Skip to main content

2020 | OriginalPaper | Buchkapitel

Branching Processes: A Personal Historical Perspective

verfasst von : Peter Jagers

Erschienen in: Statistical Modeling for Biological Systems

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This article is a slightly edited and updated version of an evening talk during the random trees week at the Mathematisches Forschungsinstitut Oberwolfach, January 2009. It gives a—personally biased—sketch of the development of branching processes, from the mid nineteenth century to 2010, emphasizing relations to bioscience and demography, and to society and culture in general.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Athreya, K. B., & Ney, P. E. (1972). Branching processes. Berlin: Springer.CrossRef Athreya, K. B., & Ney, P. E. (1972). Branching processes. Berlin: Springer.CrossRef
2.
Zurück zum Zitat Bienaymé, I. J. (1845). De la loi de multiplication et de la durée des familles. Socit philomathique de Paris Extraits, 5, 37–39. Bienaymé, I. J. (1845). De la loi de multiplication et de la durée des familles. Socit philomathique de Paris Extraits, 5, 37–39.
3.
Zurück zum Zitat Bru, B., Jongmans, F., & Seneta, E. (1992). I.J. Bienaymé: Family information and proof of the criticality theorem. International Statistical Review, 60, 177–183.CrossRef Bru, B., Jongmans, F., & Seneta, E. (1992). I.J. Bienaymé: Family information and proof of the criticality theorem. International Statistical Review, 60, 177–183.CrossRef
4.
Zurück zum Zitat Bühler, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 18, 141–152.MathSciNetCrossRef Bühler, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 18, 141–152.MathSciNetCrossRef
5.
Zurück zum Zitat Bühler, W. J. (1972). The distribution of generations and other aspects of the family structure of branching processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Probability Theory (pp. 463–480). Berkeley: University of California Press. Bühler, W. J. (1972). The distribution of generations and other aspects of the family structure of branching processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Probability Theory (pp. 463–480). Berkeley: University of California Press.
6.
Zurück zum Zitat Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Processes and their Applications, 116, 1127–1160.MathSciNetCrossRef Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Processes and their Applications, 116, 1127–1160.MathSciNetCrossRef
7.
Zurück zum Zitat Champagnat, N., Ferrière, R., & Méléard, S. (2006). Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models. Theoretical Population Biology, 69, 297–321.CrossRef Champagnat, N., Ferrière, R., & Méléard, S. (2006). Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models. Theoretical Population Biology, 69, 297–321.CrossRef
8.
Zurück zum Zitat Champagnat, N., Ferrière, R., & Méléard, S. (2008). Individual-based probabilistic models of adaptive evolution and various scaling approximations. In Seminar on stochastic analysis, random fields and applications V. Progress in probability (Vol. 59, pp. 75–113). Basel: Springer. Champagnat, N., Ferrière, R., & Méléard, S. (2008). Individual-based probabilistic models of adaptive evolution and various scaling approximations. In Seminar on stochastic analysis, random fields and applications V. Progress in probability (Vol. 59, pp. 75–113). Basel: Springer.
9.
Zurück zum Zitat Champagnat, N., & Lambert, A. (2007). Evolution of discrete populations and the canonical diffusion of adaptive dynamics. The Annals of Applied Probability, 17, 102–155.MathSciNetCrossRef Champagnat, N., & Lambert, A. (2007). Evolution of discrete populations and the canonical diffusion of adaptive dynamics. The Annals of Applied Probability, 17, 102–155.MathSciNetCrossRef
10.
Zurück zum Zitat Cournot, A. A. (1847). De l’origine et des limites de la correspondence entre l’algèbre at la géométrie. Paris: Hachette. Cournot, A. A. (1847). De l’origine et des limites de la correspondence entre l’algèbre at la géométrie. Paris: Hachette.
11.
Zurück zum Zitat Crump, K. S., & Mode, C. J. (1968). A general age-dependent branching process I. Journal of Mathematical Analysis and Applications, 24, 494–508.MathSciNetCrossRef Crump, K. S., & Mode, C. J. (1968). A general age-dependent branching process I. Journal of Mathematical Analysis and Applications, 24, 494–508.MathSciNetCrossRef
12.
