Branching Processes: A Personal Historical Perspective

  • Peter JagersEmail author


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.


Branching processes Mathematical history 


  1. 1.
    Athreya, K. B., & Ney, P. E. (1972). Branching processes. Berlin: Springer.CrossRefGoogle Scholar
  2. 2.
    Bienaymé, I. J. (1845). De la loi de multiplication et de la durée des familles. Socit philomathique de Paris Extraits, 5, 37–39.Google Scholar
  3. 3.
    Bru, B., Jongmans, F., & Seneta, E. (1992). I.J. Bienaymé: Family information and proof of the criticality theorem. International Statistical Review, 60, 177–183.CrossRefGoogle Scholar
  4. 4.
    Bühler, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 18, 141–152.MathSciNetCrossRefGoogle Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Processes and their Applications, 116, 1127–1160.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Champagnat, N., Ferrière, R., & Méléard, S. (2006). Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models. Theoretical Population Biology, 69, 297–321.CrossRefGoogle Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    Champagnat, N., & Lambert, A. (2007). Evolution of discrete populations and the canonical diffusion of adaptive dynamics. The Annals of Applied Probability, 17, 102–155.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cournot, A. A. (1847). De l’origine et des limites de la correspondence entre l’algèbre at la géométrie. Paris: Hachette.Google Scholar
  11. 11.
    Crump, K. S., & Mode, C. J. (1968). A general age-dependent branching process I. Journal of Mathematical Analysis and Applications, 24, 494–508.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Crump, K. S., & Mode, C. J. (1969). A general age-dependent branching process II. Journal of Mathematical Analysis and Applications, 25, 8–17.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dieckmann, U., & Doebeli, M. (1999). On the origin of species by sympatric speciation. Nature, 400, 354–357.CrossRefGoogle Scholar
  14. 14.
    Dieckmann, U., & Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology, 34, 579–612.MathSciNetCrossRefGoogle Scholar
  15. 15.
    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.Google Scholar
  16. 16.
    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.Google Scholar
  17. 17.
    Haccou, P., Jagers, P., & Vatutin, V. A. (2005). Branching processes: Variation, growth, and extinction of populations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  18. 18.
    Harris, T. E. (1963). The Theory of Branching Processes. Berlin: Springer. Reprinted by Courier Dover Publications, 1989.Google Scholar
  19. 19.
    Heyde, C. C., & Seneta, E. (1977). I.J. Bienaymé: Statistical theory anticipated. New York: Springer.CrossRefGoogle Scholar
  20. 20.
    Iosifescu, M., Limnios, N., & Oprisan, G. (2007). Modèles stochastiques. Paris: Hermes Lavoisier.zbMATHGoogle Scholar
  21. 21.
    Jagers, P. (1969). A general stochastic model for population development. Scandinavian Actuarial Journal, 1969, 84–103.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Jagers, P. (1975). Branching processes with biological applications. London: Wiley.zbMATHGoogle Scholar
  23. 23.
    Jagers, P. (1982). How probable is it to be firstborn? and other branching process applications to kinship problems. Mathematical Biosciences, 59, 1–15.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jagers, P. (1989). General branching processes as Markov fields. Stochastic Processes and their Applications, 32, 183–212.MathSciNetCrossRefGoogle Scholar
  25. 25.
    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.CrossRefGoogle Scholar
  26. 26.
    Jagers, P., & Klebaner, F. C. (2011). Population-size-dependent, age-structured branching processes linger around. Journal of Applied Probability, 48, 249–260.CrossRefGoogle Scholar
  27. 27.
    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.Google Scholar
  28. 28.
    Jagers, P., & Nerman, O. (1996). The asymptotic composition of supercritical, multi-type branching populations. In Séminaire de Probabilités XXX (pp. 40–54). Berlin: Springer.CrossRefGoogle Scholar
  29. 29.
    Jagers, P., & Sagitov, S. (2008). General branching processes in discrete time as random trees. Bernoulli, 14, 949–962.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Joffe, A., & Waugh, W. A. O. (1982). Exact distributions of kin numbers in a Galton-Watson process. Journal of Applied Probability, 19, 767–775.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kendall, D. G. (1948). On the generalized “birth-and-death” process. Annals of Mathematical Statistics, 19, 1–15.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kersting, G. (1992). Asymptotic Gamma distributions for stochastic difference equations. Stochastic Processes and their Applications, 40, 15–28.MathSciNetCrossRefGoogle Scholar
  33. 33.
    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.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lotka, A. J. (1934). Théorie analytique des associations biologiques (Vol. 1). Paris: Hermann.zbMATHGoogle Scholar
  35. 35.
    Lotka, A. J. (1939). Théorie analytique des associations biologiques (Vol. 2). Paris: Hermann.zbMATHGoogle Scholar
  36. 36.
    Méléard, S., & Tran, C. V. (2009). Trait substitution sequence process and canonical equation for age-structured populations. Journal of Mathematical Biology, 58, 881–921.MathSciNetCrossRefGoogle Scholar
  37. 37.
    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.zbMATHGoogle Scholar
  38. 38.
    Mode, C. J. (1971). Multitype branching processes: Theory and applications. New York: Elsevier.zbMATHGoogle Scholar
  39. 39.
    Nerman, O., & Jagers, P. (1984). The stable doubly infinite pedigree process of supercritical branching populations. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65, 445–460.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Neveu, J. (1986). Arbres et processus de Galton-Watson. Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 2, 199–207.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Sevastyanov, B. A. (1971). Vetvyashchiesya Protsessy (Branching processes). Moscow: Nauka.Google Scholar
  42. 42.
    Steffensen, J. F. (1930). Om Sandssynligheden for at Afkommet uddør. Matematisk Tidsskrift B, 19–23.Google Scholar
  43. 43.
    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.CrossRefGoogle Scholar
  44. 44.
    Yakovlev, A. Y., & Yanev, N. M. (1989). Transient processes in cell proliferation kinetics. Lecture notes in biomathematics (Vol. 82). Berlin: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of TechnologyGothenburgSweden
  2. 2.University of GothenburgGothenburgSweden

Personalised recommendations