A bayesian version of Galton–Watson for population growth and its use in the management of small population
Bertrand Cloez1 ; Tanguy Daufresne2 ; Marion Kerioui1 ; Bénédicte Fontez1
1 MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France
2 Eco&Sol, Université Montpellier, INRAE, Institut Agro, Montpellier, France
MathematicS In Action, Tome 12 (2023) no. 1, pp. 49-64.

Sharp prediction of extinction times is needed in biodiversity monitoring and conservation management. The Galton–Watson process is a classical stochastic model for describing population dynamics. Its evolution is like the matrix population model where offspring numbers are random. Extinction probability, extinction time, abundance are well known and given by explicit formulas. In contrast with the deterministic model, it can be applied to small populations. Parameters of this model can be estimated through the Bayesian inference framework. This enables to consider various scenarios. We show how coupling Bayesian inference with the Galton–Watson model provides several features: (i) a flexible modelling approach with easily understandable parameters (ii) compatibility with the classical matrix population model (Leslie type model) (iii) An approach which leads to more information with less computing iv) inclusion of expert or previous knowledge...It can be seen to go one step further than the classical matrix population model for the viability problem. To illustrate these features, we provide analysis details for a real life example with French Pyrenean brown bears.

Publié le :
DOI : 10.5802/msia.31
Mots clés : Bayesian, Galton–Watson, Population Growth, Small population
Bertrand Cloez 1 ; Tanguy Daufresne 2 ; Marion Kerioui 1 ; Bénédicte Fontez 1

1 MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France
2 Eco&Sol, Université Montpellier, INRAE, Institut Agro, Montpellier, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Bertrand Cloez; Tanguy Daufresne; Marion Kerioui; Bénédicte Fontez. A bayesian version of Galton–Watson for population growth and its use in the management of small population. MathematicS In Action, Tome 12 (2023) no. 1, pp. 49-64. doi : 10.5802/msia.31. https://msia.centre-mersenne.org/articles/10.5802/msia.31/

[1] Krishna B. Athreya; Peter E. Ney Branching processes, Courier Corporation, 2004

[2] Andrew D. Barbour; Lars Holst; Svante Janson Poisson Approximation, Oxford Science Publications, Clarendon Press, 1992

[3] Harro Bernardelli Population waves, J. Burma Res. Soc., Volume 31 (1941), pp. 3-18

[4] Irénée-Jules Bienaymé De la loi de multiplication et de la durée des familles, Soc. Philomat. Paris Extraits, Sér, Volume 5 (1845), pp. 37-39

[5] P. Bliman; M. S. Aronna; F. C. Coelho; M. A. H. B. da Silva Global stabilizing feedback law for a problem of biological control of mosquito-borne diseases, 2015 54th IEEE Conference on Decision and Control (CDC) (2015), pp. 3206-3211 | DOI

[6] Mark S. Boyce Restitution of gamma-and K-Selection as a Model of Density-Dependent Natural Selection, Annu. Rev. Ecol. System., Volume 15 (1984) no. 1, pp. 427-447 | DOI

[7] Mark S. Boyce Population Viability Analysis, Annu. Rev. Ecol. System., Volume 23 (1992) no. 1, pp. 481-497 | arXiv | DOI

[8] B. J. Cairns; J. V. Ross; T. Taimre A comparison of models for predicting population persistence, Ecol. Model., Volume 201 (2007) no. 1, pp. 19-26 (Management, Control and Decision Making for Ecological Systems) | DOI

[9] J. J. Camarra; J. Sentilles; A. Gastineau; P. Y. Quenette SUIVI DE L’OURS BRUN DANS LES PYRENEES FRANCAISES (Sous-populations occidentale et centrale) Rapport annuel Année 2016 (2016) (Technical report)

[10] F. Campillo; C. Lobry Effect of population size in a predator–prey model, Ecol. Model., Volume 246 (2012), pp. 1-10 | DOI

[11] Hal Caswell Matrix population models: construction, analysis, and interpretation, Sinauer Associates, 1989

[12] Hal Caswell; Masami Fujiwara; Solange Brault Declining survival probability threatens the North Atlantic right whale, Proc. Natl. Acad. Sci. USA, Volume 96 (1999) no. 6, pp. 3308-3313 | DOI

[13] Guillaume Chapron; Pierre-Yves Quenette; Stéphane Legendre; Jean Clobert Which future for the French Pyrenean brown bear (Ursus arctos) population? An approach using stage-structured deterministic and stochastic models, C. R. Biol., Volume 326 (2003), pp. 174-182 | DOI

[14] Daryl J. Daley Extinction conditions for certain bisexual Galton-Watson branching processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 9 (1968) no. 4, pp. 315-322 | DOI | MR | Zbl

[15] Steinar Engen; Russell Lande; Bernt-Erik Sæther; Henri Weimerskirch Extinction in relation to demographic and environmental stochasticity in age-structured models, Math. Biosci., Volume 195 (2005) no. 2, pp. 210-227 | DOI | MR | Zbl

[16] Arnaud Estoup; Eric Lombaert; Jean-Michel Marin; Thomas Guillemaud; Pierre Pudlo; Christian Robert; Jean-Marie Cornuet Estimation of demo-genetic model probabilities with Approximate Bayesian Computation using linear discriminant analysis on summary statistics, Mol. Ecol. Resour., Volume 12 (2012) no. 5, pp. 846-855 | DOI