Zurück zum Zitat Crump, K. S., & Mode, C. J. (1969). A general age-dependent branching process II. Journal of Mathematical Analysis and Applications, 25, 8–17.MathSciNetCrossRef Crump, K. S., & Mode, C. J. (1969). A general age-dependent branching process II. Journal of Mathematical Analysis and Applications, 25, 8–17.MathSciNetCrossRef
13.
Zurück zum Zitat Dieckmann, U., & Doebeli, M. (1999). On the origin of species by sympatric speciation. Nature, 400, 354–357.CrossRef Dieckmann, U., & Doebeli, M. (1999). On the origin of species by sympatric speciation. Nature, 400, 354–357.CrossRef
14.
Zurück zum Zitat Dieckmann, U., & Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology, 34, 579–612.MathSciNetCrossRef Dieckmann, U., & Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology, 34, 579–612.MathSciNetCrossRef
15.
Zurück zum Zitat Euler, L. (1767). Recherches génerales sur la mortalité et la multiplication du genre humain. Memoires de l’academie des sciences de Berlin, 16, 144–164. Euler, L. (1767). Recherches génerales sur la mortalité et la multiplication du genre humain. Memoires de l’academie des sciences de Berlin, 16, 144–164.
16.
Zurück zum Zitat Fahlbeck, P. E. (1898). Sveriges adel: statistisk undersökning öfver de å Riddarhuset introducerade ätterna (The Swedish nobility, a statistical investigation of the families of the house of nobility) (Vols. 1–2). Lund: C. W. K. Gleerup. Fahlbeck, P. E. (1898). Sveriges adel: statistisk undersökning öfver de å Riddarhuset introducerade ätterna (The Swedish nobility, a statistical investigation of the families of the house of nobility) (Vols. 1–2). Lund: C. W. K. Gleerup.
17.
Zurück zum Zitat Haccou, P., Jagers, P., & Vatutin, V. A. (2005). Branching processes: Variation, growth, and extinction of populations. Cambridge: Cambridge University Press.CrossRef Haccou, P., Jagers, P., & Vatutin, V. A. (2005). Branching processes: Variation, growth, and extinction of populations. Cambridge: Cambridge University Press.CrossRef
18.
Zurück zum Zitat Harris, T. E. (1963). The Theory of Branching Processes. Berlin: Springer. Reprinted by Courier Dover Publications, 1989. Harris, T. E. (1963). The Theory of Branching Processes. Berlin: Springer. Reprinted by Courier Dover Publications, 1989.
19.
Zurück zum Zitat Heyde, C. C., & Seneta, E. (1977). I.J. Bienaymé: Statistical theory anticipated. New York: Springer.CrossRef Heyde, C. C., & Seneta, E. (1977). I.J. Bienaymé: Statistical theory anticipated. New York: Springer.CrossRef
20.
Zurück zum Zitat Iosifescu, M., Limnios, N., & Oprisan, G. (2007). Modèles stochastiques. Paris: Hermes Lavoisier.MATH Iosifescu, M., Limnios, N., & Oprisan, G. (2007). Modèles stochastiques. Paris: Hermes Lavoisier.MATH
21.
Zurück zum Zitat Jagers, P. (1969). A general stochastic model for population development. Scandinavian Actuarial Journal, 1969, 84–103.MathSciNetCrossRef Jagers, P. (1969). A general stochastic model for population development. Scandinavian Actuarial Journal, 1969, 84–103.MathSciNetCrossRef
22.
Zurück zum Zitat Jagers, P. (1975). Branching processes with biological applications. London: Wiley.MATH Jagers, P. (1975). Branching processes with biological applications. London: Wiley.MATH
23.
Zurück zum Zitat Jagers, P. (1982). How probable is it to be firstborn? and other branching process applications to kinship problems. Mathematical Biosciences, 59, 1–15.MathSciNetCrossRef Jagers, P. (1982). How probable is it to be firstborn? and other branching process applications to kinship problems. Mathematical Biosciences, 59, 1–15.MathSciNetCrossRef
24.
