Self-organized criticality and pattern emergence through the lens of tropical geometry

Abstract

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.

Human reproductive system microbiomes exhibited significantly different heterogeneity scaling with gut microbiome, but the intra-system scaling is invariant

“…The more recent hologenome and holobiont theory should also predict the same notions. Taylor (1961, 1984, 1986) Taylor & Taylor (1977), Taylor et al (1983, 1988) had long been arguing that the aggregation parameter ( b ) of Taylor’s power law is a species-specific characteristic determined by a species’ evolutionary history, and recent theoretical and experimental studies have validated their early conjectures (Eisler et al 2008; Cohen et al 2012, 2015; Stumpf & Porter 2012; Zhang et al 2014; Giometto et al 2015; Oh et al 2016; Tippett & Cohen (2016); Plank & Pitchford 2017; Reuman et al 2017; Kalinin et al 2018). The recent extensions of Taylor’s power law from population to community level should have preserved this important characteristic of parameter ( b ) (Ma 2012, 2015, Zhang et al 2014, Oh et al 2016, Li & Ma 2019).…”

“…Taylor’s power law (Taylor 1961, 1984, 2007, Taylor & Taylor 1977, Taylor et al 1983, 1988) is one of the classic mathematical models that have reached the rare status of the ecological law. It has been validated by hundreds, if not thousands of field observations in macro-ecology of plants and animals, and its theoretical implications and practical applications have extended well beyond ecology and biology, reaching fields such as epidemiology, natural catastrophe prediction, human migration, financing, and computational science (Taylor 1961, 1984, 2007, Taylor & Taylor 1977, Taylor et al 1983, 1988, Cohen et al 2012, 2015; Eisler et al 2008, Stumpf & Porter 2012; Giometto et al 2015; Ma 2012, 2013, 2015, Oh et al 2016, Tippett & Cohen (2016), Plank & Pitchford 2017, Reuman et al 2017, Kalinin et al 2018). In its original form, Taylor’s power law describes the relationship between population variance ( V ) and population mean (abundance) ( m ) in the following power function: where parameter a is primarily influenced by the sampling scheme and environmental factors and is of limited ecological implications, and parameter b is of rich ecological and evolutionary implications.…”

“…At the community level, we use heterogeneity to refer to the uneven or heterogeneous nature of species abundances among different species within a community and/or between communities, which can be quantitatively measured with an extension to the classic Taylor’s power law (Ma 2015, Li & Ma 2019). Taylor’s power law has been extensively investigated both theoretically and practically and found applications in many fields beyond its original domain of population ecology (Taylor 1961, 1984, 2007, Taylor & Taylor 1977, Taylor et al 1983, 1988, Cohen et al 2012, 2015; Eisler et al 2008, Stumpf & Porter 2012; Giometto et al 2015; Ma 2015, Oh et al 2016; Tippett & Cohen (2016); Plank & Pitchford 2017; Reuman et al 2017; Kalinin et al 2018).…”

Human reproductive system microbiomes exhibited significantly different heterogeneity scaling with gut microbiome, but the intra‐system scaling is invariant

“…Taylor's power law (Taylor , , , Taylor and Taylor , Taylor et al , ) is one of the few mathematical models to have reached the rare status of ecological law. It has been validated by numerous field observations in macro‐ecology of plants and animals, and its theoretical implications and practical applications have extended well beyond ecology and biology, reaching fields such as epidemiology, natural catastrophe prediction, human migration, financing and computational science (Taylor , , , Taylor and Taylor , Taylor et al , , Eisler et al , Cohen et al , Cohen and Xu , Ma , b, , , Stumpf and Porter , Giometto et al , Oh et al , Tippett and Cohen , Plank and Pitchford , Reuman et al , Kalinin et al ). In its original form, Taylor's power law describes the relationship between population variance ( V ) and population mean (abundance) ( m ) in the following power function: …”

“…At the community level, we use heterogeneity to characterize the uneven or heterogeneous nature of species abundances among different species within a community and/or across communities, which can be quantitatively measured with an extension to the classic Taylor's power law (Ma , Li and Ma ). Taylor's power law has been extensively studied both theoretically and practically and has found applications in many fields beyond its original domain of population ecology (Taylor , , , Taylor and Taylor , Taylor et al , , Eisler et al , Cohen et al , Stumpf and Porter , Cohen and Xu , Giometto et al , Ma , Oh et al , Tippett and Cohen , Plank and Pitchford , Reuman et al , Kalinin et al , Li and Ma ).…”