Evolutionary games on graphs

A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

A family of mimetic finite difference methods on polygonal and polyhedral meshes

Control of waves, patterns and turbulence in chemical systems

Control of spatiotemporal chaos is one of the central problems of nonlinear dynamics. We report on suppression of chemical turbulence by global delayed feedback using, as an example, catalytic carbon monoxide oxidation on a platinum (110) single-crystal surface and carbon monoxide partial pressure as the controlled feedback variable. When feedback intensity was increased, spiral-wave turbulence was transformed into new intermittent chaotic regimes with cascades of reproducing and annihilating local structures on the background of uniform oscillations. The global feedback further led to the development of cluster patterns and standing waves and to the stabilization of uniform oscillations. These findings are reproduced by theoretical simulations.

https://science.sciencemag.org/content/sci/292/5520/1357.full.pdf

Controlling Chemical Turbulence by Global Delayed Feedback: Pattern Formation in Catalytic CO Oxidation on Pt(110)

Stochasticity pervades life at the cellular level. Cells receive stochastic signals, perform detection and transduction with stochastic biochemistry, and grow and die in stochastic environments. Here we review progress in going from the molecular details to the information-processing strategies cells use in their decision-making. Such strategies are fundamentally influenced by stochasticity. We argue that the cellular decision-making can only be probabilistic and occurs at three levels. First, cells must infer from noisy signals the probable current and anticipated future state of their environment. Second, they must weigh the costs and benefits of each potential response, given that future. Third, cells must decide in the presence of other, potentially competitive, decision-makers. In this context, we discuss cooperative responses where some individuals can appear to sacrifice for the common good. We believe that decision-making strategies will be conserved, with comparatively few strategies being implemented by different biochemical mechanisms in many organisms. Determining the strategy of a decision-making network provides a potentially powerful coarse-graining that links systems and evolutionary biology to understand biological design.

https://www.embopress.org/doi/epdf/10.1038/msb.2009.83

Strategies for cellular decision‐making

Resonance regions similar to the Arnol'd tongues found in single oscillator frequency locking are observed in experiments using a spatially extended periodically forced Belousov-Zhabotinsky system. We identify six distinct 2:1 subharmonic resonant patterns and describe them in terms of the position-dependent phase and magnitude of the oscillations. Some experimentally observed features are also found in numerical studies of a forced Brusselator reaction-diffusion model.

/pdf/resonant-phase-patterns-in-a-reaction-diffusion-system-4nd0gggy6q.pdf

Resonant Phase Patterns in a Reaction-Diffusion System

http://chaos.utexas.edu/manuscripts/1093898977.pdf

Localization and extinction of bacterial populations under inhomogeneous growth conditions.

We investigate pattern formation in self-oscillating systems forced by an external periodic perturbation. Experimental observations and numerical studies of reaction-diffusion systems and an analysis of an amplitude equation are presented. The oscillations in each of these systems entrain to rational multiples of the perturbation frequency for certain values of the forcing frequency and amplitude. We focus on the subharmonic resonant case where the system locks at one-fourth the driving frequency, and four-phase rotating spiral patterns are observed at low forcing amplitudes. The spiral patterns are studied using an amplitude equation for periodically forced oscillating systems. The analysis predicts a bifurcation (with increasing forcing) from rotating four-phase spirals to standing two-phase patterns. This bifurcation is also found in periodically forced reaction-diffusion equations, the FitzHugh-Nagumo and Brusselator models, even far from the onset of oscillations where the amplitude equation analysis is not strictly valid. In a Belousov-Zhabotinsky chemical system periodically forced with light we also observe four-phase rotating spiral wave patterns. However, we have not observed the transition to standing two-phase patterns, possibly because with increasing light intensity the reaction kinetics become excitable rather than oscillatory.

/pdf/four-phase-patterns-in-forced-oscillatory-systems-2ay2xnbke4.pdf

Four-phase patterns in forced oscillatory systems

Various resonant and near-resonant patterns form in a light-sensitive Belousov-Zhabotinsky (BZ) reaction in response to a spatially homogeneous time-periodic perturbation with light. The regions (tongues) in the forcing frequency and forcing amplitude parameter plane where resonant patterns form are identified through analysis of the temporal response of the patterns. Resonant and near-resonant responses are distinguished. The unforced BZ reaction shows both spatially uniform oscillations and rotating spiral waves, while the forced system shows patterns such as standing-wave labyrinths and rotating spiral waves. The patterns depend on the amplitude and frequency of the perturbation, and also on whether the system responds to the forcing near the uniform oscillation frequency or the spiral wave frequency. Numerical simulations of a forced FitzHugh-Nagumo reaction-diffusion model show both resonant and near-resonant patterns similar to the BZ chemical system.

/pdf/resonance-tongues-and-patterns-in-periodically-forced-4bt8v1ilg2.pdf

Resonance tongues and patterns in periodically forced reaction-diffusion systems

Experiments on a quasi-two-dimensional Belousov-Zhabotinsky (BZ) reaction-diffusion system, pe- riodically forced at approximately twice its natural frequency, exhibit resonant labyrinthine patterns that develop through two distinct mechanisms. In both cases, large amplitude labyrinthine patterns f orm that consist ofinterpenetrating fingers off requency-locked regions differing in phase by π. Analysis of a forced complex Ginzburg-Landau equation captures both mechanisms observed for the f ormation ofthe labyrinths in the BZ experiments: a transverse instability off ront structures and a nucleation of stripes from unlocked oscillations. The labyrinths are found in the experiments and in the model at a similar location in the forcing amplitude and frequency parameter plane.

/pdf/development-of-standing-wave-labyrinthine-patterns-1w5v5bs4cg.pdf

Anna L. Lin

Papers

Resonant Phase Patterns in a Reaction-Diffusion System

Localization and extinction of bacterial populations under inhomogeneous growth conditions.

Four-phase patterns in forced oscillatory systems

Resonance tongues and patterns in periodically forced reaction-diffusion systems

Development of Standing-Wave Labyrinthine Patterns ∗