Large population limit and time behaviour of a stochastic particle model describing an age-structured population
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 345-386.

We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the super-critical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.

DOI : https://doi.org/10.1051/ps:2007052
Classification : 60J80,  60K35,  92D25,  60F10
Mots clés : age-structured population, interacting measure-valued process, large population approximation, large deviations, exit time estimates, Gurtin-McCamy PDE, extinction time
@article{PS_2008__12__345_0,
     author = {Tran, Viet Chi},
     title = {Large population limit and time behaviour of a stochastic particle model describing an age-structured population},
     journal = {ESAIM: Probability and Statistics},
     pages = {345--386},
     publisher = {EDP-Sciences},
     volume = {12},
     year = {2008},
     doi = {10.1051/ps:2007052},
     mrnumber = {2404035},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007052/}
}
TY  - JOUR
AU  - Tran, Viet Chi
TI  - Large population limit and time behaviour of a stochastic particle model describing an age-structured population
JO  - ESAIM: Probability and Statistics
PY  - 2008
DA  - 2008///
SP  - 345
EP  - 386
VL  - 12
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007052/
UR  - https://www.ams.org/mathscinet-getitem?mr=2404035
UR  - https://doi.org/10.1051/ps:2007052
DO  - 10.1051/ps:2007052
LA  - en
ID  - PS_2008__12__345_0
ER  - 
Tran, Viet Chi. Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 345-386. doi : 10.1051/ps:2007052. http://www.numdam.org/articles/10.1051/ps:2007052/

[1] D. Aldous, Stopping times and tightness. Ann. Probab. 6 (1978) 335-340. | MR 474446 | Zbl 0391.60007

[2] K.B. Athreya and P.E. Ney, Branching Processes. Springer edition (1970). | MR 373040 | Zbl 0259.60002

[3] R. Bellman and T.E. Harris, On age-dependent binary branching processes. Ann. Math. 55 (1952) 280-295. | MR 45971 | Zbl 0046.35502

[4] P. Billingsley, Convergence of Probability Measures. John Wiley & Sons (1968). | MR 233396 | Zbl 0172.21201

[5] E. Bishop and R.R. Phelps, The support functionals of a convex set, in Proc. Sympos. Pure Math. Amer. Math. Soc., Ed. Providence 7 (1963) 27-35. | MR 154092 | Zbl 0149.08601

[6] S. Busenberg and M. Iannelli, A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Anal. 7 (1983) 501-529. | MR 698362 | Zbl 0528.92016

[7] N. Champagnat, R. Ferrière and S. Méléard, Individual-based probabilistic models of adpatative evolution and various scaling approximations, in Proceedings of the 5th seminar on Stochastic Analysis, Random Fields and Applications, Probability in Progress Series, Ascona, Suisse (2006). Birkhauser. | Zbl 1140.92017

[8] N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation. Theoretical Population Biology (2006). | Zbl 1118.92039

[9] K.S. Crump and C.J. Mode, A general age-dependent branching process i. J. Math. Anal. Appl. 24 (1968) 494-508. | Zbl 0192.54301

[10] K.S. Crump and C.J. Mode, A general age-dependent branching process ii. J. Math. Anal. Appl. 25 (1969) 8-17. | MR 237005 | Zbl 0201.19202

[11] D.A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987) 247-308. | MR 885876 | Zbl 0613.60021

[12] A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications. Jones and Bartlett Publishers, Boston (1993). | MR 1202429 | Zbl 0793.60030

[13] R.A. Doney, Age-dependent birth and death processes. Z. Wahrscheinlichkeitstheorie verw. 22 (1972) 69-90. | MR 312581 | Zbl 0215.53702

[14] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). | MR 463994 | Zbl 0322.90046

[15] L.C. Evans, Partial Differential Equations, Grad. Stud. Math. 19 American Mathematical Society (1998). | Zbl 0902.35002

[16] H. Von Foerster, Some remarks on changing populations, in The Kinetics of Cellular Proliferation, Grune & Stratton Ed., New York (1959) 382-407.

[17] N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004) 1880-1919. | MR 2099656 | Zbl 1060.92055

[18] M.I. Freidlin and A. Ventzell, Random Perturbations of Dynamical Systems. Springer-Verlag (1984). | MR 722136 | Zbl 0522.60055

[19] F. Galton and H.W. Watson, On the probability of the extinction of families. J. Anthropol. Inst. Great B. and Ireland 4 (1874) 138-144.

