Search Results (17)

Total leaf area per shoot (A_T) can reflect the photosynthetic capacity of a shoot. A prior study hypothesized that A_T is proportional to the product of the sum of the individual leaf widths per shoot (L_KS) and the maximum individual leaf length per shoot (W_KS), referred to as the Montgomery–Koyama–Smith equation (MKSE). However, empirical evidence does not support such a proportional relationship hypothesis, as A_T was found to allometrically scale with L_KSW_KS, i.e., $A_{\mathrm{T}}\propto {\left(L_{\mathrm{K}\mathrm{S}}{W}_{\mathrm{K}\mathrm{S}}\right)}^{\mathsf{\alpha }}$, where $\mathsf{\alpha }\ne 1$, referred to as the power law equation (PLE). Given that there is variation in the total number of leaves per shoot (n), little is known about whether the leaf area distribution has an explicit mathematical link with the sorted leaf area sequence per shoot, and it is unknown whether the mathematical link can affect the prediction accuracy of the MKSE and PLE. In the present study, the leaves of 500 shoots of a dwarf bamboo (Sasaella kongosanensis ‘Aureostriatus’) were scanned, and the leaf area, length, and width values were obtained by digitizing the leaf images. We selected the shoots with n ranging from 3 to 10, which accounted for 76.6% of the totally sampled shoots (388 out of 500 shoots). We used the formula for the sum of the first j terms (j ranging from 1 to n) of a geometric series (GS), with the mean of the quotients of any adjacent two terms (denoted as ${\overline{q}}_A$) per shoot as the common ratio of the GS, to fit the cumulative leaf area observations. Mean absolute percentage error (MAPE) was used to measure the goodness of fit of the GS. We found that there were 367 out of 388 shoots (94.6%) where 1 < ${\overline{q}}_A$ < 1.618 and MAPE < 15%, and these 367 shoots were defined as valid samples. The GS hypothesis for leaf area distribution was supported by the result that the MAPE values for most valid samples (349 out of 367, i.e., 95.1%) were smaller than 5%. Here, we provide a theoretical basis using the GS hypothesis to demonstrate the validity of the MKSE and PLE. The MAPE values for the two equations to predict A_T were smaller than 5%. This work demonstrates that the leaf area sequence per shoot follows a GS and provides a useful tool for the calculation of total leaf area per shoot, which is helpful to assess the photosynthetic capacity of plants. Full article

Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such as genetic variation and environmental fluctuations. In this study, we extend a general model for ontogenetic growth proposed by West et al. to stochastic models for ontogenetic growth by incorporating stochasticity using white noise. According to data variance fitting for stochasticity, we propose two stochastic models for ontogenetic growth, one is for determinate growth and one is for indeterminate growth. To develop a universal stochastic process for ontogenetic growth across diverse species, we approximate stochastic trajectories of two stochastic models, apply random time change, and obtain a geometric Brownian motion with a multiplier of an exponential time factor. We conduct detailed mathematical analysis and numerical analysis for our stochastic models. Our stochastic models not only predict average growth well but also variations in growth within species. This stochastic framework may be extended to studies of other growth phenomena. Full article

Accurately estimating land-use demand is essential for urban models to predict the evolution of urban spatial morphology. Due to the uncertainties inherent in socioeconomic development, the accurate forecasting of urban land-use demand remains a daunting challenge. The present study proposes a modeling framework to determine the scaling relationship between the population and urban area and simulates the spatiotemporal dynamics of land use and land cover (LULC). An allometric scaling (AS) law and a Markov (MK) chain are used to predict variations in LULC. Random forest (RF) and cellular automata (CA) serve to calibrate the transition rules of change in LULC and realize its micro-spatial allocation (MKCA_RF-AS). Furthermore, this research uses several shared socioeconomic pathways (SSPs) as scenario storylines. The MKCA_RF-AS model is used to predict changes in LULC under various SSP scenarios in Jinjiang City, China, from 2020 to 2065. The results show that the figure of merit (FoM) and the urban FoM of the MKCA_RF-AS model improve by 3.72% and 4.06%, respectively, compared with the MKCA_ANN model during the 2005–2010 simulation period. For a 6.28% discrepancy between the predicted urban land-use demand and the actual urban land-use demand over the period 2005–2010, the urban FoM degrades by 21.42%. The growth of the permanent urban population and urban area in Jinjiang City follows an allometric scaling law with an exponent of 0.933 for the period 2005–2020, and the relative residual and R² are 0.0076 and 0.9994, respectively. From 2020 to 2065, the urban land demand estimated by the Markov model is 19.4% greater than the urban area predicted under scenario SSP5. At the township scale, the different SSP scenarios produce significantly different spatial distributions of urban expansion rates. By coupling random forest and allometric scaling, the MKCA_RF-AS model substantially improves the simulation of urban land use. Full article

