Richards Growth Model Research Articles

Use of a Carbon Density Growth Model to Assess the Potential Carbon Sink Function of a Mongolian Pine Plantation in Heilongjiang Province, Northeast China

Accurate estimation of the potential increase in the carbon (C) sink function of forests is required for climate mitigation and C neutrality assessments. Also, accurate forest carbon density estimates are critical for understanding national- and global-level carbon cycling and storage and can inform climate change mitigation. This study established a stand C density growth model to further analyze the C sink potential of planted Mongolian pine (Pinus sylvestris var. mongolica) forests. Samples (390) from fixed plots of Mongolian pine were collected in Heilongjiang Province, Northeast China. The site index (SCI) and stand density index (SDI) were introduced to a constructed stand C density growth model, with an optimal model selected through model fitting. The effect of SDI on stand C density in different SCI grouping intervals was assessed. Total C sequestration of Mongolian pine was calculated using the established C density model. Sample plots with higher C density in each forest age stand were selected to establish a model of potential C sequestration for Mongolian pine, and the difference between this rate and the average was compared to obtain the potential increase in C sink capacity of the forest stand. Slightly different fitting accuracies among the different C density growth models were observed, with the Richards model showing the best performance, which improved through the introduction of the SCI and SDI. Stand C density was associated with an increasing trend in SCI, which within each SCI subgroup was related to the increasing SDI trend. The potential C sequestration rate of the stand was close to the average between years 5 and 13. The average C sequestration rate peaked at 3.86 Mg·ha−1·year−1 at year 13, whereas the potential C sequestration rate peaked at 4.42 Mg·ha−1·year−1 in year 15. A gap between the potential and average C sequestration rate existed between ages 13 and 45, indicating the possibility for an increased C sink function in this forest age range. The Richards growth model incorporating SCI and SDI provided a better reflection of the C density of the Mongolian pine plantation, and the established stand C sequestration rate model showed that the optimal increment in the plantation C sink function can be obtained between years 13 and 45. The results of this study can guide C sink management in the Mongolian pine plantation.

A Bertalanffy–Richards growth model perturbed by a time-dependent pattern, statistical analysis and applications

We analyze a modification of the Richards growth model by introducing a time-dependent perturbation in the growth rate. This modification becomes effective at a special switching time, which represents the first-crossing-time of the Richards growth curve through a given constant boundary. The relevant features of the modified growth model are studied and compared with those of the original one. A sensitivity analysis on the switching time is also performed. Then, we define two different stochastic processes, i.e. a non-homogeneous linear birth–death process and a lognormal diffusion process, such that their means identify to the growth curve under investigation. For the diffusion process, we address the problem of parameters estimation through the maximum likelihood method. The estimates are obtained via meta-heuristic algorithms (namely, Simulated Annealing and Ant Lion Optimizer). A simulation study to validate the estimation procedure is also presented, together with a real application to oil production in France. Special attention is devoted to the approximation of switching time density, viewed as the first-passage-time density for the lognormal process.

A fractional-order generalized Richards growth model and its implementation to COVID-19 data

In this article, we include the memory effect in the generalized Richards model (GRM), which is proposed in the form of a fractional-order GRM. The fractional-order GRM is developed by replacing the first-order derivative in the GRM with a fractional-order derivative and then taking care of model homogeneity of dimension. We consider two fractional-order differential operators: Caputo and Atangana–Baleanu in the Caputo sense (ABC). The proposed fractional-order models are then implemented to fit the COVID-19 data in East Java, Indonesia from March 25 until October 31, 2020. The fitting is performed by minimizing the sum of the squared residual between the numerical solutions of each fractional-order model and the daily data of a cumulative number of COVID-19 cases. The numerical solutions of both fractional-order models are determined by the predictor-corrector method. The performance of the two fractional models is measured by two performance metrics: coefficient of determination (R 2) and root mean square error (RMSE). By considering that the order of fractional derivative as an extra degree of freedom, we perform data fitting for several orders of fractional derivative and evaluate the two performance metrics. It is observed that the fractional-order model with the ABC operator generally has the best performance in calibrating and forecasting both the cumulative number of COVID-19 cases and daily new cases of COVID-19.

