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Asymptotic behaviour of ecological models: reproduction number vs Malthusian parameter

Jordi Ripoll

Universitat de Girona

Abstract
We review the procedure to derive the basic reproduction number R0 for ecological models of population dynamics. R0, defined as the expected lifetime number of offspring of a newborn individual of the population, determines the asymptotic behaviour of the population, as well as the traditional Malthusian parameter (exponential growth rate of the population). We illustrate the differences of both approaches from simple unstructured models (ODE), including trade-offs and evolutionary aspects, to more sophisticated structured models (PDE) where we need to elaborate a suitable numerical method to compute R0 when analytical expressions are not available.

Jordi Ripoll is an associate professor in the Department of Computer Science, Applied Mathematics and Statistics at the University of Girona, where he is a member of the research project entitled Mathematical Modeling, Theoretical Biology and Population Dynamics. He received his Ph.D. degree in mathematical sciences from the University of Barcelona in 2005. His main interest is in population dynamics applied to ecology and epidemiology. He has worked on structured population dynamics, models for biological evolution, and epidemic models with the formalism of the complex networks (systems of many elements with a nontrivial pattern of connections). The mathematical tools he uses are ODE and PDE (transport and reaction–diffusion equations) and also stochastic simulations when dealing with models of epidemic spreading. He made a postdoctoral research stay at the University of Trento during 2006 and 2007, and he is co-author of several papers in international journals and conferences.