"An individual-based model for allometric relationships."Brodu, VirgileAn allometric relationship is of the form $B \propto M^{\alpha} $, where $B$ and $M$ are biological parameters, $M$ being typically a mass, and $\alpha$ is called the allometric coefficient. These allometries are a key ingredient for modelling ecological dynamics. We design a simple individual-based model, structured by the mass of individuals. This gives rise to a Piecewise Deterministic Markov Process (PDMP) with allometric features. We design this model to be valid for a very broad range of living species, in the spirit of the Metabolic Theory of Ecology. In particular, we define an allometric coefficient $\alpha$ related to metabolism. Usually, experiments and dimensional argument lead to $$\beta = \delta = \gamma - 1 = \alpha- 1,$$ with $\beta$, $\delta$, $\gamma$ being allometric coefficients related to, respectively, the birth rate, the death rate, and the energy gain in the population. We show that elementary -and biologically relevant- constraints imposed to the model allow us to give bounds on $\beta$, $\delta$, $\gamma$ thanks to $\alpha$ only. For this purpose, the study of the underlying PDMP is crucial and we identify various behaviors for our process, depending on the allometric coefficients. |
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