Geometry of ‘standoffs’ in lattice models of the spatial Prisoner’s Dilemma and Snowdrift games

The Prisoner’s Dilemma and Snowdrift games are the main theoretical constructs used to study the evolutionary dynamics of cooperation. In large, well-mixed populations, mean-field models predict a stable equilibrium abundance of all defectors in the Prisoner’s Dilemma and a stable mixed-equilibrium of cooperators and defectors in the Snowdrift game. In the spatial extensions of these games, which can greatly modify the fates of populations (including allowing cooperators to persist in the Prisoner’s Dilemma, for example), lattice models are typically used to represent space, individuals play only with their nearest neighbours, and strategy replacement is a function of the differences in payoffs between neighbours. Interestingly, certain values of the cost–benefit ratio of cooperation, coupled with particular spatial configurations of cooperators and defectors, can lead to ‘global standoffs’, a situation in which all cooperator–defector neighbours have identical payoffs, leading to the development of static spatial patterns. We start by investigating the conditions that can lead to ‘local standoffs’ (i.e., in which isolated pairs of neighbouring cooperators and defectors cannot overtake one another), and then use exhaustive searches of small square lattices (4×4$4\times 4$ and 6×6$6\times 6$) of degree k=3,k=4$k=3,k=4$, and k=6$k=6$, to show that two main types of global standoff patterns–‘periodic’ and ‘aperiodic’–are possible by tiling local standoffs across entire spatially structured populations. Of these two types, we argue that only aperiodic global standoffs are likely to be potentially attracting, i.e., capable of emerging spontaneously from non-standoff conditions. Finally, we use stochastic simulation models with comparatively large lattices (100×100$100\times 100$) to show that global standoffs in the Prisoner’s Dilemma and Snowdrift games do indeed only (but not always) emerge under the conditions predicted by the small-lattice analysis.     

Empirical equations of state for solids

A comparison of three one-parameter empirical equations of state due to Lennard-Jones, Bardeen, and Birch using Bridgman's experimental compression data shows that they do not generally fit the data within experimental error. Heuristic arguments are used to derive a well known three-parameter equation of state which includes these equations as special cases. Several simple two-parameter equations of state, including the Murnaghan equation, can be obtained as special cases of the general form. All of these two-parameter equations are shown to fit Bridgman's data within experimental error with about equal success. It is also shown that the general three-parameter equation, and hence its special cases, is consistent with the observation of Grover, Getting, and Kennedy that the log of the bulk modulus of solids is a linear function of compression. Several of the two-parameter special cases are shown to give satisfactory extrapolations.     

The polarographic determination of isopropenyl esters in aqueous solution

Both mercuration and bromination have been used for the polarographic analysis of isopropenyl esters in aqueous solutions. For mercuration, no disadvantages were found in comparison with the established methods [4,5] for the estimation of vinyl esters in methanol, due to either the lower rate of mercuration [5,22] or the expected lower stability of the depolariser formed in aqueous solution, the hemiacetal mercuric acetate addition product. At concentrations below 10^?2 mol l^?1, acetone did not interfere [38] with the determinations.     

2.3 Rocks, minerals, soils

Abstracts2 Particular products and fields of application

2.3 Rocks, minerals, soils     

21Na(p,gamma)22Mg reaction and oxygen-neon novae

The 21Na(p,gamma)22Mg reaction is expected to play an important role in the nucleosynthesis of 22Na in oxygen-neon novae. The decay of 22Na leads to the emission of a characteristic 1.275 MeV gamma-ray line. This report provides the first direct measurement of the rate of this reaction using a radioactive 21Na beam, and discusses its astrophysical implications. The energy of the important state was measured to be E(c.m.)=205.7+/-0.5 keV with a resonance strength omegagamma=1.03+/-0.16(stat)+/-0.14(sys) meV.     

Observation of the 11N ground state

The ground state of the proton-rich, unbound nucleus 11N was observed, together with six excited states using the multinucleon transfer reaction 10B(14N,13B)11N at 30A MeV incident energy at Grand Accelerateur National d'Ions Lourds. Levels of 11N are observed as well defined resonances in the spectrum of the 13B ejectiles. They are localized at 1.63(5), 2.16(5), 3.06(8), 3.61(5), 4.33(5), 5.98(10), and 6.54(10) MeV above the 10C+p threshold. The ground-state resonance has a mass excess of 24.618(50) MeV; the experimental width is smaller than theoretical predictions.