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Fig. 8. World Megaurbanization Dynamics (% of the world population living in cities with > 250 thousand inhabitants), 10000 BCE – 1960 CE: the fit be-tween predictions of the hyperbolic model and empirical estimates Note: R = 0.987, R2 = 0.974, p << 0.0001. The black dots correspond to Chandler's (1987) estimates, UN Population Division (2014), Modelski (2003), and Gruebler (2006). The grey solid line has been generated by the following equation:
The best-fit values of parameters С (403.012) and t0 (1990) have been calculated with the least squares method. For comparison, the best fit (R2) obtained here for the exponential model is 0.492. 202 Mathematical Modeling of Big History Phases 
Fig. 9. Dynamics of World Urban Population Living in Cities with more than 250,000 Inhabitants (mlns), 10000 BCE – 1960 CE: the fit between predic-tions of the quadratic-hyperbolic model and the observed data Note: R = 0.998, R2 = 0.996, p << 0.0001. The black markers correspond to estimates of Chandler (1987) and UN Population Division (2014). The grey solid line has been generated by the following equation:
The best-fit values of parameters С (912057.9) and t0 (2008) have been calculated with the least squares method. For comparison, the best fit (R2) obtained here for the exponential model is 0.637. Within this context it is hardly surprising that the general macro dy-namics of the size of the largest settlement within the World System is also quadratic-hyperbolic (see Fig. 10).
Fig. 10. Dynamics of Size of the Largest Settlement of the World (thousands of inhabitants), 10000 BCE – 1950 CE: the fit between predictions of the quadratic-hyperbolic model and the observed data Note: R = 0.992, R2 = 0.984, p << 0.0001. The black markers correspond to estimates of Modelski (2003) and Chandler (1987). The grey solid line has been generated by the following equation:
The best-fit values of parameters С (104020618.5) and t0 (2040) have been calculated with the least squares method. For comparison, the best fit (R2) obtained here for the exponen-tial model is 0.747. As has been demonstrated by cross-cultural anthropologists (see, e.g., Naroll and Divale 1976; Levinson and Malone 1980: 34), for pre-agrarian, agrarian, and early industrial cultures the size of the largest settlement is a rather effective indicator of the general sociocultural complexity of a social system. This, of course, suggests that in the ‘Mal-thusian-Kuznetsian’ era the World System's general sociocultural com-plexity also increased, in a generally quadratic-hyperbolic way. As we have noted in the beginning, the dynamics of marine biodi-versity is strikingly similar to the population dynamics in China, the country with the best-known demographic history. The similarity probably stems from the fact that both curves are produced by the interference of the same three components (general 204 Mathematical Modeling of Big History Phases hyperbolic trend, as well as cyclical and stochastic dynamics). In fact, there is a lot of evidence that some aspects of biodiversity dynamics are stochastic (Raup et al. 1973; Sepkoski 1994; Markov 2001a; Markov 2001b; Cornette and Lieberman 2004), while others are periodic (Raup and Sepkoski 1984; Rohde and Müller 2005). On cyclical and stochastic components of the long-term population dynamics of China (as well as other complex agrarian societies) see, e.g., Korotayev and Khaltourina 2006; Korotayev, Malkov, and Khaltourina 2006b; Chu and Lee 1994; Nefedov 2004; Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Turchin and Nefedov 2009; Usher 1989; Komlos and Nefedov 2002; Grinin, Korotayev and Malkov 2008; Grinin et al. 2009; Grinin 2007c; Korotayev 2006; Korotayev, Khaltourina, and Bozhevolnov 2010; Koro-tayev et al. 2010; van Kessel-Hagesteijn 2009; Abel 1980; Braudel 1973; Goldstone 1991; Grinin, Korotayev 2012 etc.). In fact, similarly to what we have observed with respect to the world population dynamics, even before the start of its intensive mod-ernization, the population dynamics of China was characterized by a pronounced hyperbolic trend – as we can see below (see Figs 11 and 12), the hyperbolic model describes traditional Chinese population dynam-ics much more accurately than either linear or exponential models do: 
Fig. 11. Population Dynamics of China (million people), 57–1851 CE: fit with linear and exponential models Note: based on calculations in Korotayev, Malkov, and Khaltourina 2006b: 47–88.
