A review on spatial aggregation methods involving several time scales |
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| Affiliation: | 1. IRD UMI 209 UMMISCO, IRD France Nord, F-93143 Bondy Cedex, France;2. UPMC Univ Paris 06, UMI 209, UMMISCO, F-75005 Paris, France;3. Aix-Marseille Univ., LMGEM – UMR CNRS 6117, 13288 Marseille, France;4. Dpto. Matemática Aplicada, E.T.S. Ingenieros Industriales, c. José Gutiérrez Abascal, 2, 28006 Madrid, Spain;1. DUT-RU International School of Information Science and Technology, Dalian University of Technology, Dalian, China;2. Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, Dalian, China;3. School of Mathematical Science, Dalian University of Technology, Dalian, China;4. Department of Radiology, Beijing Children’s Hospital, Captital Medical University, National Center for Children’s Health, China;1. Department of Angiology and Vascular Surgery, Hospital Universitario La Paz-Instituto de Investigación del Hospital Universitario La Paz, Madrid, Spain;2. Department of Surgery, Faculty of Medicine, Universidad Autónoma de Madrid, Madrid, Spain;3. Department of Experimental Surgery, Hospital Universitario La Paz-Instituto de Investigación del Hospital Universitario La Paz, Madrid, Spain;4. Department of Pathology, Hospital Universitario Rey Juan Carlos, Móstoles, Madrid, Spain;5. Cell Therapy Laboratory, Research Department, Instituto de Investigación Sanitaria-Fundación Jiménez Díaz (IIS-FJD), Madrid, Spain;6. Department of Surgery, University Hospital Fundación Jiménez Díaz, Madrid, Spain;1. Belgian Cancer Registry, Rue Royale 215, Koningstraat 215 – 1210, Bruxelles, Brussel, Belgium;2. Belgian Health Care Knowledge Centre (KCE). Centre Administratif Botanique, Doorbuilding, Boulevard du Jardin Botanique 55, B-1000, Brussels, Belgium;3. Department of Respiratory Medicine, University Hospitals KU Leuven, Herestraat 49, 3000, Leuven, Belgium;4. Faculty of Medicine & Health Sciences, University of Antwerp, Antwerp University Hospital, Universiteitsplein 1, 2610, Antwerp, Belgium;5. Department of Pulmonology & Thoracic Oncology, Antwerp University Hospital, Wilrijkstraat 10, 2650, Edegem, Belgium;6. European Reference Network (ERN-LUNG/EURACAN);7. Department of Thoracic Surgery, University Hospitals KU Leuven, Herestraat 49, 3000, Leuven, Belgium |
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| Abstract: | This article is a review of spatial aggregation of variables for time continuous models. Two cases are considered. The first case corresponds to a discrete space, i.e. a set of discrete patches connected by migrations, which are assumed to be fast with respect to local interactions. The mathematical model is a set of coupled ordinary differential equations (O.D.E.). The spatial aggregation allows one to derive a global model governing the time variation of the total numbers of individuals of all patches in the long term. The second case considers a continuous space and is a set of partial differential equations (P.D.E.). In that case, we also assume that diffusion is fast in comparison with local interactions. The spatial aggregation allows us again to obtain an O.D.E. governing the total population density, which is obtained by integration all over the spatial domain, at the slow time scale. These aggregations of variables are based on time scales separation methods which have been presented largely elsewhere and we recall the main results. We illustrate the methods by examples in population dynamics and prey–predator models. |
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