Dynamics of a Conformable Fractional Order Generalized Richards Growth Model on Star Network with N=20 Nodes

In this study, we analyze dynamical behavior of the conformable fractional order Richards growth model. Before examining the analysis of the dynamical behavior of the fractional continuous time model, the model is reduced to the system of difference equations via utilizing piecewise constant functions. An algebraic condition that ensures the stability of the positive fixed point of the system is obtained. With the center manifold theory, the existence of a Neimark-Sacker bifurcation at the fixed point of the discrete-time system is proven and the direction of this bifurcation is determined. In addition, the discrete dynamical system is also studied on the star network with N=20 nodes. Analysis complex dynamics of Richards growth model into coupled dynamical network shows that the complex star network with N=20 nodes also exhibits Neimark-Sacker bifurcation about the fixed point concerning with parameter c. Numerical simulations are performed to demonstrate the stability, bifurcations and dynamic transition of the coupled network.

Richards Büyüme Modelinin Parametrelerinin Biyolojik Olarak Anlamlı Parametrelere Dönüşümü Üzerine Bir Çalışma

In this study, it is explained how the Richards model, which is one of the sigmoid growth models, is transformed into a mechanical model with biologically significant parameters such as maximum growth value, lag time and maximum specific growth rate. Detail transformations of the Richards growth model containing 4 parameters with biologically significant parameters calculated with the help of the first and second derivatives were given. In order to test the accuracy of this model, the model prediction values were analyzed by using the height data of the E. Camaldulensis Dehn tree by years. The estimated values of the biologically significant parameters of Richards growth model containing 4 parameters were compared with the estimated values of the modified Gompertz, Logistic and Bertalanffy models containing the other 3 parameters. Error Sum of Squares, Coefficients of Determination and Akaike Information Criteria were used as model evaluation criteria for all proposed models, and it was concluded that the Richards growth model containing 4-parameters predicted relatively better than other models.

A Richards growth model to predict fruit weight

SummaryThe Richards model comprises several popular sigmoidal and monomolecular growth curves. We illustrate fitting of a Bayesian Richards model by splitting the full growth model into several submodels, followed by a model selection procedure. The performance of the methodology is evaluated by Monte Carlo simulations. A double‐sigmoidal version of the Richards model is applied to model grape bunch weight based on data from a New Zealand vineyard over a single growing period.A Bayesian Richards growth model applied to grape size data. Representations of phenological processes are selected through multi‐model inference.

Application of the Richards Model in Settlement Prediction of Loess-Filled Foundations

The Richards equation has high plasticity and compatibility. Compared with other curve equations, it has higher applicability and accuracy in describing S-shaped curves. It can describe the changes in the initial growth stage, rapid growth stage, and stable growth stage of organisms well and reflect the slow-fast-slow change characteristics of growth. In this paper, the Richards model is introduced into the settlement deformation prediction of loess-filled foundation. Taking a high-loess fill site in northern Shaanxi Province as an example, continuous settlement deformation monitoring is carried out for the filling during the project operation period, and the settlement characteristics of a loess-filled foundation are studied using the monitoring results. It is considered that the settlement deformation curve of a loess-filled foundation has similar characteristics; that is, its deformation increases slowly in the early stage, rapidly in the middle stage, and is basically stable at a fixed value in the later stage. Its cumulative deformation time history curve is a smooth S-shaped curve, and the cumulative settlement increases with time and tends to converge. Analysis shows that the time history curves of cumulative settlement, the settlement deformation rate, and the acceleration of loess foundation filling are highly consistent with the Richards model. Compared with the modified exponential model and hyperbolic model, the predicted results are closer to the measured results; that is, the Richards growth model can be used to predict the settlement deformation of a fill foundation.

Genetic and phenotypic correlations between Aleutian disease tests with body weight, growth, and feed efficiency traits in mink.

