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 Probability Surveys > Vol. 9 (2012) open journal systems 

Quasi-stationary distributions and population processes

Sylvie Méléard, CMAP, École Polytechnique
Denis Villemonais, INRIA Nancy

This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.

AMS 2000 subject classifications: Primary 92D25, 60J70, 60J80, 65C50.

Keywords: Population dynamics, quasi-stationarity, Yaglom limit, birth and death process, logistic Feller diffusion, Fleming-Viot particle system.

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Méléard, Sylvie, Villemonais, Denis, Quasi-stationary distributions and population processes, Probability Surveys, 9, (2012), 340-410 (electronic). DOI: 10.1214/11-PS191.


[1]    Aldous, D., Stopping times and tightness. Ann. Probability, Volume 6, Issue 2 (1978), pages 335–340. MR0474446

[2]    Anderson, W. J., Continuous-time Markov chains. Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. MR1118840

[3]    Asselah, A., Ferrari, P. A., and Groisman, P., Quasistationary distributions and Fleming-Viot processes in finite spaces. J. Appl. Probab., Volume 48, Issue 2 (2011), pages 322–332. MR2840302

[4]    Athreya, K. B. and Ney, P. E., Branching processes. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196. MR0373040

[5]    Bansaye, V., Surviving particles for subcritical branching processes in random environment. Stochastic Process. Appl., Volume 119, Issue 8 (2009), pages 2436–2464. MR2532207

[6]    Barbour, A. D. and Pollett, P. K., Total variation approximation for quasi-stationary distributions. J. Appl. Probab., Volume 47, Issue 4 (2010), pages 934–946. MR2752899

[7]    Berezin, F. A. and Shubin, M. A., The Schrödinger equation, volume 66 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1983 Russian edition by Yu. Rajabov, D. A. Leĭtes and N. A. Sakharova and revised by Shubin, With contributions by G. L. Litvinov and Leĭtes. MR1186643

[8]    Bieniek, M., Burdzy, K., and Finch, S., Non-extinction of a fleming-viot particle model. Probab. Theory Related Fields (2011), pages 1–40. MR2925576

[9]    Billingsley, P., Convergence of probability measures. John Wiley & Sons Inc., New York, 1968. MR0233396

[10]    Burdzy, K., Holyst, R., Ingerman, D., and March, P., Configurational transition in a fleming-viot-type model and probabilistic interpretation of laplacian eigenfunctions. J. Phys. A, Volume 29 (1996), pages 2633–2642.

[11]    Burdzy, K., Hołyst, R., and March, P., A Fleming-Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys., Volume 214, Issue 3 (2000), pages 679–703. MR1800866

[12]    Carey, Liedo, P., Orozco, D., and Vaupel, J. W., Slowing of mortality rates at older ages in large medfly cohorts. Science, Volume 258 (1992), pages 457–461.

[13]    Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., and San Martín, J., Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab., Volume 37, Issue 5 (2009), pages 1926–1969. MR2561437

[14]    Cattiaux, P. and Méléard, S., Competitive or weak cooperative stochastic lotka-volterra systems conditioned to non-extinction. J. Math. Biology, Volume 6 (2010), pages 797–829. MR2606515

[15]    Collet, P., Martínez, S., Méléard, S., and San Martín, J., Quasi-stationary distributions for structured birth and death processes with mutations. Probab. Theory Related Fields, Volume 151, Issue 1 (2011), pages 191–231. MR2834717

[16]    Collet, P., Martínez, S., and San Martín, J., Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Probab., Volume 23, Issue 3 (1995), pages 1300–1314. MR1349173

[17]    Coolen-Schrijner, P. and van Doorn, E. A., Quasi-stationary distributions for a class of discrete-time Markov chains. Methodol. Comput. Appl. Probab., Volume 8, Issue 4 (2006), pages 449–465. MR2329282

[18]    Darroch, J. N. and Seneta, E., On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probability, Volume 2 (1965), pages 88–100. MR0179842

[19]    Darroch, J. N. and Seneta, E., On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability, Volume 4 (1967), pages 192–196. MR0212866

[20]    Etheridge, A. M., Survival and extinction in a locally regulated population. Ann. Appl. Probab., Volume 14, Issue 1 (2004), pages 188–214. MR2023020

[21]    Ethier, S. N. and Kurtz, T. G., Markov processes, Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. MR0838085

[22]    Ferrari, P. A., Kesten, H., Martínez, S., and Picco, P., Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab., Volume 23, Issue 2 (1995), pages 501–521. MR1334159

[23]    Ferrari, P. A. and Marić, N., Quasi stationary distributions and Fleming-Viot processes in countable spaces. Electron. J. Probab., Volume 12, Issue 24 (2007), pages 684–702. MR2318407

[24]    Ferrari, P. A., Martínez, S., and Picco, P., Some properties of quasi-stationary distributions in the birth and death chains: a dynamical approach. In Instabilities and nonequilibrium structures, III (Valparaíso, 1989), volume 64 of Math. Appl. (1991), pages 177–187. MR1177850

[25]    Fukushima, M., Dirichlet forms and Markov processes, volume 23 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1980. MR0569058

[26]    Galton, F. and Watson, H. W., On the probability of the extinction of families. Available at, 1974.

[27]    Gantmacher, F. R., The theory of matrices. Vols. 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959. MR0107649

[28]    Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. R. Soc. London, Volume 115 (1825), pages 513–583.