[20] C. Graham and S. Méléard, A large deviation principle for a large star-shaped loss network with links of capacity one. Markov Processes and Related Fields 3 (1997) 475-492. | MR 1607107 | Zbl 0907.60083

[21] C. Graham and S. Méléard, An upper bound of large deviations for a generalized star-shaped loss network. Markov Processes and Related Fields 3 (1997) 199-224. | MR 1468174 | Zbl 0906.60070

[22] M.E. Gurtin and R.C. Maccamy, Nonlinear age-dependent population dynamics. Arch. Rat. Mech. Anal. 54 (1974) 281-300. | MR 354068 | Zbl 0286.92005

[23] T.E. Harris, The Theory of Branching Processes. Springer, Berlin (1963). | MR 163361 | Zbl 0117.13002

[24] R.B. Israel, Existence of phase transitions for long-range interactions. Comm. Math. Phys. 43 (1975) 59-68. | MR 459458 | Zbl 0309.46056

[25] J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987). | MR 959133 | Zbl 0635.60021

[26] P. Jagers, A general stochastic model for population development. Skand. Aktuarietidskr 52 (1969) 84-103. | MR 431406 | Zbl 0184.40603

[27] P. Jagers and F. Klebaner, Population-size-dependent and age-dependent branching processes. Stochastic Process Appl. 87 (2000) 235-254. | MR 1757114 | Zbl 1045.60090

[28] A. Jakubowski, On the skorokhod topology. Ann. Inst. H. Poincaré 22 (1986) 263-285. | Numdam | MR 871083 | Zbl 0609.60005

[29] A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes. Advances in Applied Probability 18 (1986) 20-65. | MR 827331 | Zbl 0595.60008

[30] D.G. Kendall, Stochastic processes and population growth. J. Roy. Statist. Sec., Ser. B 11 (1949) 230-264. | MR 34977 | Zbl 0038.08803

[31] C. Kipnis and C. Léonard, Grandes déviations pour un système hydrodynamique asymétrique de particules indépendantes. Ann. Inst. H. Poincaré 31 (1995) 223-248. | Numdam | MR 1340038 | Zbl 0817.60020

[32] C. Léonard, On large deviations for particle systems associated with spatially homogeneous boltzmann type equations. Probab. Theory Related Fields 101 (1995) 1-44. | MR 1314173 | Zbl 0839.60031

[33] T.R. Malthus, An Essay on the Principle of Population. J. Johnson St. Paul's Churchyard (1798).

[34] P. Marcati, On the global stability of the logistic age dependent population growth. J. Math. Biol. 15 (1982) 215-226. | MR 684935 | Zbl 0496.92009

[35] A.G. Mckendrick, Applications of mathematics to medical problems. Proc. Edin. Math. Soc. 54 (1926) 98-130. | JFM 52.0542.04

[36] S. Méléard and S. Roelly, Sur les convergences étroite ou vague de processus à valeurs mesures. C.R. Acad. Sci. Paris, Série I 317 (1993) 785-788. | MR 1244431 | Zbl 0781.60071

[37] S. Méléard and V.C. Tran. Age-structured trait substitution sequence process and canonical equation. Submitted.

[38] S.P. Meyn and R.L. Tweedie, Stability of markovian processes iii: Foster-lyapunov criteria for continuous-time processes. Advances in Applied Probability 25 (1993) 518-548. | MR 1234295 | Zbl 0781.60053

[39] K. Oelschläger, Limit theorem for age-structured populations. Ann. Probab. (1990). | MR 1043949 | Zbl 0711.92014

[40] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge Uiversity Press (1992). | MR 1207136 | Zbl 0761.60052

[41] S.T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley & Sons (1991). | MR 1105086 | Zbl 0744.60004

[42] M.M. Rao and Z.D. Ren, Theory of Orlicz spaces. M. Dekker, New York (1991). | MR 1113700 | Zbl 0724.46032

[43] S. Roelly, A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 (1986) 43-65. | MR 878553 | Zbl 0598.60088

[44] W. Rudin, Real and Complex Analysis. McGraw-Hill International Editions, third edition (1987). | MR 924157 | Zbl 0925.00005

[45] F.R. Sharpe and A.J. Lotka, A problem in age distribution. Philos. Mag. 21 (1911) 435-438. | JFM 42.1030.02

[46] W. Solomon, Representation and approximation of large population age distributions using poisson random measures. Stochastic Process. Appl. 26 (1987) 237-255. | MR 923106 | Zbl 0663.92012

[47] V.C. Tran, Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques. Ph.D. thesis, Université Paris X - Nanterre. http://tel.archives-ouvertes.fr/tel-00125100.

[48] P.F. Verhulst, Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique 10 (1838) 113-121.

[49] C. Villani, Topics in Optimal Transportation. American Mathematical Society (2003). | MR 1964483 | Zbl 1106.90001

[50] F.J.S. Wang, A central limit theorem for age- and density-dependent population processes. Stochastic Process. Appl. 5 (1977) 173-193. | MR 443138 | Zbl 0365.92030

[51] G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied mathematics 89, Marcel Dekker, inc., New York-Basel (1985). | MR 772205 | Zbl 0555.92014

[52] C. Zuily and H. Queffélec, Éléments d'analyse pour l'agrégation. Masson (1995).

Cité par Sources :