Fig. 12. Population Dynamics of China (million people), 57–1851 CE: fit with a hyperbolic model The hyperbolic model turns out to describe mathematically the popula-tion dynamics of China in an especially accurate way with respect to the modern period (see Fig. 13). 
Fig. 13. Population Dynamics of China (million people), 57–2003 CE: fit with a hyperbolic model Note: based on calculations in Korotayev, Malkov, and Khaltourina 2006b: 47–88. 206 Mathematical Modeling of Big History Phases In a rather similar way the hyperbolic model turns out to describe the marine biodiversity (measured by number of genera) through the Phan-erozoic much more accurately than the exponential one (see Fig. 14): 
Fig. 14. Global Change in Marine Biodiversity (Number of Genera, N) through Phanerozoic Note: based on empirical data surveyed in Markov and Korotayev (2007). When measured in terms of species number the fit between the empiri-cally observed marine biodiversity dynamics and the hyperbolic model becomes even better (see Fig. 15): 
Fig. 15. Global Change in Marine Biodiversity (Number of Species, N) through Phanerozoic Note: based on empirical data surveyed in Markov and Korotayev 2007b.
The hyperbolic model describes the continental biodiversity in an espe-cially accurate way (see Fig. 16). 
Fig. 16. Global Change in Continental Biodiversity (Number of Genera, N) through Phanerozoic Note: based on empirical data surveyed in Markov and Korotayev 2007b. However, the highest fit between the hyperbolic model and the empirical data is observed when the hyperbolic model is used to describe the dy-namics of total (marine and continental) global biodiversity (see Fig. 17). 
Fig. 17. Global Change in Total (Marine + Continental) Biodiversity (Number of Genera, N) through Phanerozoic Note: based on empirical data surveyed in Markov and Korotayev 2007b. 208 Mathematical Modeling of Big History Phases As we see, the hyperbolic dynamics is most prominent when both ma-rine and continental biotas are considered together. This fact can be in-terpreted as a proof of the integrated nature of the biosphere. But why throughout the Phanerozoic did the global biodiversity tend to follow the hyperbolic trend (similarly to what we observed within social World System in general and China in particular)? As we have noted above, in macrosociological models, the hyper-bolic pattern of the world population growth arises from a non-linear second-order positive feedback (more or less identical with the mecha-nism of collective learning) between the demographic growth and tech-nological development (more people – more potential inventors – faster technological growth – the carrying capacity of the Earth grows faster – faster population growth – more people – more potential inventors, and so on). Based on the analogy with macrosociological models and diverse paleontological data, we suggest that the hyperbolic character of biodi-versity growth can be similarly accounted for by a non-linear second-order positive feedback10 between the diversity growth and community structure complexity (more genera – higher alpha diversity – the com-munities become more stable and ‘buffered’– average life span of genera grows; extinction rate decreases – faster diversity growth – more gen-era – higher alpha diversity, and so on). The growth of genus richness through the Phanerozoic was mainly due to the increase of average longevity of genera and gradual accumu-lation of long-lived (stable) genera in the biota. This pattern reveals it-self in the decrease of extinction rate. Interestingly, in both biota and humanity, growth was facilitated by the decrease in mortality rather than by the increase in birth rate. The longevity of newly arising genera was growing in a stepwise manner. The most short-lived genera ap-peared during the Cambrian; more long-lived genera appeared in Or-dovician to Permian; the next two stages correspond to the Mesozoic and Cenozoic (Markov 2001a, 2002).We suggest that diversity growth can facilitate the increase in genus longevity via the progressive step-wise changes in the structure of communities. Most authors agree that there were three major biotic changes that resulted in fundamental reorganization of community structure during the Phanerozoic: Ordovician radiation, end-Permian extinction, and end-Cretaceous extinction (Bambach 1977; Sepkoski et al. 1981; Sepkoski 1988, 1992; Markov 2001a; Bambach et al. 2002). Generally, after each