The ineffectiveness of vaccination, medicine, and culling strategy leads mink farmers to control Aleutian disease (AD) by selecting AD-resilient mink based on AD tests. However, the genetic background of AD tests and their correlations with economically important or AD-resilient traits are limited. This study estimated the genetic and phenotypic correlations between four AD tests and seven body weight (BW) traits, six growth parameters from the Richards growth model, and eight feed-related traits. Univariate models were used to test the significance (P < 0.05) of fixed effects (sex, color type, AD test year, birth year, and row-by-year), random effects (additive genetic, maternal genetic, and permanent environmental), and a covariate of age using ASReml 4.1. Likewise, pairwise bivariate analyses were conducted to estimate the phenotypic and genetic correlations among the studied traits. Both antigen- and virus capsid protein-based enzyme-linked immunosorbent assay tests (ELISA-G and ELISA-P) showed significant (P < 0.05) moderate positive genetic correlations (±SE) with maturation rate (from 0.36 ± 0.18 to 0.38 ± 0.19). ELISA-G showed a significant negative genetic correlation (±SE) with average daily gain (ADG, -0.37 ± 0.16). ELISA-P showed a significant positive moderate genetic correlation (±SE) with off-feed days (DOF, 0.42 ± 0.17). These findings indicated that selection for low ELISA scores would reduce the maturation rate, increase ADG (by ELISA-G), and minimize DOF (by ELISA-P). The iodine agglutination test (IAT) showed significant genetic correlations with DOF (0.73 ± 0.16), BW at 16 weeks of age (BW16, 0.45 ± 0.23), and BW at harvest (HW, -0.47 ± 0.20), indicating that selection for lower IAT scores would lead to lower DOF and BW16, and higher HW. These estimated genetic correlations suggested that the selection of AD tests would not cause adverse effects on the growth, feed efficiency, and feed intake of mink. The estimates from this study might strengthen the previous finding that ELISA-G could be applied as a reliable and practical indicator trait in the genetic selection of AD-resilient mink in AD-positive farms.

Importation, Local Transmission, and Model Selection in Estimating the Transmissibility of COVID-19: The Outbreak in Shaanxi Province of China as a Case Study.

Background: Since the emergence of the COVID-19 pandemic, many models have been applied to understand its epidemiological characteristics. However, the ways in which outbreak data were used in some models are problematic, for example, importation was mixed up with local transmission. Methods: In this study, five models were proposed for the early Shaanxi outbreak in China. We demonstrated how to select a reasonable model and correctly use the outbreak data. Bayesian inference was used to obtain parameter estimates. Results: Model comparison showed that the renewal equation model generates the best model fitting and the Susceptible-Exposed-Diseased-Asymptomatic-Recovered (SEDAR) model is the worst; the performance of the SEEDAR model, which divides the exposure into two stages and includes the pre-symptomatic transmission, and SEEDDAAR model, which further divides infectious classes into two equally, lies in between. The Richards growth model is invalidated by its continuously increasing prediction. By separating continuous importation from local transmission, the basic reproduction number of COVID-19 in Shaanxi province ranges from 0.45 to 0.61, well below the unit, implying that timely interventions greatly limited contact between people and effectively contained the spread of COVID-19 in Shaanxi. Conclusions: The renewal equation model provides the best modelling; mixing continuous importation with local transmission significantly increases the estimate of transmissibility.

