A spatial individual-based contact model with age structure

Dominika Jasińska

Abstract


The Markov dynamics of an infinite continuum birth-and-death system of point particles with age is studied. Each particle is characterized by its location \(x\in \mathbb{R}^d\) and age \(a_x\geq 0\). The birth and death rates of a particle are age dependent. The states of the system are described in terms of probability measures on the corresponding configuration space. The exact solution of the  evolution equation for the correlation functions of first and second orders is found.

Keywords


Correlation function; contact model; birth and death model; configuration space; spatial individual-based model; Markov evolution; age structure

Full Text:

PDF

References


Berns, Ch., Kondratiev, Y., Kozitsky, Y., Kutoviy, O., Kawasaki dynamics in continuum: micro- and mesoscopic descriptions, J. Dynam. Differential Equations 25 (4) (2013), 1027-1056.

Bogoliubov, N., Problems of a Dynamical Theory in Statistical Physics, Gostekhisdat, Moscow, 1946 (in Russian). English translation, in: J. de Boer and G. E. Uhlenbeck (editors), Studies in Statistical Mechanics, Volume 1, 1-118, North-Holland, Amsterdam, 1962.

Daletskii, A., Kondratiev, Y., Kozitsky, Y., Phase transitions in continuum ferromagnets with unbounded spins, J. Math. Phys. 56 (11) (2015), 1-20.

Finkelshtein, D., Kondratiev, Y., Oliveira, M., Markov evolutions and hierarchical equations in the continuum I. One-component systems, J. Evol. Equ. 9 (2) (2009), 197-233.

Iannelli, M., Mathematical theory of age-structured population dynamics, Applied Mathematics Monographs, Giardini Editori e Stampatori, Pisa, 1995.

Kondratiev, Y., Kuna, T., Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2) (2002), 201-233.

Kondratiev, Y., Kutoviy, O., Pirogrov, S., Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2) (2008), 231-258.

Kondratiev, Y., Lytvynov, E., Us, G., Analysis and geometry on (mathbb{R}_+) marked configuration spaces, Meth. Func. Anal. and Geometry 5 (1) (2006), 29-64.

Meleard, S., Tran, V., Trait substitution sequence process and canonical equation for age-structured populations, J. Math. Biol. 58 (6) (2009), 881-921.

Meleard, S., Tran, V., Slow and fast scales for superprocess limits of age-structured populations, Stochastic Process. Appl. 122 (1) (2012), 250-276.

Minlos, R. A., Lectures on statistical physics, Russian Mathematical Surveys 23 (1) (1968), 133-190.

Tanaś, A., A continuum individual based model of fragmentation: dynamics of correlation functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 64 (2) (2015), 73-83.




DOI: http://dx.doi.org/10.17951/a.2017.71.1.41
Data publikacji: 2017-06-30 17:33:55
Data złożenia artykułu: 2017-06-30 12:28:13

Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 Dominika Jasińska