Ulam’s spiral reveals patterns in the prime numbers by presenting positive integers in a right-angled whorl. The classic spatial prisoner’s dilemma (PD) reveals pathways to cooperation by presenting ...a model of agents interacting on a grid. This paper brings these tools together via a deterministic spatial PD model that distributes cooperators at the prime-numbered locations of Ulam’s spiral. The model focuses on a narrow boundary game variant of the PD for ease of comparison with early studies of the spatial PD. Despite constituting an initially small portion of the population, cooperators arranged in Ulam’s spiral always grow to dominance when (i) the payoff to free-riding is less than or equal to 8/6 (≈1.33) times the payoff to mutual cooperation and (ii) grid size equals or exceeds 23 × 23. As in any spatial PD model, particular formations of cooperators spur this growth and here these formations draw attention to rare configurations in Ulam’s spiral.
Plate Tectonics and the Archean Earth Brown, Michael; Johnson, Tim; Gardiner, Nicholas J
Annual review of earth and planetary sciences,
05/2020, Letnik:
48, Številka:
1
Journal Article
Recenzirano
Odprti dostop
If we accept that a critical condition for plate tectonics is the creation and maintenance of a global network of narrow boundaries separating multiple plates, then to argue for plate tectonics ...during the Archean requires more than a local record of subduction. A case is made for plate tectonics back to the early Paleoproterozoic, when a cycle of breakup and collision led to formation of the supercontinent Columbia, and bimodal metamorphism is registered globally. Before this, less preserved crust and survivorship bias become greater concerns, and the geological record may yield only a lower limit on the emergence of plate tectonics. Higher mantle temperature in the Archean precluded or limited stable subduction, requiring a transition to plate tectonics from another tectonic mode. This transition is recorded by changes in geochemical proxies and interpreted based on numerical modeling. Improved understanding of the secular evolution of temperature and water in the mantle is a key target for future research.
Higher mantle temperature in the Archean precluded or limited stable subduction, requiring a transition to plate tectonics from another tectonic mode.
Plate tectonics can be demonstrated on Earth since the early Paleoproterozoic (since c. 2.2 Ga), but before the Proterozoic Earth's tectonic mode remains ambiguous.
The Mesoarchean to early Paleoproterozoic (3.2-2.3 Ga) represents a period of transition from an early tectonic mode (stagnant or sluggish lid) to plate tectonics.
The development of a global network of narrow boundaries separating multiple plates could have been kick-started by plume-induced subduction.
•The paper reports a model of a finite population of agents constrained to strategies that alternate between activity and inactivity (a.k.a. temporal partitioning) in a social environment where ...multiple one-shot prisoner's dilemma games occur across discrete, intra-generational time points.•Numerical simulation of the model indicates that cooperators reach fixation with far greater frequency when using schedules with prime-number period lengths.•Simulation results dovetail with the findings of a recent analytic model that confirmed a longstanding, hypothesized link between the prime numbers and the evolution of cooperation.•The findings suggest that schedules with prime-number period lengths constitute a new mechanism for the evolution of cooperation.•Viewed in concert with the findings of past-predator prey models, the results raise the possibility that cyclical behavior with prime-number period lengths might serve as an adaptive solution to a range of challenges that lifeforms face.
This paper presents a model of a finite population of agents constrained to strategies that alternate between activity and inactivity (a.k.a. temporal partitioning) in a social environment where multiple one-shot prisoner's dilemma games occur across discrete, intra-generational time points. Evolutionary selection acts on agents’ behavioral dispositions to cooperate/defect and the schedules that determine when agents periodically implement that behavior. Numerical simulation of the model indicates that cooperators reach fixation with far greater frequency when using schedules with prime-number period lengths. These findings reinforce recent analytic findings that indicate a connection between the evolution of cooperation and the prime numbers, plus they offer new empirical predictions about the timing of social behavior.
•An analytic evolutionary game theory model involving the prisoner's dilemma is shown to identify the prime numbers.•The model confirms a longstanding hypothesis positing a connection between prime ...numbers and the cross-disciplinary puzzle of how cooperation evolved.•The model provides a theoretical foundation for future research into strategies for cooperation and free-riding when social interaction occurs across discrete time points with varying social partners.
The development of methods to identify prime numbers spans centuries and includes models of physical and biological systems that spot primes. This paper adds to the latter research genre by reporting a prisoner's dilemma model that identifies prime numbers greater than 2. Albeit containing unconventional features and arguable assumptions, the model nonetheless confirms a previously hypothesized connection between prime numbers and the cross-disciplinary puzzle of how cooperation evolved. In a companion paper (part II), the features and assumptions of the analytic model reported here are explored in a finite-population, computational model.
Much of the present-day volume of Earth's continental crust had formed by the end of the Archean Eon, 2.5 billion years ago, through the conversion of basaltic (mafic) crust into sodic granite of ...tonalite, trondhjemite and granodiorite (TTG) composition. Distinctive chemical signatures in a small proportion of these rocks, the so-called high-pressure TTG, are interpreted to indicate partial melting of hydrated crust at pressures above 1.5 GPa (>50 km depth), pressures typically not reached in post-Archean continental crust. These interpretations significantly influence views on early crustal evolution and the onset of plate tectonics. Here we show that high-pressure TTG did not form through melting of crust, but through fractionation of melts derived from metasomatically enriched lithospheric mantle. Although the remaining, and dominant, group of Archean TTG did form through melting of hydrated mafic crust, there is no evidence that this occurred at depths significantly greater than the ~40 km average thickness of modern continental crust.