Urban forms are human-made systems that display a close connection with fractal objects, following organisation patterns that are not as random as believed. In this context, fractal theory can be seriously considered as a powerful tool for characterizing land-use planning. By applying the box-counting method and image-processing methods, the morphology and fractal metrics of urban networks of Chilean cities were measured. This dimension shows a close correlation with area, population and gross domestic product of each entity, revealing significant asymmetries regarding their distribution throughout the country. Such asymmetries have influenced the current shape of cities, issues concerning economic and social inequalities of urban development that still remain in the territory and explained by social segregation process and the historical evolution of cities. Additionally, some interesting allometric scaling laws obtained from these urban forms are also reported. Our results suggest that the use of fractal metrics can be a meaningful and cheap tool for characterizing the complexity of urban networks, providing useful and quick information about the organisation and efficiency of urban planning in developing countries. Full article

Towns serve as the basic unit of implementation for comprehensive land consolidation and rehabilitation. The utilization of scaling law can provide a new perspective for construction land consolidation. From two perspectives of the town hierarchic system and the growth of a single town, this research applies the Rank-Size Rule and Allometric Scaling Law to analyze the scale structure, hierarchy and allometric scaling evolution relationship of population and construction land in the Loess Plateau at the town scale in 2000, 2010, and 2017. Furthermore, the consolidation potential of construction land is divided into five zones and puts forward recommendations for the comprehensive consolidation of the construction land. The results indicate: (1) The majority of towns have small or medium populations and 62% of the towns in the study show negative population growth. Geographically, the northern part has a smaller population size compared with the southern part. 96% of the towns show an increasing trend in the quantity of construction land, and the south and north parts of the study area have more construction land compared with the center part. The zone of the Valley Plain has the largest population size, and the zone of the Sandy and Desert Area has the largest quantity of construction land. (2) The rank-size distributions of both population and construction land comply with the power-law relation. The population hierarchy has changed from equilibrium to concentration, while the hierarchy of construction land shows an opposite pattern. So, the whole town hierarchic system of the Loess Plateau is gradually tending to the optimal distribution, which is the town hierarchic system gradually forming an ideal sequence structure. (3) The population-construction land relationship obeys the allometric scaling law, and the major allometric type is positive allometry. The human–land relationship tends to be coordinated, and the town system tends to be reasonable. The allometric scaling coefficient is not robust in different geographical areas, especially in Irrigated Agricultural Areas. Furthermore, 90% of the towns have weak coordination of human–land relationships, and 60% of the towns have a relatively faster growth rate of construction land than the relative growth (decline) rate of population. (4) The consolidation potential of construction land is divided into five types. High Consolidation Potential Area concentrates in the Eastern Loess Plateau, while Medium and Low Consolidation Potential Area concentrically distribute in the Western Loess Plateau. The Human–land Coordination Area has a small number and scattered spatial distribution. The land use of towns that are concentrated around prefecture-level cities or with unique resources is not intensive enough. The zoning of construction land consolidation potential based on the results of the allometric scale is in line with reality, and local governments should make use of the characteristics and trends of the town system to formulate planning schemes to promote the integrated development of urban and rural areas. Full article

Urban resilience, as an emerging research focus in urban studies, is the capability of an urban system to adapt to the uncertainties and disturbances faced by modern cities. Numerical characterization of an urban system’s resilience can be performed with urban resilience indicators. Moreover, as cities evolve with intensive socio-economic interactions, the performances of urban indicators are heavily dependent on the scale of these interactions; these relationships are conceptualized as urban scaling laws. Therefore, this study explores the scaling patterns of urban resilience, analyzing the scaling relationship between different resilience indicators and urban population size, as well as the spatial–temporal evolutions of the scaling patterns. The empirical case is based on 267 prefectural-level cities in China. The results show resilience indicators demonstrate scaling patterns on both spatial and temporal scales. Moreover, the scale-adjusted metropolitan indicator (SAMI) differs from the commonly used per capita indicator. Therefore, the scale needs to be considered when assessing urban resilience performance. Findings in this study indicate that moderate scale enhances resilience, enriching urban resilience theorization and urban scaling laws application. The empirical results in the case study also provide a reference for future urban resilience planning and management. Full article

