Brownian motors in nonlinear diffusive media

Single-species fragmentation: The role of density-dependent feedback

Internal feedback is commonly present in biological populations and can play a crucial role in the emergence of collective behavior. To describe the temporal evolution of the distribution of a single-species population, we consider a generalization of the Fisher-KPP equation. This equation includes the elementary processes of random motion, reproduction, and, importantly, nonlocal interspecific competition, which introduces a spatial scale of interaction. In addition, we take into account feedback mechanisms in diffusion and growth processes, mimicked by power-law density dependencies. This feedback includes, for instance, anomalous diffusion, reaction to overcrowding or to the rarefaction of the population, as well as Allee-like effects. We show that, depending on the kind of feedback that takes place, the population can self-organize splitting into disconnected subpopulations, in the absence of external constraints. Through extensive numerical simulations, we investigate the temporal evolution and the characteristics of the stationary population distribution in the one-dimensional case. We discuss the crucial role that density-dependence has on pattern formation, particularly on fragmentation, which can bring important consequences to processes such as epidemic spread and speciation.

Nonlinear population dynamics in a bounded habitat

A key issue in ecology is whether a population will survive long term or go extinct. This is the question we address in this paper for a population in a bounded habitat. We will restrict our study to the case of a single species in a one-dimensional habitat of length L. The evolution of the population density distribution ρ(x, t), where x is the position and t the time, is governed by elementary processes such as growth and dispersal, which, in standard models, are typically described by a constant per capita growth rate and normal diffusion, respectively. However, feedbacks in the regulatory mechanisms and external factors can produce density-dependent rates. Therefore, we consider a generalization of the standard evolution equation, which, after dimensional scaling and assuming large carrying capacity, becomes ∂_tρ=∂_x(ρ^ν-1∂_xρ)+ρ^μ, where μ,ν∈R. This equation is complemented by absorbing boundaries, mimicking adverse conditions outside the habitat. For this nonlinear problem, we obtain, analytically, exact expressions of the critical habitat size L_c for population survival, as a function of the exponents and initial conditions. We find that depending on the values of the exponents (ν, μ), population survival can occur for either L ≥ L_c, L ≤ L_c or for any L. This generalizes the usual statement that L_c represents the minimum habitat size. In addition, nonlinearities introduce dependence on the initial conditions, affecting L_c.

Nonlinear diffusion effects on biological population spatial patterns

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density u(x,t). The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form ∂tu(x,t)=D∂xxu(x,t)ν (with D,ν>0), encompassing the cases where the state-dependent diffusion coefficient either increases (ν>1) or decreases (ν<1) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.

Brownian pump in nonlinear diffusive media

A Brownian pump in nonlinear diffusive media is investigated in the presence of an unbiased external force. The pumping system is embedded in a finite region and bounded by two particle reservoirs. In the adiabatic limit, we obtain the analytical expressions of the current and the pumping capacity as a function of temperature for normal diffusion, subdiffusion, and superdiffusion. It is found that important anomalies are detected in comparison with the normal diffusion case. The superdiffusive regime, compared with the normal one, exhibits an opposite current for low temperatures. In the subdiffusive regime, the current may become forbidden for low temperatures and negative for high temperatures.