Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-12T19:44:21.147Z Has data issue: false hasContentIssue false
  • English
  • Français

Branching processes with interactions: subcritical cooperative regime

Part of: Special processes Markov processes

Published online by Cambridge University Press:  17 March 2021

Adrián González Casanova ,
Juan Carlos Pardo  and
José Luis Pérez
Adrián González Casanova*
Affiliation:
Universidad Nacional Autónoma de México
Juan Carlos Pardo*
Affiliation:
Centro de Investigación en Matemáticas
José Luis Pérez*
Affiliation:
Centro de Investigación en Matemáticas
*
*Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, 04510, México, D.F. Email address: [email protected]
**Postal address: Centro de Investigación en Matemáticas, A.C., Calle Jalisco s/n, Col. Valenciana CP 36023 Guanajuato, Gto., México, Apartado Postal 402, CP 36000.
**Postal address: Centro de Investigación en Matemáticas, A.C., Calle Jalisco s/n, Col. Valenciana CP 36023 Guanajuato, Gto., México, Apartado Postal 402, CP 36000.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interactions, such as competition, annihilation, and cooperation; and interactions among several individuals that can be viewed as catastrophes. We call such families of processes branching processes with interactions. Our aim is to study their long-term behaviour under a specific regime of the pairwise interaction parameters that we introduce as the subcritical cooperative regime. Under such a regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype, and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such a distribution exists; it is also an interesting object in its own right.

Keywords

Branching coalescing processesstabilitymoment dualitypopulation genetics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 251 - 278
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alkemper, R. and Hutzenthaler, M. (2007). Graphical representation of some duality relations in stochastic populations models. Electron. Commun. Prob. 12, 206–220.CrossRefGoogle Scholar
Athreya, S. R. and Swart, J. M. (2005). Branching–coalescing particle systems. Prob. Theory Relat. Fields 131, 376–414.CrossRefGoogle Scholar
Athreya, S. R. and Swart, J. M. (2012). Systems of branching, annihilating and coalescing particles. Electron. J. Prob. 17, 1–32.CrossRefGoogle Scholar
Blath, J., Etheridge, A. M. and Meredith, M. (2007). Coexistence in competing populations and branching annihilating random walk. Ann. Appl. Prob. 17, 1474–1507.CrossRefGoogle Scholar
Blath, J. and Kurt, N. (2011). Survival and extinction of caring double-branching annihilating random walk. Electron. Comm. Prob. 16, 271–282.CrossRefGoogle Scholar
Bramson, M. and Gray, L. (1985). The survival of branching annihilating random walk. Z. Wahrscheinlichkeitsth. 68, 447–460.CrossRefGoogle Scholar
Carinci, G., Giardina, C., Giberti, C. and Redig, F. (2015). Dualities in population genetics: a fresh look with new dualities. Stoch. Process. Appl. 125, 941–969.CrossRefGoogle Scholar
Chen, M. F. (1991). On three classical problems for Markov chains with continuous time parameters. J. Appl. Prob. 28, 305–320.CrossRefGoogle Scholar
Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Prob. 40, 813–857.CrossRefGoogle Scholar
Donnelly, P. (1991). Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics. Ann. Prob. 19, 1102–1117.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov processes. Characterization and convergence. John Wiley, New York.CrossRefGoogle Scholar
Foucart, C. (2013). The impact of selection in the $\Lambda$-Wright–Fisher model. Electron. Commun. Prob. 18, 1–10.CrossRefGoogle Scholar
Foucart, C. (2019). Continuous-state branching processes with competition: duality and reflection at infinity. Electron. J. Prob. 24, 1–38.CrossRefGoogle Scholar
González Casanova, A., Miró Pina, V. and Pardo, J. C. (2020). The Wright–Fisher model with efficiency. Theoret. Pop. Biol. 132, 33–46.CrossRefGoogle Scholar
González Casanova, A. and Spanó, D. (2018). Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection. Ann. Appl. Prob. 28, 250–284.CrossRefGoogle Scholar
Griffiths, R. C. (2014). The $\Lambda$-Fleming–Viot process and a connection with Wright–Fisher diffusion. Adv. Appl. Prob. 46, 1009–1035.CrossRefGoogle Scholar
Hairer, M. (2016). Lectures Notes on the Convergence of Markov Processes. Available at http://www.hairer.org/notes/Convergence.pdf.Google Scholar
Jagers, P. (1997). Towards dependence in general branching processes. In Classical and Modern Branching Processes, eds. K. B. Athreya and P. Jagers, Springer, New York, pp. 127–139.CrossRefGoogle Scholar
Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Prob. Surveys 11, 59–120.CrossRefGoogle Scholar
Krone, S. and Neuhauser, C. (1997). The genealogy of samples in models with selection. Genetics 145, 519–534.Google ScholarPubMed
Krone, S. and Neuhauser, C. (1997). Ancestral processes with selection. Theoret. Pop. Biol. 51, 210–237.CrossRefGoogle Scholar
Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Prob. 15, 1506–1535.CrossRefGoogle Scholar
Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Processes and Mixing Times. American Mathematical Society, Providence, RI.Google Scholar
Li, Z. and Pu, F. (2012). Strong solutions of jump-type stochastic equations. Electron. Commun. Prob. 17, 1–13.CrossRefGoogle Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
López, F. J., Martínez, S. and Sanz, G. (2000). Stochastic domination and Markovian couplings. Adv. Appl. Prob. 32, 1064–1076.CrossRefGoogle Scholar
Matis, J. H. and Kiffe, T. R. (2004). On stochastic logistic population growth models with immigration and multiple births. Theoret. Pop. Biol. 65, 89–104.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548.CrossRefGoogle Scholar
Mijatović, A. and Urusov, M. (2012). Convergence of integral functionals of one-dimensional diffusions. Electron. Commun. Prob. 17, 1–13.CrossRefGoogle Scholar
Möhle, M. (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5, 761–777.CrossRefGoogle Scholar
Nåsell, I. (2001). Extinction and quasi-stationarity in the Verhulst logistic model. J. Theoret. Biol. 211, 11–27.CrossRefGoogle Scholar
Norris, J. R. (1997). Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.CrossRefGoogle Scholar
Sturm, A. and Swart, J. M. (2015). A particle system with cooperative branching and coalescence. Ann. Appl. Prob. 25, 1616–1649.CrossRefGoogle Scholar
Tweedie, R. L. (1975). Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. Math. Proc. Camb. Phil. Soc. 78, 125–136.CrossRefGoogle Scholar