Rhythmic natural selection over intertidal and brackish water genotypes: simple formulations for testing hypothesis.

Tidal cycles at benthic habitats induce a set of periodic environmental changes in variables like salinity, temperature and sediment water content which are able to stress benthic organisms. Consequently, a natural selection temporally correlated with tides affects the fitness of genotypes (wi) depending on their adaptation degree. Classic population genetics demonstrate that (1) rhythmic wi is more restrictive than equivalent spatial variations to preserve genetic variance, and (2) mean fitness of the population (w¯) does not have to be enhanced by genetic variance (σ(2)w). The present study develops a simple replicator dynamics-based model of continuous selection, where wi of multiple asexual genotypes fluctuates as a sinusoid. The amplitude of w was set as 0.5 (1-wmin), whereas the ratio of tide period to generation time (h) was defined. Overall, the model shows that if h>1, then the success of an advantageous genotype is exposed to randomness, and w¯ may decrease over generations. In contrast, if h<1 the success is deterministic, is limiting co-dominance, and only depends on wmin. The amount of different genotypes buffers the decay of σ(2)w and hence increases cohesiveness. Finally, the reliability of the model is analyzed for a set of target intertidal and brackish water organisms.

Entropy rate estimates from mutual information.

We show how to estimate the Kolmogorov-Sinai entropy rate for chaotic systems using the mutual information function, easily obtainable from experimental time series. We state the conditions under which the relationship is exact, and explore the usefulness of the approach for both maps and flows. We also explore refinements of the method, and study its convergence properties as a function of time series length.

Geometry of repeated measurements in chaotic systems.

We use joint probability matrices for measurements at different times to describe chaotic systems. By coarse graining the range of the measured variable into uniformly sized bins we can generate matrices that contain both topological and metric information about the systems being studied. Armed with this tool we examine two extreme families of chaotic systems. In the case of one-dimensional piecewise linear maps, we can construct transfer matrices that depend on the map and partition used, and which allow us to generate the respective joint probability matrices for all times as well as the exact time evolution of the mutual information function. We find that the mutual information decays linearly or exponentially depending on whether the second-largest eigenvalue of the transfer matrix is zero or not. In the case of three-dimensional, continuous-time chaotic systems we generate the joint probability matrices directly from numerical data. We show that these matrices directly provide attractor reconstructions with information about the attractor's probability measure.

Determinism test for very short time series.

A test for determinism suitable for time series shorter than 100 points is presented, and applied to numerical and observed data. The method exploits the linear d(d(0)) dependence in the expression d(t) approximately d(0)e(lambda t) which describes the growth of small separations between trajectories in chaotic systems.

Unstable periodic orbits and discretization cycles.

Limit cycles that arise from discretizing the variable(s) of a nonlinear map are generally found to shadow individual unstable periodic orbits (UPOs) of the corresponding continuous map. In a few cases the discretization cycles can only be explained with other mechanisms, such as the near-occurrence of an UPO, or crossover between two or more UPOs.

Absence of determinism in El Niño Southern Oscillation.

We perform two direct determinism tests on the El Niño Southern Oscillation index monthly average series. The results indicate that, for timescales over 1 month, the series does not exhibit determinism, an essential feature of chaos.

Reducing noise in discretized time series.

We show that applying a noise-reduction algorithm to a discretized time series increases its average error, compared to the original series. We find that adding external noise comparable to the discretization step before noise reduction limits the increase of the average error and improves the estimation of Lyapunov exponents.

Multiscale analysis of complex systems.

We calculate block information versus size profiles for two-symbol strings generated by several dynamical processes: random, periodic, regular language, and substitutive. The profiles provide a good diagnostic of the complexity of the strings.

Finite statistical complexity for sofic systems.

We propose a measure of complexity for symbolic sequences, which is based on conditional probabilities, and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata (DFA). Moreover, if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known, we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures (causal states).

Observation of structure in the Lorenz map.

Through massive numerical integration of the Lorenz system, we are able to discern structure in its Poincare map. We are also able to estimate its capacity dimension; our result is consistent with previous measurements of the correlation dimension of the Lorenz attractor. (c) 1999 American Institute of Physics.