TY  - INPR
A1  - Méléard, Sylvie
A1  - Roelly, Sylvie
T1  - A host-parasite multilevel interacting process and continuous approximations
N2  - We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2011, 01 
KW  - two-level interacting processes
KW  - birth-death-mutation-competition point process
KW  - host-parasite stochastic particle system
Y1  - 2011
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51694
ER  - 
TY  - INPR
A1  - Méléard, Sylvie
A1  - Roelly, Sylvie
T1  - Evolutive two-level population process and large population approximations
N2  - We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 8 
KW  - Two-level interacting process
KW  - birth-death-mutation-competition point process
KW  - non-linear integro-differential equations
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64604
SN  - 2193-6943
ER  -