Zurück zum Zitat Jagers, P. (1989). General branching processes as Markov fields. Stochastic Processes and their Applications, 32, 183–212.MathSciNetCrossRef Jagers, P. (1989). General branching processes as Markov fields. Stochastic Processes and their Applications, 32, 183–212.MathSciNetCrossRef
25.
Zurück zum Zitat Jagers, P. (2011). Extinction, persistence, and evolution. In F. A. Chalub & J. F. Rodrigues (Eds.) The mathematics of Darwin’s legacy (pp. 91–104). Basel: Springer.CrossRef Jagers, P. (2011). Extinction, persistence, and evolution. In F. A. Chalub & J. F. Rodrigues (Eds.) The mathematics of Darwin’s legacy (pp. 91–104). Basel: Springer.CrossRef
26.
Zurück zum Zitat Jagers, P., & Klebaner, F. C. (2011). Population-size-dependent, age-structured branching processes linger around. Journal of Applied Probability, 48, 249–260.CrossRef Jagers, P., & Klebaner, F. C. (2011). Population-size-dependent, age-structured branching processes linger around. Journal of Applied Probability, 48, 249–260.CrossRef
27.
Zurück zum Zitat Jagers, P., Klebaner, F. C., & Sagitov, S. (2007). On the path to extinction. Proceedings of the National Academy of Sciences of the United States of America, 104, 6107–6111. Jagers, P., Klebaner, F. C., & Sagitov, S. (2007). On the path to extinction. Proceedings of the National Academy of Sciences of the United States of America, 104, 6107–6111.
28.
Zurück zum Zitat Jagers, P., & Nerman, O. (1996). The asymptotic composition of supercritical, multi-type branching populations. In Séminaire de Probabilités XXX (pp. 40–54). Berlin: Springer.CrossRef Jagers, P., & Nerman, O. (1996). The asymptotic composition of supercritical, multi-type branching populations. In Séminaire de Probabilités XXX (pp. 40–54). Berlin: Springer.CrossRef
29.
Zurück zum Zitat Jagers, P., & Sagitov, S. (2008). General branching processes in discrete time as random trees. Bernoulli, 14, 949–962.MathSciNetCrossRef Jagers, P., & Sagitov, S. (2008). General branching processes in discrete time as random trees. Bernoulli, 14, 949–962.MathSciNetCrossRef
30.
Zurück zum Zitat Joffe, A., & Waugh, W. A. O. (1982). Exact distributions of kin numbers in a Galton-Watson process. Journal of Applied Probability, 19, 767–775.MathSciNetCrossRef Joffe, A., & Waugh, W. A. O. (1982). Exact distributions of kin numbers in a Galton-Watson process. Journal of Applied Probability, 19, 767–775.MathSciNetCrossRef
31.
Zurück zum Zitat Kendall, D. G. (1948). On the generalized “birth-and-death” process. Annals of Mathematical Statistics, 19, 1–15.MathSciNetCrossRef Kendall, D. G. (1948). On the generalized “birth-and-death” process. Annals of Mathematical Statistics, 19, 1–15.MathSciNetCrossRef
32.
Zurück zum Zitat Kersting, G. (1992). Asymptotic Gamma distributions for stochastic difference equations. Stochastic Processes and their Applications, 40, 15–28.MathSciNetCrossRef Kersting, G. (1992). Asymptotic Gamma distributions for stochastic difference equations. Stochastic Processes and their Applications, 40, 15–28.MathSciNetCrossRef
33.
Zurück zum Zitat Klebaner, F. C., Sagitov, S., Vatutin, V. A., Haccou, P., & Jagers, P. (2011). Stochasticity in the adaptive dynamics of evolution: The bare bones. Journal of Biological Dynamics, 5, 147–162.MathSciNetCrossRef Klebaner, F. C., Sagitov, S., Vatutin, V. A., Haccou, P., & Jagers, P. (2011). Stochasticity in the adaptive dynamics of evolution: The bare bones. Journal of Biological Dynamics, 5, 147–162.MathSciNetCrossRef
34.
