# Contact process with random slowdowns: phase transition and hydrodynamic limits

Abstract : In this thesis, we study an interacting particle system that generalizes a contact process, evolving in a random environment. The contact process can be interpreted as a spread of a population or an infection. The motivation of this model arises from behavioural ecology and evolutionary biology via the \textsl{sterile insect technique} ; its aim is to control a population by releasing sterile individuals of the same species: the progeny of a female and a sterile male does not reach sexual maturity, so that the population is reduced or potentially dies out. To understand this phenomenon, we construct a stochastic spatial model on a lattice in which the evolution of the population is governed by a contact process whose growth rate is slowed down in presence of sterile individuals, shaping a dynamic random environment. A first part of this document investigates the construction and the properties of the process on the lattice $\mathbb Z^d$. One obtains monotonicity conditions in order to study the survival or the extinction of the process. We exhibit the existence and uniqueness of a phase transition with respect to the release rate. On the other hand, when $d=1$ and now fixing initially the random environment, we get further survival and extinction conditions which yield explicit numerical bounds on the phase transition. A second part concerns the macroscopic behaviour of the process by studying its hydrodynamic limit when the microscopic evolution is more intricate. We add movements of particles to births and deaths. First on the $d$-dimensional torus, we derive a system of reaction-diffusion equations as a limit. Then, we study the system in infinite volume in $\mathbb Z^d$, and in a bounded cylinder whose boundaries are in contact with stochastic reservoirs at different densities. As a limit, we obtain a non-linear system, with additionally Dirichlet boundary conditions in bounded domain.
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Contributor : Kevin Kuoch <>
Submitted on : Wednesday, January 7, 2015 - 12:09:55 PM
Last modification on : Friday, September 20, 2019 - 4:34:03 PM
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• HAL Id : tel-01100894, version 1

### Citation

Kevin Kuoch. Contact process with random slowdowns: phase transition and hydrodynamic limits. Probability [math.PR]. Université Paris Descartes, 2014. English. ⟨tel-01100894⟩

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