment process; extinction probability

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jap/1261670684
Digital Object Identifier: doi:10.1239/jap/1261670684
Zentralblatt MATH identifier: 05665451
Mathematical Reviews number (MathSciNet): MR2582702

back to Table of Contents

References

Agresti, A. (1975). On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39--46.

Mathematical Reviews (MathSciNet): MR365733

Zentralblatt MATH: 0306.60052

Digital Object Identifier: doi:10.2307/3212405

JSTOR: links.jstor.org

Alsmeyer, G. and Rösler, U. (1996). The bisexual Galton--Watson process with promiscuous mating: extinction probabilities in the supercritical case. Ann. Appl. Prob. 6, 922--939.

Mathematical Reviews (MathSciNet): MR1410122

Zentralblatt MATH: 0869.60076

Digital Object Identifier: doi:10.1214/aoap/1034968234

Project Euclid: euclid.aoap/1034968234

Alsmeyer, G. and Rösler, U. (2002). Asexual versus promiscuous bisexual Galton--Watson processes: the extinction probability ratio. Ann. Appl. Prob. 12, 125--142.

Mathematical Reviews (MathSciNet): MR1890059

Zentralblatt MATH: 1020.60073

Digital Object Identifier: doi:10.1214/aoap/1015961158

Project Euclid: euclid.aoap/1015961158

Bagley, J. H. (1986). On the asymptotic properties of a supercritical bisexual branching process. J. Appl. Prob. 23, 820--826.

Mathematical Reviews (MathSciNet): MR855387

Zentralblatt MATH: 0608.60078

Digital Object Identifier: doi:10.2307/3214019

JSTOR: links.jstor.org

Bruss, F. T. (1984). A note on extinction criteria for bisexual Galton--Watson processes. J. Appl. Prob. 21, 915--919.

Mathematical Reviews (MathSciNet): MR766827

Zentralblatt MATH: 0568.60073

Digital Object Identifier: doi:10.2307/3213707

JSTOR: links.jstor.org

Daley, D. J. (1968). Extinction conditions for certain bisexual Galton--Watson branching processes. Z. Wahrscheinlichkeitsth. 9, 315--322.

Mathematical Reviews (MathSciNet): MR230383

Digital Object Identifier: doi:10.1007/BF00531755

Daley, D. J., Hull, D. M. and Taylor, J. M. (1986). Bisexual Galton--Watson branching processes with superadditive mating functions. J. Appl. Prob. 23, 585--600.

Mathematical Reviews (MathSciNet): MR855367

Zentralblatt MATH: 0611.60083

Digital Object Identifier: doi:10.2307/3213999

JSTOR: links.jstor.org

González, M., Molina, M. and Mota, M. (2000). Limit behaviour of a subcritical bisexual Galton--Watson branching process with immigration. Statist. Prob. Lett. 49, 19--24.

Mathematical Reviews (MathSciNet): MR1789660

González, M., Molina, M. and Mota, M. (2001). On the limit behavior of a supercritical bisexual Galton--Watson branching process with immigration of mating units. Stoch. Anal. Appl. 19, 933--945.

Mathematical Reviews (MathSciNet): MR1865640

Zentralblatt MATH: 0988.60083

Digital Object Identifier: doi:10.1081/SAP-120000755

Haccou, P., Jagers, P. and Vatutin, V. A. (2007). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.

Mathematical Reviews (MathSciNet): MR2429372

Zentralblatt MATH: 1118.92001

Hille, E. and Phillips, R. S. (1957). Functional Analysis and Semi-Groups (Amer. Math. Soc. Coll. Publ. 31). American Mathematical Society, Providence, RI.

Mathematical Reviews (MathSciNet): MR89373

Holzheimer, J. (1984). $\varphi$-branching processes in a random environment. Zastos. Mat. 18, 351--358.

Mathematical Reviews (MathSciNet): MR757885

Hull, D. M. (1982). A necessary condition for extinction in those bisexual Galton--Watson branching processes governed by superadditive mating functions. J. Appl. Prob. 19, 847--850.

