Abstract
We propose a novel measure valued process which models the behaviour of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species and continuous movement of molecules in space. Reactions rates at a spatial location are proportional to the mass of different species present locally and to a location specific chemical rate, which may be a function of the local or global species mass as well. We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types. In particular, when the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.
Funding Statement
LP was supported by NSERC (Natural Sciences and Engineering Research Council of Canada) and the CRM (Centre de Recherches Mathématiques) -UMI travel allowance. AV was also supported in part by the chaire Modélisation Mathématique et Biodiversité of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X.
Acknowledgments
The authors are grateful to the two reviewers for their particularly useful and constructive comments on previous versions of the manuscript, which helped to improve readability and to simplify some technical aspects of the construction of the process.
Citation
Lea Popovic. Amandine Véber. "A spatial measure-valued model for chemical reaction networks in heterogeneous systems." Ann. Appl. Probab. 33 (5) 3706 - 3754, October 2023. https://doi.org/10.1214/22-AAP1904
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