Although the sizes of business firms have been a subject of intensive research, the definition of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems. Full article

This article uses a fractal observation to help delineate the constraints placed by multiple city walls on the growth of historical East Asian cities. By applying advanced technologies from economic geography and fractal indices, a staged scaling process within urban dimension coherence can be applied to both indices. In this study, a discovery is proposed based on the urban organism concept that is capable of indicating a proportional intra-urban structure from a fundamental wall-bounded urban element (local specificity) to other greater walled spatial properties (global variables). This local specificity potentially performs approximate scaling regularities, and spatially denotes an average historical threshold of urban growth for its overall size, with similar scaling law constraints. This finding involves territorial, urban planning, and ancient architectural perspectives, providing a historical and local response to the expansion of contemporary cities. By employing growing fractal estimation, data processing enables the logarithmic city size to be obtained by measuring each wall’s specific features using the Ordinary Least Squares (OLS) method. On the basis of two-dimensional allometric scaling patches, a spatial unfolding mechanism is utilized to reproduce these dynamic changes with city walls as a result of the human trajectories in time geography. Full article

Scaling and dimensional analysis is applied to networks that describe various physical systems. Some of these networks possess fractal, scale-free, and small-world properties. The amount of information contained in a network is found by calculating its Shannon entropy. First, we consider networks arising from granular and colloidal systems (small colloidal and droplet clusters) due to pairwise interaction between the particles. Many networks found in colloidal science possess self-organizing properties due to the effect of percolation and/or self-organized criticality. Then, we discuss the allometric laws in branching vascular networks, artificial neural networks, cortical neural networks, as well as immune networks, which serve as a source of inspiration for both surface engineering and information technology. Scaling relationships in complex networks of neurons, which are organized in the neocortex in a hierarchical manner, suggest that the characteristic time constant is independent of brain size when interspecies comparison is conducted. The information content, scaling, dimensional, and topological properties of these networks are discussed. Full article

Complexity and information theory are two very valuable but distinct fields of research, yet sharing the same roots. Here, we develop a complexity framework inspired by the allometric scaling laws of living biological systems in order to evaluate the structural features of networks. This is done by aligning the fundamental building blocks of information theory (entropy and mutual information) with the core concepts in network science such as the preferential attachment and degree correlations. In doing so, we are able to articulate the meaning and significance of mutual information as a comparative analysis tool for network activity. When adapting and applying the framework to the specific context of the business ecosystem of Japanese firms, we are able to highlight the key structural differences and efficiency levels of the economic activities within each prefecture in Japan. Moreover, we propose a method to quantify the distance of an economic system to its efficient free market configuration by distinguishing and quantifying two particular types of mutual information, total and structural. Full article

Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many natural and social phenomena such as cities, earthquakes, and rivers. This paper reveals the power law behaviors in systems of natural cities by reconstructing the urban hierarchy with cascade structure. Cities of the U.S.A., Britain, France, and Germany are taken as examples to perform empirical analyses. The hierarchical scaling relations can be well fitted to the data points within the scaling ranges of the number, size and area of the natural cities. The size-number and area-number scaling exponents are close to 1, and the size-area allometric scaling exponent is slightly less than 1. The results show that natural cities follow hierarchical scaling laws very well. The principle of entropy maximization of urban evolution is then employed to explain the hierarchical scaling laws, and differences entropy maximizing processes are used to interpret the scaling exponents. This study is helpful for scientists to understand the power law behavior in the development of cities and systems of cities. Full article

