Population Dynamics At High Reynolds Number

Abstract

We study the statistical properties of population dynamics evolving in a realistic two-dimensional compressible turbulent velocity field. We show that the interplay between turbulent dynamics and population growth and saturation leads to quasi-localization and a remarkable reduction in the carrying capacity. The statistical properties of the population density are investigated and quantified via multifractal scaling analysis. We also investigate numerically the singular limit of negligibly small growth rates and delocalization of population ridges triggered by uniform advection. where C(x, t) is a continuous variable describing the concentration of micro-organisms, D is the diffusion coefficient and µ is the growth rate. As an example of "life at high Reynolds number", we could take Eq.(1) to represent the density of the marine cyanobacterium Synechococcus [3] under conditions of abundant nutrients, so that µ ∼ constant. As discussed in [4], an advecting compressible turbulent flow leads to highly non-trivial dynamics. Although the results of [4] were obtained only in one dimension using a synthetic advecting flow from a shell model of turbulence, two striking effects were observed: the concentration field C(x, t) is strongly localized near transient but long-lived sinks of the turbulent flows for small enough growth rate µ; in the same limit, the space-time average concentration (denoted in the following as carrying capacity) becomes much smaller than its maximum value 1. Both effects are relevant in biological applications [5].In this Letter, we present new numerical results for more realistic two dimensional turbulent flows. We assume that the microorganism concentration field C(x, t), whose dynamics is described by Eq. (1), lies on a planar surface of constant height in a three dimensional fully developed turbulent flow with periodic boundary conditions. Such a system could be a rough approximation to photosynthetic microorganisms that actively control their bouyancy to manitain a fixed depth below the surface of a turbulent fluid [6]. As a consequence of this choice, the flow field in the two dimensional slice becomes compressible [7]. We consider here a turbulent advecting field u(x, t) described by the Navier-Stokes equations, and nondimensionalize time by the Kolmogorov dissipation time-scale τ η ≡ (ν/ǫ) 1/2 and space by the Kolmogorov length-scale η ≡ (ν 3 /ǫ) 1/4 , where ǫ is the mean rate of energy dissipation and ν is the kinematic viscosity. The non-dimensional numbers charecterizing the evolution of the scalar field C(x, t) are then the Schmidt number Sc = ν/D and the non-dimensional time µτ η . A particularly interesting regime arises when the doubling time τ g ≡ µ −1 is somewhere in the middle of the inertial range of eddy turnover times (τ r = r/δ r u, where δ r u is the typical velocity difference across length scale r) that characterize the turbulence. Although the underlying turbulent energy cascade is somewhat different [8], this situation arises for oceanic cyanobacteria and phytoplankton, ...

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