Adaptation Noise And Self Organizing Systems

We introduce a model for a sandpile, with N sites, critical height N and each site connected to every other site. It is thus a mean-field model in the spin-glass sense. We find an exact solution for the steady state probability distribution of avalanche sizes, and discuss its asymptotics for large N.

We describe a simple framework for micro simulation of city traffic. A medium sized excerpt of Dallas was used to examine different levels of simulation fidelity of a cellular automaton method for the traffic flow simulation and a simple intersection model. We point out problems arising with the granular structure of the underlying rules of motion.

This paper reports experiences with iterated traffic microsimulations in the context of a Dallas study. ``Iterated microsimulations'' here means that the information generated by a microsimulation is fed back into the route planner so that the simulated individuals can adjust their routes to circumvent congestion. This paper gives an overview over what has been done in the Dallas context to better understand the relaxation process, and how to judge the robustness of the results.

A model for large-scale evolution recently introduced by Amaral and Meyer is studied analytically and numerically. Species are located at different trophic levels and become extinct if their prey becomes extinct. It is proved that this model is self-organized critical in the thermodynamic limit, with an exponent 2 characterizing the size distribution of extinction events. The lifetime distribution of species, cutoffs due to finite-size effects, and other quantities are evaluated. The relevance of this model to biological evolution is critically assessed.

Using data drawn from large-scale databases, a number of interesting trends in the fossil record have been observed in recent years. These include the average decline in extinction rates throughout the Phanerozoic, the average increase in standing diversity, correlations between rates of origination and extinction, and simple laws governing the form of survivorship curves and the distribution of the lifetimes of taxa. In this paper we derive mathematically a number of relations between these quantities and show how these different trends are inter-related. We also derive a variety of constraints on the possible forms of these trends, such as limits on the rate at which extinction may decline and limits on the allowed difference between extinction and origination rates at any given time.

We propose a model for co-evolving ecosystems that takes into account two levels of description of an organism, for instance genotype and phenotype. Performance at the macroscopic level forces mutations at the microscopic level. These, in turn, affect the dynamics of the macroscopic variables. In some regions of parameter space, the system self-organises into a state with localised activity and power law distributions.

We investigate a system of harmonically coupled identical nonlinear constituents subject to noise in different spatial arrangements. For global coupling we find for infinitely many constituents the coexistence of several ergodic components and a bifurcation behaviour like in first order phase transitions. These results are compared with simulations for finite systems both for global coupling and for nearest neighbour coupling on two- and three-dimensional cubic lattices. The mean-field type results for global coupling provide a better understanding of the more complex behaviour in the latter case.

We examine the dynamics of an age-structured population model in which the life expectancy of an offspring may be mutated with respect to that of the parent. While the total population of the system always reaches a steady state, the fitness and age characteristics exhibit counter-intuitive behavior as a function of the mutational bias. By analytical and numerical study of the underlying rate equations, we show that if deleterious mutations are favored, the average fitness of the population reaches a steady state, while the average population age is a decreasing function of the overall fitness. When advantageous mutations are favored, the average population fitness grows linearly with time t, while the average age is independent of fitness. For no mutational bias, the average fitness grows as t^{2/3}.

The problem of flame propagation is studied as an example of unstable fronts that wrinkle on many scales is studied. The analytic tool of pole expansion in the complex plane is emloyed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that explain these results in terms of the noisy pole dynamics.

This is a brief introduction to fractals, multifractals and wavelets in an accessible way, in order that the founding ideas of those strange and intriguing newcomers to science as fractals may be communicated to a wider public. Fractals are the geometry of the wildness of nature, where the euclidian geometry fails. The structures of nonlinear dynamics associated with chaos are fractal. Fractals may also be used as the geometry of social systems. Wavelets are introduced as a tool for fractal analysis. As an example of its application on a social system, we use wavelet fractal analysis to compare electrical power demand of two different places, a touristic city and a whole country.