major crisis the communities became more complex, diverse and stable. The stepwise increase of alpha diversity (average number of species or genera in a community) through the Phanerozoic was demonstrated by Bambach (1977) and Sepkoski (1988). Although Powell and Kowalewski (2002) argued that the observed increase in alpha diversity might be an artifact caused by several specific biases that influenced the taxonomic richness of different parts of the fossil record, there is evidence that these biases largely compensated each other, so that the observed in-crease in alpha diversity was probably underestimated rather than overestimated (Bush and Bambach 2004). Another important symptom of progressive development of com-munities is the increase in evenness of distribution of species (or genus) abundances. In the primitive, pioneer or suppressed communities, this distribution is strongly uneven (community is overwhelmingly domi-nated by a few very abundant species). In more advanced, climax or flourishing communities, this distribution is more even (Magurran 1988). The former type of community is generally more vulnerable. Evenness of distribution of species richness in communities increased substantially during the Phanerozoic (Powell and Kowalewski 2002; Bush and Bambach 2004). Most probably there was also an increase in habitat utilization, total biomass and rate of trophic flow in biota through the Phanerozoic (Powell and Kowalewski 2002). The more complex the community, the more stable it is due to the development of effective interspecies interactions and homeostatic mechanisms based on the negative feedback principle. In a complex community, when the abundance of a species decreases, many factors arise that facilitate its recovery (e.g., there will be more food and fewer predators). Even if a species becomes extinct, its vacant niche may ‘re-cruit’ another species, most probably a related one that may acquire morphological similarity with its predecessor and thus, the taxonomists will assign it to the same genus. So a complex community can facilitate the stability (and longevity) of its components, such as niches, taxa and morphotypes. This effect reveals itself in the phenomenon of ‘coordi-nated stasis’: the fossil record shows many examples of persistence of particular communities for many million years while the rates of extinc-tion and taxonomic turnover are minimized (Brett et al. 1996, 2007). Selective extinction leads to accumulation of ‘extinction-tolerant’ taxa in the biota (Sepkoski 1991b). Although there is evidence that mass extinctions can be non-selective in some aspects (Jablonski 2005), they are obviously highly selective with respect to the ability of taxa to en-dure unpredictable environmental changes. This can be seen, for in- 210 Mathematical Modeling of Big History Phases stance, from the selectivity of the end-Cretaceous mass extinction with respect to the time of the first occurrence of genera. In younger cohorts the extinction level was higher compared to the older cohorts (see Markov and Korotayev 2007a: Fig. 2) . The same pattern can be observed during the periods of ‘background’ extinction as well (Markov 2000). This means that genera differ in their ability to survive the extinction events, and that in the course of time the extinction-tolerant genera ac-cumulate in each cohort. Thus, taxa generally become more stable and long-lived in the course of evolution, apart from the effects of communi-ties. The communities composed of more stable taxa would be, in turn, more stable themselves, thus creating a positive feedback. The stepwise change of dominant taxa plays a major role in biotic evolution. This pattern is maintained not only by the selectivity of ex-tinction (discussed above), but also by the selectivity of the recovery after crises (Bambach et al. 2002). The taxonomic structure of the Phanero-zoic biota was changing in a stepwise way, as demonstrated by the concept of three sequential ‘evolutionary faunas’ (Sepkoski 1992). There were also stepwise changes in the proportion of major groups of animals with differ-ent ecological and physiological parameters. There was a stepwise growth in proportion of motile genera compared to non-motile; ‘physiologically buffered’ genera compared to ‘unbuffered’, and predators compared to prey (Bambach et al. 2002). All these trends should have facilitated the stability of communities (e.g., diversification of predators implies that they become more specialized; a specialized predator regulates its prey's abundance more effectively than a non-specialized predator). There is also another possible mechanism of the second-order posi-tive feedback between the diversity and its growth rate. Recent research has demonstrated a shift in typical relative-abundance distributions in paleocommunities after the Paleozoic (Wagner et al. 2006). One possible interpretation of this shift is that the community structure and the inter-actions between species in the communities became more complex. In the post-Paleozoic communities, new species probably increase eco-logical space more efficiently, either by facilitating opportunities for ad-ditional species or by niche construction (Wagner et al. 2006; Solé et al. 2002; Laland et al. 1999). This possibility makes the mechanisms under-lying the hyperbolic growth of biodiversity and human population even more similar, because the total ecological space of the biota is analogous to the ‘carrying capacity of the Earth’ in demography. As far as new species can increase ecological space and facilitate opportunities for ad-ditional species entering the community, they are analogous to the ‘in-
