Statistical Fluctuations in Evolutionary and Population Dynamics

Abstract

In this thesis, we study collective phenomena that arise from microscopic fluctuations at the individual level of two different living populations. First, we study evolutionary dynamics of two-species competitions in a well-mixed environment subject to population size fluctuations. We demonstrate a mechanism for neutral evolution such that population size fluctuations favor a fixation of one species over the other. An effective evolutionary dynamics for fluctuation-induced selection is derived. We then investigate strong mutualism, in a limit where a varying population size can strongly influence the evolutionary dynamics. We determine fixation probabilities as well as mean fixation times taking into account the population size degree of freedom. The results elucidate the interplay between population size fluctuations and evolutionary dynamics in well-mixed systems. Second, we investigate single species marine population subject to a constant flow field and quenched random spatially fluctuating growth rates. We show that the non-equilibrium steady-state population density of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation develops a flow-driven striation pattern. The striations are highly asymmetric with a longitudinal correlation length that diverges linearly with the flow speed and a transverse correlation length that approaches a finite velocity-independent value. The findings suggest that, although the growth disorder can be spatially uncorrelated, correlated population structures with striations emerge naturally at sufficiently strong advection.