Population dynamics advected by chaotic flows: A discrete-time map approach

Connectivity Analysis Applied to Mesoscale Eddies in the Western Mediterranean Basin

The Western Mediterranean basin (WMED) is characterized by the presence of energetic and dynamic mesoscale cyclonic and anticyclonic eddies. They mainly originate along the Algerian and the Northern currents and have a large influence on the basin circulation. Eddies can last for months, with longer lifetimes associated with the anticyclones, which can move far from their areas of origin. As they partially isolate and transfer water masses, they also have an impact on water properties (physical, chemical and biological), pollutant’s dispersion and transport of eggs, larvae and planktonic organisms. In this study, a connectivity analysis method is applied to the anticyclonic eddies (AEs) identified by an automated hybrid detection and tracking algorithm south of 42°N in the WMED. The same methodology is also applied to the trajectories of Lagrangian surface drifters available in the study area. The purpose is to highlight the connections between different areas of the basin linked to eddy activities in addition to the connectivity due to the mean surface circulation. Drifter data analysis showed that all the WMED sub-basins are strongly interconnected, with the mean surface circulation allowing a shortcut connection among many areas of the basin. The connectivity analysis of the AEs tracks shows that although AEs are ubiquitous in the WMED, their connectivity is limited to well-defined regions, depending on their origin location. Three main regions: the south-western, the south-eastern and the northern parts of the basin are characterized by AEs recirculation, with sporadic export of eddies to the other WMED zones.

Discrete time model for chemical or biological decay in chaotic flows: reentrance phase transitions

We consider a discrete time model of advection, reaction, and diffusion on a lattice to investigate the steady-state spatial structure of chemically decaying substances. The time discretization of the dynamics has a considerable impact on these structures. Additional smooth-filamental phase transitions, nonexistent in the continuous-time description, appear. We show how these structures and their scaling properties depend on the time step of the discrete dynamics. Exploiting the analogies of this discrete model with the logistic map, some general features are discussed.

Noise- and inertia-induced inhomogeneity in the distribution of small particles in fluid flows

The dynamics of small spherical neutrally buoyant particulate impurities immersed in a two-dimensional fluid flow are known to lead to particle accumulation in the regions of the flow in which vorticity dominates over strain, provided that the Stokes number of the particles is sufficiently small. If the flow is viewed as a Hamiltonian dynamical system, it can be seen that the accumulations occur in the nonchaotic parts of the phase space: the Kolmogorov-Arnold-Moser tori. This has suggested a generalization of these dynamics to Hamiltonian maps, dubbed a bailout embedding. In this paper we use a bailout embedding of the standard map to mimic the dynamics of neutrally buoyant impurities subject not only to drag but also to fluctuating forces modeled as white noise. We find that the generation of inhomogeneities associated with the separation of particle from fluid trajectories is enhanced by the presence of noise, so that they appear in much broader ranges of the Stokes number than those allowing spontaneous separation. (c) 2002 American Institute of Physics.

Small-scale structure of nonlinearly interacting species advected by chaotic flows

We study the spatial patterns formed by interacting biological populations or reacting chemicals under the influence of chaotic flows. Multiple species and nonlinear interactions are explicitly considered, as well as cases of smooth and nonsmooth forcing sources. The small-scale structure can be obtained in terms of characteristic Lyapunov exponents of the flow and of the chemical dynamics. Different kinds of morphological transitions are identified. Numerical results from a three-component plankton dynamics model support the theory, and they serve also to illustrate the influence of asymmetric couplings. (c) 2002 American Institute of Physics.

Metabolic network dynamics in open chaotic flow

We have analyzed the dynamics of metabolically coupled replicators in open chaotic flows. Replicators contribute to a common metabolism producing energy-rich monomers necessary for replication. The flow and the biological processes take place on a rectangular grid. There can be at most one molecule on each grid cell, and replication can occur only at localities where all the necessary replicators (metabolic enzymes) are present within a certain neighborhood distance. Due to this finite metabolic neighborhood size and imperfect mixing along the fractal filaments produced by the flow, replicators can coexist in this fluid system, even though coexistence is impossible in the mean-field approximation of the model. We have shown numerically that coexistence mainly depends on the metabolic neighborhood size, the kinetic parameters, and the number of replicators coupled through metabolism. Selfish parasite replicators cannot destroy the system of coexisting metabolic replicators, but they frequently remain persistent in the system. (c) 2002 American Institute of Physics.

Finite-size effects on active chaotic advection

A small (but finite-size) spherical particle advected by fluid flows obeys equations of motion that are inherently dissipative, due to the Stokes drag. The dynamics of the advected particle can be chaotic even with a flow field that is simply time periodic. Similar to the case of ideal tracers, whose dynamics is Hamiltonian, chemical or biological activity involving such particles can be analyzed using the theory of chaotic dynamics. Using the example of an autocatalytic reaction, A+Bright arrow2B, we show that the balance between dissipation in the particle dynamics and production due to reaction leads to a steady state distribution of the reagent. We also show that, in the case of coalescence reaction, B+Bright arrowB, the decay of the particle density obeys a universal scaling law as approximately t(minus sign1) and that the particle distribution becomes restricted to a subset with fractal dimension D2, where D2 is the correlation dimension of the chaotic attractor in the particle dynamics.

Geometry of reaction interfaces in chaotic flows

The analysis of transport-controlled reactions in chaotic flows provides a physical frame to extend the concept of the intermaterial contact area (ICA)--introduced in the purely kinematic case--to mixing systems with diffusion, where the ICA is identified through the reaction interface between segregated reactants. We show that the dynamics of the ICA undergoes a crossover from kinematics-dominated exponential growth to a persistent oscillatory regime resulting from the intertwined action of advection and diffusion. The scaling of the crossover length versus the Peclet number is analyzed.

Analytical and numerical studies of noise-induced synchronization of chaotic systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of one-dimensional maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. (c) 2001 American Institute of Physics.