Abstract
The coexistence of divergent (spreading ridge) and convergent (subduction zone) plate boundaries at which lithosphere is respectively generated and destroyed is the hallmark of plate ...tectonics. Here, we document temporally- and spatially-associated Neoarchean (2.55–2.51 Ga) rock assemblages with mid-ocean ridge and supra-subduction-zone origins from the Angou Complex, southern North China Craton. These assemblages record seafloor spreading and contemporaneous subduction initiation and mature arc magmatism, respectively, analogous to modern divergent and convergent plate boundary processes. Our results provide direct evidence for lateral plate motions in the late Neoarchean, and arguably the operation of plate tectonics, albeit with warmer than average Phanerozoic subduction geotherms. Further, we surmise that plate tectonic processes played an important role in shaping Earth’s surficial environments during the Neoarchean and Paleoproterozoic.
Abstract
Unlike many Archean diorites and granitoids that arguably formed in different geodynamic settings, their post-Archean counterparts are commonly regarded to have formed at convergent margins, ...although in detail their petrogenesis remains contentious. Here we present new whole-rock data and zircon Hf–O isotope analyses from dioritic (750–730 Ma), granitic (810–790 Ma) and tonalite–trondhjemite–granodiorite (TTG)-like intrusions (800–740 Ma) from the Panxi and Hannan regions, which form part of an extensive Neoproterozoic convergent margin exposed in South China. The dioritic rocks from the Panxi region exhibit high zircon εHf(t) (+10.1 to +13.1) and sub-mantle to mantle-like δ18O (3.1–6.3 ‰) values, whereas those from the Hannan region preserve low εHf(t) (+4.1 to +8.1) and high δ18O values (5.9–6.6 ‰), indicating that the dioritic melts were derived from subduction-modified lithospheric mantle sources and experienced variable degrees of lower crustal contamination. Zircons within granite and TTG from the Panxi region show a narrow range of Hf isotopic compositions generally spanning 2–4 εHf units (+3.1 to +7.9 for most felsic intrusions). By contrast, those from the Hannan region show a much wider range of zircon εHf(t) values spanning almost 10 εHf units (+1.1 to +10.9). Based on their O–Hf–Nd isotopic signatures, we propose that the granite and TTG from both areas were derived through partial melting of hydrated basaltic rocks in the arc root, and that the isotopic variability between the intrusions mirrors spatial and temporal chemical variations in these deep crustal source rocks. In both regions, the granites, along with mantle-derived mafic–ultramafic and intermediate rocks, show a coupled evolution associated with increasing εNd(t) and εHf(t) and decreasing δ18O with decreasing ages, whereas the TTGs formed during late-stage arc magmatism and preserve relatively homogeneous Nd–Hf isotopes and mantle-like δ18O values. Combined, these data record continuous crustal thickening through underplating of juvenile magmas and a progressive increase in the depth of melting, along with a decrease in the degree of interaction between the melts and basement rocks within the arc root. Our results suggest that slab melting was not required to produce post-Archean TTG signatures. Further, we suggest that the variability in the Hf–O–Nd isotopic compositions of metaluminous (I-type) granites mostly does not reflect a heterogeneity in upper mantle signatures, and that there is no conclusive evidence for the involvement of partial melts of subducted sediment based on Hf–O isotope signatures in zircon.
Abstract
The bulk rock composition of granitoids reflects the composition of their source and the conditions of partial melting, which are functions of the geodynamic setting in which they formed. ...Granitoids in active continental margins (continental arcs) are dominated by calc-alkaline rocks with subordinate alkaline compositions, although how these different magma compositions formed is not well understood. Neoproterozoic magmatic rocks are widely distributed along the western margin of the Yangtze Block in South China to form the >1000-km long Panxi continental arc system, which is dominated by granitoids with minor mafic–ultramafic and intermediate plutons. The granitoids are subdivided into sodic and potassic variants that occur as belts along the western and eastern sides of the continental arc, respectively. Sodic granitoids from the western part consist of tonalite, granodiorite, and monzogranite with crystallisation ages ranging from 870 Ma to 740 Ma. They have low K2O/Na2O ratios (0.1–1.0) and high Na2O contents (3.5–6.7 wt%), high but variable SiO2 (61–75 wt%) concentrations, and negative to positive whole-rock εNd(t) values (−1.7 to +2.9). Zircon grains from the sodic granitoids have εHf(t) values ranging from +0.3 to +9.6 and δ18O from 3.90‰ to 7.71‰. The potassic granitoids from the eastern side consist of monzogranite and syenogranite with crystallisation ages from 820 Ma to 790 Ma. They have high K2O/Na2O ratios (0.6–2.2), K2O (2.6–5.9 wt%) and SiO2 contents (69–78 wt%), but whole-rock εNd(t) (−0.9 to +2.9) and zircon εHf(t) (+1.8 to +12.9), and δ18O values (2.98‰ to 6.41‰) similar to those of the sodic granitoids. The isotopic compositions of both the sodic and potassic granitoids are similar to those of spatially- and temporally-related mantle-derived (mafic to ultramafic) rocks, and are considered to have been derived from juvenile mafic continental crust. Phase equilibrium modelling shows that the H2O content of the granitoid source rocks played a key role in their petrogenesis, both in lowering solidus temperatures and in controlling the compositions of the derived partial melts. Our results indicate that calc-alkaline sodic granitoids can be formed by water-fluxed melting of juvenile mafic crust at 750–900°C and 9–12 kbar in which the required H2O was derived from the dewatering of underplating mafic arc magmas. By contrast, the potassic granitoids were generated by fluid-absent (H2O-undersaturated) partial melting of a similar juvenile mafic source at 725–900°C and 6–9 kbar. We conclude that the sodic granitoids were derived from partial melting of the newly-formed mafic lower crust in the continental arc, whereas the potassic granitoids were likely generated in the back-arc setting induced by upwelling of asthenospheric mantle.
PDF