Zurück zum Zitat Lotka, A. J. (1934). Théorie analytique des associations biologiques (Vol. 1). Paris: Hermann.MATH Lotka, A. J. (1934). Théorie analytique des associations biologiques (Vol. 1). Paris: Hermann.MATH
35.
Zurück zum Zitat Lotka, A. J. (1939). Théorie analytique des associations biologiques (Vol. 2). Paris: Hermann.MATH Lotka, A. J. (1939). Théorie analytique des associations biologiques (Vol. 2). Paris: Hermann.MATH
36.
Zurück zum Zitat Méléard, S., & Tran, C. V. (2009). Trait substitution sequence process and canonical equation for age-structured populations. Journal of Mathematical Biology, 58, 881–921.MathSciNetCrossRef Méléard, S., & Tran, C. V. (2009). Trait substitution sequence process and canonical equation for age-structured populations. Journal of Mathematical Biology, 58, 881–921.MathSciNetCrossRef
37.
Zurück zum Zitat Metz, J. A., Geritz, S. A., Meszéna, G., Jacobs, F. J., & Van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In S. J. van Strien & S. M. Verduyn Lunel (Eds.), Stochastic and spatial structures of dynamical systems (Vol. 45, pp. 183–231). Amsterdam: North-Holland.MATH Metz, J. A., Geritz, S. A., Meszéna, G., Jacobs, F. J., & Van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In S. J. van Strien & S. M. Verduyn Lunel (Eds.), Stochastic and spatial structures of dynamical systems (Vol. 45, pp. 183–231). Amsterdam: North-Holland.MATH
38.
Zurück zum Zitat Mode, C. J. (1971). Multitype branching processes: Theory and applications. New York: Elsevier.MATH Mode, C. J. (1971). Multitype branching processes: Theory and applications. New York: Elsevier.MATH
39.
Zurück zum Zitat Nerman, O., & Jagers, P. (1984). The stable doubly infinite pedigree process of supercritical branching populations. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65, 445–460.MathSciNetCrossRef Nerman, O., & Jagers, P. (1984). The stable doubly infinite pedigree process of supercritical branching populations. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65, 445–460.MathSciNetCrossRef
40.
Zurück zum Zitat Neveu, J. (1986). Arbres et processus de Galton-Watson. Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 2, 199–207.MathSciNetMATH Neveu, J. (1986). Arbres et processus de Galton-Watson. Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 2, 199–207.MathSciNetMATH
41.
Zurück zum Zitat Sevastyanov, B. A. (1971). Vetvyashchiesya Protsessy (Branching processes). Moscow: Nauka. Sevastyanov, B. A. (1971). Vetvyashchiesya Protsessy (Branching processes). Moscow: Nauka.
42.
Zurück zum Zitat Steffensen, J. F. (1930). Om Sandssynligheden for at Afkommet uddør. Matematisk Tidsskrift B, 19–23. Steffensen, J. F. (1930). Om Sandssynligheden for at Afkommet uddør. Matematisk Tidsskrift B, 19–23.
43.
Zurück zum Zitat Watson, H. W., & Galton, F. (1875). On the probability of the extinction of families. Journal of the Anthropological Institute of Great Britain and Ireland, 4, 138–144.CrossRef Watson, H. W., & Galton, F. (1875). On the probability of the extinction of families. Journal of the Anthropological Institute of Great Britain and Ireland, 4, 138–144.CrossRef
44.
Zurück zum Zitat Yakovlev, A. Y., & Yanev, N. M. (1989). Transient processes in cell proliferation kinetics. Lecture notes in biomathematics (Vol. 82). Berlin: Springer.CrossRef Yakovlev, A. Y., & Yanev, N. M. (1989). Transient processes in cell proliferation kinetics. Lecture notes in biomathematics (Vol. 82). Berlin: Springer.CrossRef
Metadaten
Titel
Branching Processes: A Personal Historical Perspective
verfasst von
Peter Jagers
Copyright-Jahr
2020
DOI
https://doi.org/10.1007/978-3-030-34675-1_18