Finite size effects in stochastic spatio-temporal models

The classical approach to population dynamics is deterministic in nature: the concentrations of the interacting constituents are assumed to be continuous variables. Alternatively, one can proceed with a stochastic modelling, respecting the intrinsic granularity of the scrutinized system. Starting for a microscopic, hence inherently stochastic, formulation of the inspected model, one can rigorously recover the underlying deterministic picture and assess the role played by finite size corrections via the celebrated van Kampen expansion. At the conventional order of approximation, also called linear noise approximation, the van Kampen analysis yields a Fokker-Planck equation for the distribution of fluctuations. This machinery can be for instance applied to explain the emergence of the so-called quasi-cycles, regular time oscillations of the concentrations, which result from a resonant amplification of the stochastic noise. This possibility is discussed in the first part of the thesis with reference to problem of intracellular calcium dynamics. Then, we have considered the van Kampen expansion beyond the Gaussian order of approximation. The method has been challenged for a selection models and its predictive ability tested versus numerical simulations. In practice, when accounting for higher order corrections, one obtains a generalized Fokker-Planck equation which depends explicitly on N, the size of the system. This enables us to write down a system of ODEs for the unknown moments of the distribution of fluctuations. The sought distribution is eventually recovered by Fourier inversion. The analysis that we have carried out testifies on the adequacy of the van Kampen expansion also when the weak noise hypothesis breaks down or, equivalently, when the characteristic size of the population is small. To complete the study we have also drawn a quantitative comparison between the WKB and the generalized van Kampen expansion, working with a stochastic version of the logistic equations. In the second part of the thesis we have studied reaction-diffusion systems of both stochastic and deterministic inspiration. As an interesting ingredient we have considered the effect of the microscopic competition for the available space, imposing a finite carrying capacity constraint into the microscopic formulation of the models. This is presumably relevant when the interacting species are densely packed in space, in a regime that is often referred to as to the molecular crowding. We studied in particular the conditions for the emergence of stochastic spatially extended patterns and the interplay between Turing like patterns and the cross diffusion due to crowding.