ventors’ of the demographic models whose inventions increase the car-rying capacity of the Earth. Exponential and logistic models of biodiversity imply several possi-ble ways in which the rates of origination and extinction may change through time (Sepkoski 1991a). For instance, exponential growth can be derived from constant per-taxon extinction and origination rates the latter being higher than the former. However, actual paleontological data suggest that origination and extinction rates did not follow any distinct trend through the Phanerozoic, and their changes over time look very much like chaotic fluctuations (Cornette and Lieberman 2004). Therefore, it is more difficult to find a simple mathematical approxima-tion for origination and extinction rates than for the total diversity. In fact, the only critical requirement of the exponential model is that the difference between the origination and extinction through time should
where No and Ne are the numbers of genera with, respectively, first and last occurrences within the time interval t, and N is mean diversity level in the interval. The same is true for the hyperbolic model. It does not predict the exact way in which origination and extinction should change, but it does predict that their difference should be roughly pro-portional to the square of the current diversity level:
In demographic models discussed above, the hyperbolic growth of the world population was not decomposed into separate trends of birth and death rates. The main driving force of this growth is presumably the increase of the Earth's carrying capacity and the way this capacity is realized – either by decreasing death rate, or by increasing birth rate, or both – depends upon many factors and may vary from time to time. The same is probably true for biodiversity. The overall shape of the diversity curve depends mostly on the differences in the mean rates of diversity growth in the Paleozoic (low), Mesozoic (moderate), and Ce-nozoic (high). The Mesozoic increase was mainly due to lower extinc-tion rate (compared to the Paleozoic), while the Cenozoic increase was largely due to higher origination rate (compared to the Mesozoic) (see Markov and Korotayev 2007a: 316, Figs 3a, 3b). This probably means that the acceleration of diversity growth during the last two eras was driven by different mechanisms of positive feedback between diversity and its growth rate. Generally, the increment rate ((No −Ne)/ t) was changing in a more regular way than the origination rate No/ t and extinction rate Ne/ t. The large-scale changes in the increment rate cor- 212 Mathematical Modeling of Big History Phases relate better with N2 than with N (Ibid.: figs 3c and 3d), thus supporting the hyperbolic rather than the exponential model. Conclusion In macrosociological models the hyperbolic pattern of the world popu-lation growth arises from a non-linear second-order positive feedback between the demographic growth and technological development (more people – more potential inventors – faster technological growth – the carrying capacity of the Earth grows faster – faster population growth – more people – more potential inventors, and so on, which is more or less identical with the working of the collective learning mechanism). Based on the analogy with macrosociological models and diverse paleontological data, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by a non-linear sec-ond-order positive feedback between the diversity growth and commu-nity structure complexity (which suggests the presence within the bio-sphere of a certain analogue of the collective learning mechanism). The feedback can work via two parallel mechanisms: 1) decreasing extinc-tion rate (more taxa – higher is the alpha diversity, or mean number of taxa in a community – communities become more complex and stable – extinction rate decreases – more taxa, and so on), and 2) increasing origination rate (new taxa facilitate niche construction; newly formed niches can be occupied by the next ‘generation’ of taxa). The latter makes the mechanisms underlying the hyperbolic growth of biodiver-sity and human population even more similar, because the total eco-space of the biota is analogous to the ‘carrying capacity of the Earth’ in demography. As far as new species can increase ecospace and facilitate opportunities for additional species entering the community, they are analogous to the ‘inventors’ in the demographic models whose inven-tions increase the carrying capacity of the Earth. The hyperbolic growth of the Phanerozoic biodiversity suggests that ‘cooperative’ interactions between taxa can play an important role in evolution, along with gener-ally accepted competitive interactions. Due to this ‘cooperation’ (~ ‘col-lective learning’?), the evolution of biodiversity acquires some features of a self-accelerating process. The same naturally refers to coopera-tion/collective learning as regards the global social evolution. The dis-cussed above suggests that we can trace rather similar macropatterns within both the biological and social phases of Big History that produce rather similar curves in diagrams and that can be described in rather accurate way with rather simple mathematical models.
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