[17] Esther Gerber; Christine Krebs; Craig Murrell; Marco Moretti; Remy Rocklin; Urs Schaffner Exotic invasive knotweeds (Fallopia spp.) negatively affect native plant and invertebrate assemblages in European riparian habitats, Biol. Conserv., Volume 141 (2008) no. 3, pp. 646-654 | DOI

[18] Lev R. Ginzburg; Lawrence B. Slobodkin; Keith Johnson; Andrew G. Bindman Quasiextinction Probabilities as a Measure of Impact on Population Growth, Risk Analysis, Volume 2 (1982) no. 3, pp. 171-181 | DOI

[19] Miguel González; Jacinto Martín; Rodrigo Martínez; Manuel Mota Non-parametric Bayesian estimation for multitype branching processes through simulation-based methods, Comput. Stat. Data Anal., Volume 52 (2008) no. 3, pp. 1281-1291 | DOI | MR

[20] Frédéric Gosselin; Jean-Dominique Lebreton Potential of branching processes as a modeling tool for conservation biology, Quantitative methods for conservation biology, Springer, 2000, pp. 199-225 | DOI

[21] Patsy Haccou; Patricia Haccou; Peter Jagers; Vladimir A. Vatutin Branching processes: variation, growth, and extinction of populations, Cambridge University Press, 2005 | DOI

[22] Theodore E. Harris The theory of branching processes, Courier Corporation, 2002

[23] Elizabeth Eli Holmes; John L. Sabo; Steven Vincent Viscido; William Fredric Fagan A statistical approach to quasi-extinction forecasting, Ecol. Lett., Volume 10 (2007) no. 12, pp. 1182-1198 | arXiv | DOI

[24] Marek Kimmel; David E Axelrod Branching Processes in Biology, Interdisciplinar Applied Mathematics, 19, Springer, 2002 | DOI

[25] Jean-Dominique Lebreton; Frédéric Gosselin; Colin Niel Extinction and viability of populations: paradigms and concepts of extinction models, Ecoscience, Volume 14 (2007) no. 4, pp. 472-481 | DOI

[26] Patrick H Leslie On the use of matrices in certain population mathematics, Biometrika (1945), pp. 183-212 | DOI | MR | Zbl

[27] E. G. Lewis On the Generation and Growth of a Population, Sankhyā, Ser. A, Volume 6 (1942) no. 1, pp. 93-96

[28] Sarah Lowe; Michael Browne; Souyad Boudjelas; Maj De Poorter 100 of the world’s worst invasive alien species: a selection from the global invasive species database, 12, Invasive Species Specialist Group Auckland, 2000

[29] Sylvie Méléard Modèles aléatoires en Ecologie et Evolution, Springer, 2016

[30] Manuel Mendoza; Eduardo Gutiérrez-Peña Bayesian conjugate analysis of the Galton–Watson process, Test, Volume 9 (2000) no. 1, pp. 149-171 | DOI | MR | Zbl

[31] William F. Morris; Daniel F. Doak et al. Quantitative conservation biology. Theory and Practice of Population Viability Analysis, Sinauer, 2002

[32] Otso Ovaskainen; Baruch Meerson Stochastic models of population extinction, Trends Ecol. Evol., Volume 25 (2010) no. 11, pp. 643-652 | DOI

[33] Steven L. Peck A Tutorial for Understanding Ecological Modeling Papers for the Nonmodeler, Am. Entomol., Volume 46 (2000) no. 1, pp. 40-49 | DOI

[34] Christian Robert The Bayesian choice: from decision-theoretic foundations to computational implementation, Springer, 2007

[35] Judith Rousseau Statistique Bayésienne, Notes de cours (2009) (http://www.crest.fr/ckfinder/userfiles/files/pageperso/mandre/sb_cours.pdf)

[36] Sabrina Servanty; Jean-Michel Gaillard; Francesca Ronchi; Stefano Focardi; Eric Baubet; Olivier Gimenez Influence of harvesting pressure on demographic tactics: implications for wildlife management, J. Appl. Ecol., Volume 48 (2011) no. 4, pp. 835-843 | DOI

[37] Nikolay Sirakov Modélisation de la dynamique de population d’une plante native (palmier babaçu) dans le cadre d’un projet de gestion durable au Brésil, Ph. D. Thesis, Université de Montpellier (2018) https://tel.archives-ouvertes.fr/tel-01807940/document

[38] M. Soulé Viable Populations for Conservation, Cambridge University Press, 1987 | DOI

[39] Shripad Tuljapurkar; Hal Caswell Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Population and Community Biology Series, Springer, 1997 | DOI

[40] Henry William Watson; Francis Galton On the probability of the extinction of families, Journal of the Anthropological Institute of Great Britain and Ireland, Volume 4 (1875), pp. 138-144 | DOI

[41] Thorsten Wiegand; Javier Naves; Thomas Stephan; Alberto Fernandez Assessing the risk of extinction for the brown bear (Ursus arctos) in the Cordillera Cantabrica, Spain, Ecol. Monogr., Volume 68 (1998) no. 4, pp. 539-570 | DOI

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