Influence of Soil Wetting and Drying Cycles on Soil Detachment

Agricultural soils undergo periods of saturation followed by desiccation throughout the course of a growing season. It is believed that these periods of wetting and drying influence soil structure and may affect the rate of soil detachment. Thus, an experiment was conducted to investigate the influence of a disturbed soil (soil sieved to simulate tillage) subjected to various wetting and drying cycles, on soil bulk density and the resistance to soil detachment with runoff. Seven treatments consisting of wetting and drying cycles ranging from 0 to 6 cycles were evaluated under laboratory conditions using an experimental flume apparatus. A Richards growth model proposed for predicting the influence of wetting and drying on soil detachment was also evaluated. Results showed that the soil bulk density increased as the number of wetting and drying cycles increased. The soil detachment rate decreased as the number of wetting and drying cycles increased. Moreover, initial soil detachment (occurring as soon as runoff began) rates were high for 1 to 3 wetting and drying cycles, while the rate of initial detachment decreased after the third cycle. For example, soils with two and three wetting and drying cycles took 6.5 and 7 min to reach the maximum 1 cm souring depth, respectively, while the soils subjected to four or more wetting and drying cycles did not reach the maximum 1 cm depth during the 15 min runoff experiment. In addition, the proposed S-Shaped Richards growth model was a good predictor for estimating the soil detachment of soils experiencing various wetting and drying cycles. Findings from this study suggest that more attention should be given to the influence that soil wetting and drying have on the prediction of soil detachment. Information from this study is expected to be useful for improving soil management strategies for reducing soil erosion.

Early detection of tree encroachment in high voltage powerline corridor using growth model and UAV-borne LiDAR

UAV-borne LiDAR is an innovative and effective technique for tree encroachment detection in high voltage powerline corridor. However, the periodical inspection of the whole powerline corridor is inefficient, as the powerline segments occurring the tree encroachment only account for a very small part. In this paper, taking one segment of the powerline corridor in Taining County, Fujian province, China as the test site, we acquired the point cloud data using a UAV-borne LiDAR, and then combined the tree growth model and two-phase encroachment detection algorithm to realize efficiently early detection of tree encroachment. First, the points of powerlines and trees were classified from the point cloud, and then the individual tree heights and the belonging tree species were extracted. Based on tree plot data, the relationships between tree heights and tree ages were established using Richards growth model. Secondly, the individual tree heights were predicted at given time points, and the tree encroachments were detected in advance according to the required safe distance between powerlines and trees. To tackle the huge amount of point cloud data and calculation, the two-phase tree encroachment detection algorithm based on bounding boxes was applied to replace the traditional point traversal algorithm. As a result, the exact locations of tree encroachment were early detected and the specific encroaching trees were also pre-identified. Lastly, the pre-detected tree encroachment should be verified through field survey, and treated accordingly. Thus, the inspection efficiency would be greatly improved. In accuracy assessment, the coefficient of determination (R2) and the root mean square error (RMSE) of fitted growth model for Masson pine were 0.812 and 2.308 m, and 0.861 and 2.556 m for Eucalyptus, respectively. Compared with point traversal algorithm, the calculation efficiency of the two-phase tree encroachment detection algorithm was improved by nearly 76 times on average.

Controlling anodization time to monitor film thickness, phase composition and crystal orientation during anodic growth of TiO2 nanotubes

Anodic TiO2 nanotube (TNT) films show promises for photon-driven catalytic, electricity storage and chemical processes. The film thickness of anodic TNT is known to affect its performance in optical and electronic applications. Also, factors affecting the morphology and dimensions of anodic TNT films are rather well-known. However, the knowledge on phase transition and composition in the growth of anodic TiO2 from the titanium metal is very limited. In this work, the anodization time is controlled in intervals of 10, 60, 300, 1000, 2000 and 5000 s to investigate its effect on phase composition and transition, and the morphology of the anodic TNT during the growth process. Even though the mechanism of anodic TNT formation is still under debate, the scanning electron microscope results support bottom-up tube growth with evidence of a compact layer. It was also found that the Richards growth model is applicable to correlate growth time and film thickness. Finally, the phase transition, crystal orientation and pore formation during the anodic process are further discussed.

A Comparative Analysis between a SIRD Compartmental Model and the Richards Growth Model

We propose a compartmental SIRD model with time-dependent parameters that can be used to give epidemiological interpretations to the phenomenological parameters of the Richards growth model. We illustrate the use of the map between these two models by fitting the fatality curves of the COVID-19 epidemic data in Italy, Germany, Sweden, Netherlands, Cuba, and Japan, up to July 30, 2020.

