Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T17:36:07.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Large population limit and time behaviour of a stochastic particle model describing an age-structured population

Published online by Cambridge University Press:  08 May 2008

Viet Chi Tran
Viet Chi Tran*
Affiliation:
Université Paris X-Nanterre, Équipe Modal'X, bâtiment G, 200 avenue de la République, 92100 Nanterre Cedex, France; vtran@u-paris10.fr
Get access

Abstract


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.


Keywords

Age-structured populationinteracting measure-valued processlarge population approximationlarge deviationsexit time estimatesGurtin-McCamy PDEextinction time
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 345 - 386
Copyright
© EDP Sciences, SMAI, 2008

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

Aldous, D., Stopping times and tightness. Ann. Probab. 6 (1978) 335–340. CrossRef
K.B. Athreya and P.E. Ney, Branching Processes. Springer edition (1970).
Bellman, R. and Harris, T.E., On age-dependent binary branching processes. Ann. Math. 55 (1952) 280–295. CrossRef
P. Billingsley, Convergence of Probability Measures. John Wiley & Sons (1968).
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.
Busenberg, S. and Iannelli, M., A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Anal. 7 (1983) 501–529. CrossRef
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.
N. Champagnat, R. Ferrière and S. Méléard, Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation. Theoretical Population Biology (2006).
Crump, K.S. and Mode, C.J., A general age-dependent branching process i. J. Math. Anal. Appl. 24 (1968) 494–508. CrossRef
Crump, K.S. and Mode, C.J., A general age-dependent branching process ii. J. Math. Anal. Appl. 25 (1969) 8–17. CrossRef
Dawson, D.A. and Gärtner, J., Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987) 247–308. CrossRef
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications. Jones and Bartlett Publishers, Boston (1993).
Doney, R.A., Age-dependent birth and death processes. Z. Wahrscheinlichkeitstheorie verw. 22 (1972) 69–90. CrossRef
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
L.C. Evans, Partial Differential Equations, Grad. Stud. Math. 19 American Mathematical Society (1998).
H. Von Foerster, Some remarks on changing populations, in The Kinetics of Cellular Proliferation, Grune & Stratton Ed., New York (1959) 382–407.
Fournier, N. and Méléard, S., A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004) 1880–1919. CrossRef
M.I. Freidlin and A. Ventzell, Random Perturbations of Dynamical Systems. Springer-Verlag (1984).
Galton, F. and Watson, H.W., On the probability of the extinction of families. J. Anthropol. Inst. Great B. and Ireland 4 (1874) 138–144.
Graham, C. and Méléard, S., 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.
Graham, C. and Méléard, S., An upper bound of large deviations for a generalized star-shaped loss network. Markov Processes and Related Fields 3 (1997) 199–224.
Gurtin, M.E. and MacCamy, R.C., Nonlinear age-dependent population dynamics. Arch. Rat. Mech. Anal. 54 (1974) 281–300. CrossRef
T.E. Harris, The Theory of Branching Processes. Springer, Berlin (1963).
Israel, R.B., Existence of phase transitions for long-range interactions. Comm. Math. Phys. 43 (1975) 59–68. CrossRef
J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987).
Jagers, P., A general stochastic model for population development. Skand. Aktuarietidskr 52 (1969) 84–103.
Jagers, P. and Klebaner, F., Population-size-dependent and age-dependent branching processes. Stochastic Process Appl. 87 (2000) 235–254. CrossRef
Jakubowski, A., On the skorokhod topology. Ann. Inst. H. Poincaré 22 (1986) 263–285.
Joffe, A. and Métivier, M., Weak convergence of sequences of semimartingales with applications to multitype branching processes. Advances in Applied Probability 18 (1986) 20–65. CrossRef
D.G. Kendall, Stochastic processes and population growth. J. Roy. Statist. Sec., Ser. B 11 (1949) 230–264.
Kipnis, C. and Léonard, C., Grandes déviations pour un système hydrodynamique asymétrique de particules indépendantes. Ann. Inst. H. Poincaré 31 (1995) 223–248.
Léonard, C., On large deviations for particle systems associated with spatially homogeneous boltzmann type equations. Probab. Theory Related Fields 101 (1995) 1–44. CrossRef
T.R. Malthus, An Essay on the Principle of Population. J. Johnson St. Paul's Churchyard (1798).
Marcati, P., On the global stability of the logistic age dependent population growth. J. Math. Biol. 15 (1982) 215–226. CrossRef
McKendrick, A.G., Applications of mathematics to medical problems. Proc. Edin. Math. Soc. 54 (1926) 98–130.
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.
S. Méléard and V.C. Tran. Age-structured trait substitution sequence process and canonical equation. Submitted.
Meyn, S.P. and Tweedie, R.L., Stability of markovian processes iii: Foster-lyapunov criteria for continuous-time processes. Advances in Applied Probability 25 (1993) 518–548.
K. Oelschläger, Limit theorem for age-structured populations. Ann. Probab. (1990).
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge Uiversity Press (1992).
S.T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley & Sons (1991).
M.M. Rao and Z.D. Ren, Theory of Orlicz spaces. M. Dekker, New York (1991).
Roelly, S., A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 (1986) 43–65. CrossRef
W. Rudin, Real and Complex Analysis. McGraw-Hill International Editions, third edition (1987).
Sharpe, F.R. and Lotka, A.J., A problem in age distribution. Philos. Mag. 21 (1911) 435–438. CrossRef
Solomon, W., Representation and approximation of large population age distributions using poisson random measures. Stochastic Process. Appl. 26 (1987) 237–255. CrossRef
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.
Verhulst, P.F., Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique 10 (1838) 113–121.
C. Villani, Topics in Optimal Transportation. American Mathematical Society (2003).
Wang, F.J.S., A central limit theorem for age- and density-dependent population processes. Stochastic Process. Appl. 5 (1977) 173–193. CrossRef
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).
C. Zuily and H. Queffélec, Éléments d'analyse pour l'agrégation. Masson (1995).