Early studies showed the metabolic rate (MR) of different-sized animals was not directly related to body mass. The initial explanation of this difference, the “surface law”, was replaced by the suggestion that MR be expressed relative to massⁿ, where the scaling exponent “n” be empirically determined. Basal metabolic rate (BMR) conditions were developed and BMR became important clinically, especially concerning thyroid diseases. Allometry, the technique previously used to empirically analyse relative growth, showed BMR of endotherms varied with 0.73–0.74 power of body mass. Kleiber suggested that mass^3/4 be used, partly because of its easy calculation with a slide rule. Later studies have produced a range of BMR scaling exponents, depending on species measured. Measurement of maximal metabolism produced scaling exponents ranging from 0.80 to 0.97, while scaling of mammalian MR during growth display multi-phasic allometric relationships with scaling exponents >3/4 initially, followed by scaling exponents <3/4. There is no universal metabolic scaling exponent. The fact that “allometry” is an empirical technique to analyse relative change and not a biological law is discussed. Relative tissue size is an important determinant of MR. There is also a need to avoid simplistic assumptions regarding the allometry of surface area. Full article

The ultimate goal of our multi-article series is to demonstrate the Allometric Scaling and Resource Limitation (ASRL) approach for mapping tree heights and biomass. This third article tests the feasibility of the optimized ASRL model over China at both site (14 meteorological stations) and continental scales. Tree heights from the Geoscience Laser Altimeter System (GLAS) waveform data are used for the model optimizations. Three selected ASRL parameters (area of single leaf, α; exponent for canopy radius, η; and root absorption efficiency, γ) are iteratively adjusted to minimize differences between the references and predicted tree heights. Key climatic variables (e.g., temperature, precipitation, and solar radiation) are needed for the model simulations. We also exploit the independent GLAS and in situ tree heights to examine the model performance. The predicted tree heights at the site scale are evaluated against the GLAS tree heights using a two-fold cross validation (RMSE = 1.72 m; R² = 0.97) and bootstrapping (RMSE = 4.39 m; R² = 0.81). The modeled tree heights at the continental scale (1 km spatial resolution) are compared to both GLAS (RMSE = 6.63 m; R² = 0.63) and in situ (RMSE = 6.70 m; R² = 0.52) measurements. Further, inter-comparisons against the existing satellite-based forest height maps have resulted in a moderate degree of agreements. Our results show that the optimized ASRL model is capable of satisfactorily retrieving tree heights over continental China at both scales. Subsequent studies will focus on the estimation of woody biomass after alleviating the discussed limitations. Full article

The modeling of the cardiovascular system of mammals is discussed within the framework of governing allometric relations and related scaling laws for mammals. An earlier theory of the writer for resting-state cardiovascular function is reviewed and standard solutions discussed for reciprocal quarter-power relations for heart rate and cardiac output per unit body mass. Variation in the basic cardiac process controlling heart beat is considered and shown to allow alternate governing relations. Results have potential application in explaining deviations from the noted quarter-power relations. The work thus indicates that the cardiovascular systems of all mammals are designed according to the same general theory and, accordingly, that it provides a quantitative means to extrapolate measurements of cardiovascular form and function from small mammals to the human. Various illustrations are included. Work described here also indicates that the basic scaling laws from the theory apply to children and adults, with important applications such as the extrapolation of therapeutic drug dosage requirements from adults to children. Full article

A methodology to generate spatially continuous fields of tree heights with an optimized Allometric Scaling and Resource Limitations (ASRL) model is reported in this first of a multi-part series of articles. Model optimization is performed with the Geoscience Laser Altimeter System (GLAS) waveform data. This methodology is demonstrated by mapping tree heights over forested lands in the continental USA (CONUS) at 1 km spatial resolution. The study area is divided into 841 eco-climatic zones based on three forest types, annual total precipitation classes (30 mm intervals) and annual average temperature classes (2 °C intervals). Three model parameters (area of single leaf, α, exponent for canopy radius, η, and root absorption efficiency, γ) were selected for optimization, that is, to minimize the difference between actual and potential tree heights in each of the eco-climatic zones over the CONUS. Tree heights predicted by the optimized model were evaluated against GLAS heights using a two-fold cross validation approach (R² = 0.59; RMSE = 3.31 m). Comparison at the pixel level between GLAS heights (mean = 30.6 m; standard deviation = 10.7) and model predictions (mean = 30.8 m; std. = 8.4) were also performed. Further, the model predictions were compared to existing satellite-based forest height maps. The optimized ASRL model satisfactorily reproduced the pattern of tree heights over the CONUS. Subsequent articles in this series will document further improvements with the ultimate goal of mapping tree heights and forest biomass globally. Full article