Coupled map lattice

According to the Wikipedia article it is:

A

coupled map lattice (CML)is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverge as the size of the system increases.

Features of the CML are discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables.[2] Studied systems include populations, chemical reactions, convection, fluid flow and biological networks. Even recently, CMLs have been applied to computational networks identifying detrimental attack methods and cascading failures.

Some open source software is

- Yade. From the Yade page:

Yade is an extensible open-source framework for discrete numerical models, focused on Discrete Element Method. The project started as an offspring from SDEC at Grenoble University, now is being developed at multiple research institutes and has active and helpful user community.

The computation parts are written in c++ using flexible object model, allowing independent implementation of new alogrithms, interfaces with other software packages (e.g. flow simulation), data import/export routines. Python can be used to create and manipulate the simulation or for postprocessing.

- A simulation and Analysis Tool for Dynamical Systems (ANT). From the ANT page:

It is a gerneric simulation and Analysis Tool for dynamical systems. This software package can be applied in several scientific disciplines in both research and education areas. Hence, the target group may be scientists and engineers as well as lecturers, teachers and students.

*Open access*

- Coupled map lattice, Wikipedia.

*Closed*

- Karen Mustin, Tim G. Benton, Calvin Dytham, Justin M. J. Travis Oikos 2009 The dynamics of climate-induced range shifting; perspectives from simulation modelling.

Abstract: Predicted future climate change will alter species’ distributions as they attempt to track the most suitable ‘climate window’. Climate envelope models indicate the direction of likely range changes but do not incorporate population dynamics, therefore observed responses may differ greatly from these projections. We use simulation modelling to explore the consequences of a period of environmental change for a species structured across an environmental gradient. Results indicate that a species’ range may lag behind its climate envelope and demonstrate that the rate of movement of a range can accelerate during a period of climate change. We conclude that the inclusion of both population dynamics and spatial environmental variability is vital to develop models that can both predict, and be used to manage, the impact of changing climate on species’ biogeography.

Abstract : We present a study of ocean convection parameterization based on a novel approach which includes both eddy diffusion and advection and consists of a two-dimensional lattice of bistable maps. This approach retains important features of usual grid models and allows to assess the relative roles of diffusion and advection in the spreading of convective cells. For large diffusion our model exhibits a phase transition from convective patterns to a homogeneous state over the entire lattice. In hysteresis experiments we find staircase behavior depending on stability thresholds of local convection patterns. This nonphysical behavior is suspected to induce spurious abrupt changes in the spreading of convection in ocean models. The final steady state of convective cells depends not only on the magnitude of the advective velocity but also on its direction, implying a possible bias in the development of convective patterns. Such bias points to the need for an appropriate choice of grid geometry in ocean modeling.

- Derin B. Wysham, Alan Hastings Bulletin of Mathematical Biology May 2008, Sudden Shifts in Ecological Systems: Intermittency and Transients in the Coupled Ricker Population Model.

Abstract: Many real ecological systems show sudden changes in behavior, phenomena sometimes categorized as regime shifts in the literature. The relative importance of exogenous versus endogenous forces producing regime shifts is an important question. These forces’ role in generating variability over time in ecological systems has been explored using tools from dynamical systems. We use similar ideas to look at transients in simple ecological models as a way of understanding regime shifts. Based in part on the theory of crises, we carefully analyze a simple two patch spatial model and begin to understand from a mathematical point of view what produces transient behavior in ecological systems. In particular, since the tools are essentially qualitative, we are able to suggest that transient behavior should be ubiquitous in systems with overcompensatory local dynamics, and thus should be typical of many ecological systems.

category: computational methods