Age and Growth of Yellowfin Tuna in the U.S. Gulf of Mexico and Western Atlantic

AbstractAge, growth, and mortality were estimated for Yellowfin Tuna Thunnus albacares from the U.S. Gulf of Mexico and western Atlantic Ocean. During 2004–2017, 3,443 Yellowfin Tuna were sampled, primarily from recreational landings off the coast of Louisiana (90%). Based on reading otoliths, ages ranged from 1 to 18 years, with younger fish (&lt;4 years) representing the majority (78%) of the age‐classes. Otolith weight was allometrically related to fish age (r2 = 0.91), which suggests that it may be a useful tool in indirectly estimating age given the challenges associated with directly aging tropical tunas. Based on Akaike’s information criterion (AIC), the Richards growth model had the most parsimonious fit to the length‐at‐age data (average maximum length L∞ = 1,658 mm, growth coefficient k = 0.23 year−1, a = 1.04, b = 0.45) compared to the von Bertalanffy growth model (L∞ = 1,589 mm, k = 0.36 year−1, theoretical age at zero length t0 = −0.8 year; AIC difference = 26.21), which had a relatively poor fit. The size‐modified Richards model, which assumed a truncated error structure at the minimum size limit (686 mm curved fork length [CFL]) in fishery‐dependent collections, improved the fit in the smallest individuals and was therefore the preferred model. Males and females had similar maximum ages (18 and 17 years, respectively) but showed significant differences in growth, with males reaching a larger L∞ than females (size‐modified Richards: 1,706 versus 1,568 mm CFL, respectively). Baseline natural mortality (M) using a maximum age of 18 years was 0.346 and was then scaled across age‐classes. These growth and mortality estimates and the individual age data have been used in improving stock assessments for Yellowfin Tuna to support scientific management.

Ecological genetics of Juglans nigra: Differences in early growth patterns of natural populations.

Many boreal and temperate forest tree species distributed across large geographic ranges are composed of populations adapted to the climate they inhabit. Forestry provenance studies and common gardens provide evidence of local adaptation to climate when associations between fitness traits and the populations' home climates are observed. Most studies that evaluate tree height as a fitness trait do so at a specific point in time. In this study, we elucidate differences in early growth patterns in black walnut (Juglans nigra L.) populations by modeling height growth from seed up to age 11. The data comprise tree height measurements between ages 2 and 11 for 52 natural populations of black walnut collected through its geographic range and planted in one or more of 3 common gardens. We use the Chapman–Richards growth model in a mixed effects framework and test whether populations differ in growth patterns by incorporating populations' home climate into the model. In addition, we evaluate differences in populations' absolute growth and relative growth based on the fitted model. Models indicated that populations from warmer climates had the highest cumulative growth through time, with differences in average tree height between populations from home climates with a mean annual temperature (MAT) of 13°C and of 7°C estimated to be as high as 80% at age 3. Populations from warmer climates were also estimated to have higher and earlier maximum absolute growth rate than populations from colder climates. In addition, populations from warm climates were predicted to have higher relative growth rates at any given tree size. Results indicate that natural selection may shape early growth patterns of populations within a tree species, suggesting that fast early growth rates are likely selected for in relatively mild environments where competition rather than tolerance to environmental stressors becomes the dominant selection pressure.

Real time epidemic modeling using Richards model: application for the Covid-19 outbreak in East Kalimantan, Indonesia

The coronavirus pandemic has spread to all provinces in Indonesia within a few months. Coronavirus has entered East Kalimantan province in early March 2020. The number of people confirmed as COVID-19 increased dramatically after March. In this paper, we propose a simple mathematical model to predict coronavirus cases in East Kalimantan province. We applied the Richards growth model to the active case and probable case (PDP case) curves. We used the initial parameter values obtained from China’s Jiangsu province. This means that the strategy for handling coronavirus in East Kalimantan province is assumed to be the same as in China’s Jiangsu province. We have presented the final prediction for the coronavirus pandemic over a range of periods.