Mathematical Reviews (MathSciNet): MR675149

Zentralblatt MATH: 0501.60086

Digital Object Identifier: doi:10.2307/3213838

JSTOR: links.jstor.org

Hull, D. M. (2003). A survey of the literature associated with the bisexual Galton--Watson branching process. Extracta Math. 18, 321--343.

Mathematical Reviews (MathSciNet): MR2056600

Zentralblatt MATH: 1051.60084

Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.

Mathematical Reviews (MathSciNet): MR488341

Zentralblatt MATH: 0356.60039

Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology. Springer, New York.

Mathematical Reviews (MathSciNet): MR1903571

Zentralblatt MATH: 0994.92001

Ma, S. and Xing, Y. (2006). The asymptotic properties of supercritical bisexual Galton--Watson branching processes with immigration of mating units. Acta Math. Sci. B. 26, 603--609.

Mathematical Reviews (MathSciNet): MR2265189

Digital Object Identifier: doi:10.1016/S0252-9602(06)60086-6

Molina, M. and Yanev, N. (2003). Continuous time bisexual branching processes. C. R. Acad. Bulgare Sci. 56, 5--10.

Mathematical Reviews (MathSciNet): MR1999843

Molina, M., Mota, M. and Ramos, A. (2002). Bisexual Galton--Watson branching process with population-size-dependent mating. J. Appl. Prob. 39, 479--490.

Mathematical Reviews (MathSciNet): MR1928884

Zentralblatt MATH: 1028.60088

Digital Object Identifier: doi:10.1239/jap/1034082121

Project Euclid: euclid.jap/1034082121

Molina, M., Mota, M. and Ramos, A. (2003). Bisexual Galton--Watson branching process in varying environments. Stoch. Anal. Appl. 21, 1353--1367.

Mathematical Reviews (MathSciNet): MR2014561

Zentralblatt MATH: 1050.60081

Digital Object Identifier: doi:10.1081/SAP-120026110

Molina, M., Mota, M. and Ramos, A. (2004a). Limiting behaviour for superadditive bisexual Galton--Watson process in varying environments. Test 13, 481--499.

Mathematical Reviews (MathSciNet): MR2154010

Zentralblatt MATH: 1072.60069

Digital Object Identifier: doi:10.1007/BF02595783

Molina, M., Mota, M. and Ramos, A. (2004b). Limit behaviour for a supercritical bisexual Galton--Watson branching process with population-size-dependent mating. Stoch. Process. Appl. 112, 309--317.

Mathematical Reviews (MathSciNet): MR2073416

Zentralblatt MATH: 1078.60070

Digital Object Identifier: doi:10.1016/j.spa.2004.02.003

Molina, M., Mota, M. and Ramos, A. (2006). On the $L^\alpha$-convergence ($1\leq\alpha\leq 2$) for a bisexual branching process with population-size dependent mating. Bernoulli 12, 457--468.

Mathematical Reviews (MathSciNet): MR2232726

Digital Object Identifier: doi:10.3150/bj/1151525130

Project Euclid: euclid.bj/1151525130

Pakes, A. C. (2003). Biological applications of branching processes. In Stochastic Processes: Modelling and Simulation (Handbook Statist. 21), eds D. N. Shanbhag and C. R. Rao, North-Holland, Amsterdam, pp. 693--773.

Mathematical Reviews (MathSciNet): MR1973557

Zentralblatt MATH: 1019.92022

Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814--827.

Mathematical Reviews (MathSciNet): MR246380

Zentralblatt MATH: 0184.21103

Digital Object Identifier: doi:10.1214/aoms/1177697589

Project Euclid: euclid.aoms/1177697589

Xing, Y. and Wang, Y. (2005). On the extinction of a class of population-size-dependent bisexual branching processes. J. Appl. Prob. 42, 175--184.

Mathematical Reviews (MathSciNet): MR2144902

Zentralblatt MATH: 1083.60072

Digital Object Identifier: doi:10.1239/jap/1110381379

Project Euclid: euclid.jap/1110381379