Self-propelled non-linearly diffusing particles. Aggregation and continuum description

We introduce a model of self-propelled particles carrying out a Brownian motion with a diffusion coefficient which depends on the local density of particles within a certain finite radius. Numerical simulations show that in a range of parameters the long-time spatial distribution of particles is non-homogeneous, and clusters can be observed. A number density equation, which explains the emergence of the aggregates and indicates the values of the parameters for which they appear, is derived. Numerical results of this continuum density equation are also shown.Comment: 5 pages, 5 figures. Major modifications. A new figure and some references added. Final version accepted for publication in Phys. Rev.

Sustained plankton blooms under open chaotic flows

We consider a predator-prey model of planktonic population dynamics, of excitable character, living in an open and chaotic fluid flow, i.e., a state of fluid motion in which fluid trajectories are unbounded but a chaotic region exists that is restricted to a localized area. Despite that excitability is a transient phenomenon and that fluid trajectories are continuously leaving the system, there is a regime of parameters where the excitation remains permanently in the system, given rise to a persistent plankton bloom. This regime is reached when the time scales associated to fluid stirring become slower than the ones associated to biological growth.Comment: 14 pages, 3 figure

Efficiency of a stirred chemical reaction in a closed vessel

We perform a numerical study of the reaction efficiency in a closed vessel. Starting with a little spot of product, we compute the time needed to complete the reaction in the container following an advection-reaction-diffusion process. Inside the vessel it is present a cellular velocity field that transports the reactants. If the size of the container is not very large compared with the typical length of the velocity field one has a plateau of the reaction time as a function of the strength of the velocity field, $U$. This plateau appears both in the stationary and in the time-dependent flow. A comparison of the results for the finite system with the infinite case (for which the front speed, $v_f$, gives a simple estimate of the reacting time) shows the dramatic effect of the finite size.Comment: 4 pages, 4 figure

Spatial Patterns in Chemically and Biologically Reacting Flows

We present here a number of processes, inspired by concepts in Nonlinear Dynamics such as chaotic advection and excitability, that can be useful to understand generic behaviors in chemical or biological systems in fluid flows. Emphasis is put on the description of observed plankton patchiness in the sea. The linearly decaying tracer, and excitable kinetics in a chaotic flow are mainly the models described. Finally, some warnings are given about the difficulties in modeling discrete individuals (such as planktonic organisms) in terms of continuous concentration fields.Comment: 41 pages, 10 figures; To appear in the Proceedings of the 2001 ISSAOS School on 'Chaos in Geophysical Flows

Spatial patterns of competing random walkers

We review recent results obtained from simple individual-based models of biological competition in which birth and death rates of an organism depend on the presence of other competing organisms close to it. In addition the individuals perform random walks of different types (Gaussian diffusion and L\'{e}vy flights). We focus on how competition and random motions affect each other, from which spatial instabilities and extinctions arise. Under suitable conditions, competitive interactions lead to clustering of individuals and periodic pattern formation. Random motion has a homogenizing effect and then delays this clustering instability. When individuals from species differing in their random walk characteristics are allowed to compete together, the ones with a tendency to form narrower clusters get a competitive advantage over the others. Mean-field deterministic equations are analyzed and compared with the outcome of the individual-based simulations.Comment: 38 pages, including 6 figure

Species clustering in competitive Lotka-Volterra models

We study the properties of Lotka-Volterra competitive models in which the intensity of the interaction among species depends on their position along an abstract niche space through a competition kernel. We show analytically and numerically that the properties of these models change dramatically when the Fourier transform of this kernel is not positive definite, due to a pattern forming instability. We estimate properties of the species distributions, such as the steady number of species and their spacings, for different types of kernels.Comment: 4 pages, 3 figure

Online games: a novel approach to explore how partial information influences human random searches

Many natural processes rely on optimizing the success ratio of a search process. We use an experimental setup consisting of a simple online game in which players have to find a target hidden on a board, to investigate the how the rounds are influenced by the detection of cues. We focus on the search duration and the statistics of the trajectories traced on the board. The experimental data are explained by a family of random-walk-based models and probabilistic analytical approximations. If no initial information is given to the players, the search is optimized for cues that cover an intermediate spatial scale. In addition, initial information about the extension of the cues results, in general, in faster searches. Finally, strategies used by informed players turn into non-stationary processes in which the length of each displacement evolves to show a well-defined characteristic scale that is not found in non-informed searches.Comment: 17 pages, 10 figure

Spatial patterns in mesic savannas: the local facilitation limit and the role of demographic stochasticity

We propose a model equation for the dynamics of tree density in mesic savannas. It considers long-range competition among trees and the effect of fire acting as a local facilitation mechanism. Despite short-range facilitation is taken to the local-range limit, the standard full spectrum of spatial structures obtained in general vegetation models is recovered. Long-range competition is thus the key ingredient for the development of patterns. The long time coexistence between trees and grass, and how fires affect the survival of trees as well as the maintenance of the patterns is studied. The influence of demographic noise is analyzed. The stochastic system, under the parameter constraints typical of mesic savannas, shows irregular patterns characteristics of realistic situations. The coexistence of trees and grass still remains at reasonable noise intensities.Comment: 12 pages, 7 figure