Ecological chaos in the wake of invasion.

Irregularities in observed population densities have traditionally been attributed to discretization of the underlying dynamics. We propose an alternative explanation by demonstrating the evolution of spatiotemporal chaos in reaction-diffusion models for predator-prey interactions. The chaos is generated naturally in the wake of invasive waves of predators. We discuss in detail the mechanism by which the chaos is generated. By considering a mathematical caricature of the predator-prey models, we go on to explain the dynamical origin of the irregular behavior and to justify our assertion that the behavior we present is a genuine example of spatiotemporal chaos.

Order and disorder in fluid motion.

The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characterized statistically, and methods for doing so are evolving. Transport and mixing phenomena can also lead to spatial complexity, but the degree depends on the significance of molecular or thermal diffusion.

Spatiotemporal dynamics of guanosine 3′,5′-cyclic monophosphate revealed by a genetically encoded, fluorescent indicator

To investigate the dynamics of guanosine 3′,5′-cyclic monophosphate (cGMP) in single living cells, we constructed genetically encoded, fluorescent cGMP indicators by bracketing cGMP-dependent protein kinase (cGPK), minus residues 1–77, between cyan and yellow mutants of green fluorescent protein. cGMP decreased fluorescence resonance energy transfer (FRET) and increased the ratio of cyan to yellow emissions by up to 1.5-fold with apparent dissociation constants of ≈2 μM and >100:1 selectivity for cGMP over cAMP. To eliminate constitutive kinase activity, Thr516 of cGPK was mutated to Ala. Emission ratio imaging of the indicators transfected into rat fetal lung fibroblast (RFL)-6 showed cGMP transients resulting from activation of soluble and particulate guanylyl cyclase, respectively, by nitric oxide (NO) and C-type natriuretic peptide (CNP). Whereas all naive cells tested responded to CNP, only 68% responded to NO. Both sets of signals showed large and variable (0.5–4 min) latencies. The phosphodiesterase (PDE) inhibitor 3-isobutyl-1-methylxanthine (IBMX) did not elevate cGMP on its own but consistently amplified responses to NO or CNP, suggesting that basal activity of guanylate cyclase is very low and emphasizing the importance of PDEs in cGMP recycling. A fraction of RFL cells showed slowly propagating tides of cGMP spreading across the cell in response to delocalized application of NO. Biolistically transfected Purkinje neurons showed cGMP responses to parallel fiber activity and NO donors, confirming that single-cell increases in cGMP occur under conditions appropriate to cause synaptic plasticity.

Chaos, but in voting and apportionments?

Mathematical chaos and related concepts are used to explain and resolve issues ranging from voting paradoxes to the apportioning of congressional seats.

Titration of chaos with added noise

Deterministic chaos has been implicated in numerous natural and man-made complex phenomena ranging from quantum to astronomical scales and in disciplines as diverse as meteorology, physiology, ecology, and economics. However, the lack of a definitive test of chaos vs. random noise in experimental time series has led to considerable controversy in many fields. Here we propose a numerical titration procedure as a simple “litmus test” for highly sensitive, specific, and robust detection of chaos in short noisy data without the need for intensive surrogate data testing. We show that the controlled addition of white or colored noise to a signal with a preexisting noise floor results in a titration index that: (i) faithfully tracks the onset of deterministic chaos in all standard bifurcation routes to chaos; and (ii) gives a relative measure of chaos intensity. Such reliable detection and quantification of chaos under severe conditions of relatively low signal-to-noise ratio is of great interest, as it may open potential practical ways of identifying, forecasting, and controlling complex behaviors in a wide variety of physical, biomedical, and socioeconomic systems.