Fuzzy Modelling of Covid-19 in Turkey and Some Countries in The World

Coronaviruses are a large family of viruses that are found in many different species of animals and are deadly illnesses for human. In late December 2019, China first announced the outbreak of a new coronavirus: Corona Virus Disease 2019 (or COVID-19), in which the symptoms are similar to common colds and flu. However it can sometimes be more serious, particularly for the elderly as well as patients with weak immune systems. The World Health Organization declared COVID-19 a global pandemic on March 11, 2020. As of date \changemarker{October 14}, 2020, confirmed coronavirus cases exceeded \changemarker{38} million including more than \changemarker{one million} deaths worldwide.In this paper, we use dynamical modelling approach, namely Fuzzyfied Richards Growth Model, to understand the dynamic behaviour of the COVID-19 based on the real data and to predict possible future scenarios applying fuzzy approaches for some countries around the world including China, the United States, the top five countries with the highest population in Europe and Turkey.

Modelling fatality curves of COVID-19 and the effectiveness of intervention strategies.

The main objective of the present article is twofold: first, to model the fatality curves of the COVID-19 disease, as represented by the cumulative number of deaths as a function of time; and second, to use the corresponding mathematical model to study the effectiveness of possible intervention strategies. We applied the Richards growth model (RGM) to the COVID-19 fatality curves from several countries, where we used the data from the Johns Hopkins University database up to May 8, 2020. Countries selected for analysis with the RGM were China, France, Germany, Iran, Italy, South Korea, and Spain. The RGM was shown to describe very well the fatality curves of China, which is in a late stage of the COVID-19 outbreak, as well as of the other above countries, which supposedly are in the middle or towards the end of the outbreak at the time of this writing. We also analysed the case of Brazil, which is in an initial sub-exponential growth regime, and so we used the generalised growth model which is more appropriate for such cases. An analytic formula for the efficiency of intervention strategies within the context of the RGM is derived. Our findings show that there is only a narrow window of opportunity, after the onset of the epidemic, during which effective countermeasures can be taken. We applied our intervention model to the COVID-19 fatality curve of Italy of the outbreak to illustrate the effect of several possible interventions.

Richards Growth Model Driven by Multiplicative and Additive Colored Noises: Steady-State Analysis

We consider a Richards growth model (modified logistic model) driven by correlated multiplicative and additive colored noises, and investigate the effects of noises on the eventual distribution of population size with the help of steady-state analysis. An approximative Fokker–Planck equation is first derived for the stochastic model. By performing detailed theoretical analysis and numerical simulation for the steady-state solution of the Fokker–Planck equation, i.e., stationary probability distribution (SPD) of the stochastic model, we find that the correlated noises have complex effects on the statistical property of the stochastic model. Specifically, the phenomenological bifurcation may be caused by the noises. The position of extrema of the SPD depends on the model parameter and the characters of noises in different ways.

Short-term Forecasts of the COVID-19 Epidemic in Guangdong and Zhejiang, China: February 13-23, 2020.

The ongoing COVID-19 epidemic continues to spread within and outside of China, despite several social distancing measures implemented by the Chinese government. Limited epidemiological data are available, and recent changes in case definition and reporting further complicate our understanding of the impact of the epidemic, particularly in the epidemic’s epicenter. Here we use previously validated phenomenological models to generate short-term forecasts of cumulative reported cases in Guangdong and Zhejiang, China. Using daily reported cumulative case data up until 13 February 2020 from the National Health Commission of China, we report 5- and 10-day ahead forecasts of cumulative case reports. Specifically, we generate forecasts using a generalized logistic growth model, the Richards growth model, and a sub-epidemic wave model, which have each been previously used to forecast outbreaks due to different infectious diseases. Forecasts from each of the models suggest the outbreaks may be nearing extinction in both Guangdong and Zhejiang; however, the sub-epidemic model predictions also include the potential for further sustained transmission, particularly in Zhejiang. Our 10-day forecasts across the three models predict an additional 65–81 cases (upper bounds: 169–507) in Guangdong and an additional 44–354 (upper bounds: 141–875) cases in Zhejiang by February 23, 2020. In the best-case scenario, current data suggest that transmission in both provinces is slowing down.