Abstract
Motivated by the study of a parasite infection in a cell line, we introduce a general class of Markov processes for the modelling of population dynamics. The population process evolves as a diffusion with positive jumps whose rate is a function of the population size. It also undergoes catastrophic events which kill a fraction of the population, at a rate depending on the population state. We study the long time behaviour of this class of processes.
Highlights
We introduce a general class of non-negative continuous-time and space Markov processes, including diffusive terms, as well as negative and positive jumps
The processes studied in the current work may be interpreted as the dynamics of the quantity of parasites in a cell line
Theorem 3.3 shows that the behaviour of the noise around zero determines the fate of the process in terms of absorption: if it is large enough compared to the growth rate of the parasites around zero, the probability of being absorbed in finite time is positive
Summary
We introduce a general class of non-negative continuous-time and space Markov processes, including diffusive terms, as well as negative and positive jumps. Models where the interactions between individuals result from the fact that the whole population is subject to the variations of the same environment have been intensively studied recently, in particular in the framework of CSBPs in random environment This class of models, initially introduced by Keiding and Kurtz [12, 13] in the case of Feller diffusions in a Brownian environment, have been generalised and studied by many authors during the last decade [7, 2, 23, 21, 9, 22, 19, 1].
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