[29]    Gong, G. L., Qian, M. P., and Zhao, Z. X., Killed diffusions and their conditioning. Probab. Theory Related Fields, Volume 80, Issue 1 (1988), pages 151–167. MR0970476

[30]    Good, P., The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc., Volume 8 (1968), pages 716–722. MR0240879

[31]    Greenwood, M. and Irwin, J., The biostatistics of senility. Human Biology, Volume 11, Issue 1 (1939), pages 1–23.

[32]    Grigorescu, I. and Kang, M., Hydrodynamic limit for a Fleming-Viot type system. Stochastic Process. Appl., Volume 110, Issue 1 (2004), pages 111–143. MR2052139

[33]    Grigorescu, I. and Kang, M., Immortal particle for a catalytic branching process. Probab. Theory Related Fields (2011), pages 1–29. 10.1007/s00440-011-0347-6. MR2925577

[34]    Hart, A. G. and Pollett, P. K., New methods for determining quasi-stationary distributions for Markov chains. Math. Comput. Modelling, Volume 31, Issue 10-12 (2000), pages 143–150. MR1768777

[35]    Huillet, T., On Wright Fisher diffusion and its relatives. J. Stat. Mech.-Theory E., Volume 11 (2007), pages 6–+.

[36]    Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 1989. MR1011252

[37]    Joffe, A. and Métivier, M., Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab., Volume 18, Issue 1 (1986), pages 20–65. MR0827331

[38]    Karatzas, I. and Shreve, S. E., Brownian motion and stochastic calculus, Volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1988. MR0917065

[39]    Karlin, S. and McGregor, J. L., The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc., Volume 85 (1957), pages 489–546. MR0091566

[40]    Knobloch, R. and Partzsch, L., Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Analysis, Volume 33 (2010), pages 107–136. MR2658978

[41]    Kolb, M. and Steinsaltz, D., Quasilimiting behavior for one-dimensional diffusions with killing. To appear in Ann. Probab.. MR2917771

[42]    Lambert, A., The branching process with logistic growth. Ann. Appl. Probab., Volume 15, Issue 2 (2005), pages 1506–1535. MR2134113

[43]    Lambert, A., Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab., Volume 12, Issue 14 (2007), pages 420–446. MR2299923

[44]    Lladser, M. and San Martín, J., Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process. J. Appl. Probab., Volume 37, Issue 2 (2000), pages 511–520. MR1781008

[45]    Mandl, P., Spectral theory of semi-groups connected with diffusion processes and its application. Czechoslovak Math. J., Volume 11, Issue 86 (1961), pages 558–569. MR0137143

[46]    Martínez, S., Picco, P., and San Martín, J., Domain of attraction of quasi-stationary distributions for the Brownian motion with drift. Adv. in Appl. Probab., Volume 30, Issue 2 (1998), pages 385–408. MR1642845

[47]    Martínez, S. and San Martín, J., Classification of killed one-dimensional diffusions. Ann. Probab., Volume 32, Issue 1 (2004), pages 530–552. MR2040791

[48]    Pakes, A. G. and Pollett, P. K., The supercritical birth, death and catastrophe process: limit theorems on the set of extinction. Stochastic Process. Appl., Volume 32, Issue 1 (1989), pages 161–170. MR1008915

[49]    Pinsky, R. G., On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab., Volume 13, Issue 2 (1985), pages 363–378. MR0781410

[50]    Pollett, P., Quasi-stationary distributions: a bibliography.

[51]    Pollett, P. K. and Stewart, D. E., An efficient procedure for computing quasi-stationary distributions of Markov chains with sparse transition structure. Adv. in Appl. Probab., Volume 26, Issue 1 (1994), pages 68–79. MR1260304

[52]    Renault, O., Ferrière, R., and Porter, J., The quasi-stationary route to extinction. Private communication.

[53]    Revuz, D. and Yor, M., Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. MR1725357

[54]    Seneta, E. and Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability, Volume 3 (1966), pages 403–434. MR0207047

[55]    Serre, D., Matrices, volume 216 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. Theory and applications, Translated from the 2001 French original. MR1923507

[56]    Steinsaltz, D. and Evans, S. N., Markov mortality models: Implications of quasistationarity and varying initial conditions. Theo. Pop. Bio., Volume 65 (2004), 319–337.

[57]    Steinsaltz, D. and Evans, S. N., Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc., Volume 359, Issue 3 (2007), pages 1285–1324. MR2262851

[58]    van Doorn, E. A., Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. in Appl. Probab., Volume 23, Issue 4 (1991), pages 683–700. MR1133722

[59]    van Doorn, E. A., Conditions for the existence of quasi-stationary distributions for birth-death processes with killing. Memorandum No. 1949, Department of Applied Mathematics, University of Twente, 2011.

[60]    van Doorn, E. A. and Pollett, P. K., Quasi-stationary distributions for reducible absorbing Markov chains in discrete time. Markov Process. Related Fields, Volume 15, Issue 2 (2009), pages 191–204. MR2538313

[61]    van Doorn, E. A. and Pollett, P. K., Quasi-stationary distributions. Memorandum No. 1945, Department of Applied Mathematics, University of Twente, 2011.

[62]    Vere-Jones, D., Some limit theorems for evanescent processes. Austral. J. Statist., Volume 11 (1969), pages 67–78. MR0263165

[63]    Verhulst, P. F., Notice sur la loi que la population suit dans son accroissement. Corr. Math. et Phys., 1938.

[64]    Villemonais, D., Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electronic Journal of Probability, Volume 16 (2011), pages 1663–1692. MR2835250

[65]    Villemonais, D. Interacting particle processes and approximation of Markov processes conditioned to not be killed. ArXiv e-prints, 2011. MR2835250

[66]    Yaglom, A. M., Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.), Volume 56 (1947), pages 795–798. MR0022045

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