×

Quasi-stationary distributions for structured birth and death processes with mutations. (English) Zbl 1245.92062

Authors’ abstract: We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the actions of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely.
Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.

MSC:

92D40 Ecology
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Billingsley, P., Convergence of probability measures (1968), New York: Wiley, New York · Zbl 0172.21201
[2] Cattiaux, P.; Collet, P.; Lambert, A.; Martinez, S.; Méléard, S.; San Martín, J., Quasi-stationary distributions and diffusions models in population dynamics, Ann. Probab., 37, 5, 1926-1969 (2009) · Zbl 1176.92041 · doi:10.1214/09-AOP451
[3] Cattiaux, P.; Méléard, S., Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction, J. Math. Biol., 6, 797-829 (2010) · Zbl 1202.92082 · doi:10.1007/s00285-009-0285-4
[4] Collet, P.; Martínez, S.; San Martín, J., Asymptotic laws for one-dimensional diffusions conditioned to non-absorption, Ann. Probab., 23, 3, 1300-1314 (1995) · Zbl 0867.60046 · doi:10.1214/aop/1176988185
[5] Champagnat, N.; Ferrière, R.; Méléard, S., Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models, Theoret. Pop. Biol., 69, 297-321 (2006) · Zbl 1118.92039 · doi:10.1016/j.tpb.2005.10.004
[6] Darroch, J. N.; Seneta, E., On quasi-stationary distributions in absorbing discrete-time finite Markov chains, J. Appl. Prob., 2, 88-100 (1965) · Zbl 0134.34704 · doi:10.2307/3211876
[7] Daley, D. J.; Vere-Jones, D., An Introduction to the theory of point processes. In: Springer Series in Statistics (1988), New York: Springer, New York · Zbl 0657.60069
[8] Dunford, N., Schwartz, J.: Linear Operators, vol. I. Interscience (1958) · Zbl 0084.10402
[9] Feller, W., An Introduction to probability Theory and its Applications (1968), New York: Wiley, New York · Zbl 0155.23101
[10] Ferrari, P. A.; Kesten, H.; Martinez, S.; Picco, P., Existence of quasi-stationary distributions. A renewal dynamical approach, Ann. Probab., 23, 2, 501-521 (1995) · Zbl 0827.60061 · doi:10.1214/aop/1176988277
[11] Fournier, N.; Méléard, S., A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab., 14, 4, 1880-1919 (2004) · Zbl 1060.92055 · doi:10.1214/105051604000000882
[12] Gosselin, F., Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology, Ann. Appl. Probab., 11, 1, 261-284 (2001) · Zbl 1019.60082 · doi:10.1214/aoap/998926993
[13] Mandl, P., Spectral theory of semi-groups connected with diffusion processes and its applications, Czech. Math. J., 11, 558-569 (1961) · Zbl 0115.13503
[14] Martínez, S.; San Martín, J., Classification of killed one-dimensional diffusions, Ann. Probab., 32, 1, 530-552 (2004) · Zbl 1045.60083 · doi:10.1214/aop/1078415844
[15] Pinsky, R., On the convergence of diffusion processes conditioned to remain in bounded region for large time to limiting positive recurrent diffusion processes, Ann. Probab., 13, 363-378 (1985) · Zbl 0567.60076 · doi:10.1214/aop/1176992996
[16] Nair, M. G.; Pollett, P., On the relationship between μ−invariant measures and quasi-stationary distributions for continuous-time Markov chains, Adv. Appl. Probab., 25, 1, 82-102 (1993) · Zbl 0774.60070 · doi:10.2307/1427497
[17] Renault, O., Ferrière, R., Porter, J.: The quasi-stationary route to extinction. Private communication
[18] Steinsaltz, D.; Evans, S., Markov mortality models: implications of quasi-stationarity and varying initial distributions, Theor. Popul. Biol., 65, 319-337 (2004) · Zbl 1109.92036 · doi:10.1016/j.tpb.2003.10.007
[19] Steinsaltz, D.; Evans, S., Quasi-stationary distributions for one-dimensional diffusions with killing, Trans. Am. Math. Soc., 359, 3, 1285-1324 (2007) · Zbl 1107.60048 · doi:10.1090/S0002-9947-06-03980-8
[20] Tychonov, A., Ein Fixpunktsatz, Math. Ann., 117, 767-776 (1935) · Zbl 0012.30803 · doi:10.1007/BF01472256
[21] Van Doorn, E., Quasi-stationary distributions and convergence to quasi-stationary of birth-death processes, Adv. Appl. Probab., 23, 4, 683-700 (1991) · Zbl 0736.60076 · doi:10.2307/1427670
[22] Vere-Jones, D., Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxf. (2), 13, 7-28 (1962) · Zbl 0104.11805 · doi:10.1093/qmath/13.1.7
[23] Yaglom, A. M., Certain limit theorems of the theory of branching processes (in Russian), Dokl. Akad. Nauk. SSSR, 56, 795-798 (1947) · Zbl 0041.45602
[24] Yosida, K., Functional Analysis (1968), Berlin: Springer, Berlin · Zbl 0126.11504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.