Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U₄ and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U₄ models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U₄ model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.     

Modelling avian growth with the Unified‐Richards: as exemplified by wader‐chick growth

Postnatal growth in birds is traditionally modelled by fitting three‐parameter models, namely the logistic, the Gompertz, or the von Bertalanffy models. The purpose of this paper is to address the utility of the Unified‐Richards (U‐Richards) model. We draw attention to two forms of the U‐Richards and lay down a set of recommendations for the analysis of bird growth, in order to make this model and the methods more accessible. We examine the behaviour of the four parameters in each model form and the four derived measurements, and we show that all are easy to interpret, and that each parameter controls a single curve characteristic. The two parameters that control the inflection point, enable us to compare its placement in two dimensions, 1) inflection value (mass or length at inflection) and 2) inflection time (time since hatching), between data sets (e.g. between biometrics or between species). We also show how the parameter controlling growth rate directly presents us with the relative growth rate at inflection, and we demonstrate how one can compare growth rates across data sets. The three traditional models, where the inflection value is fixed (to a specific percentage of the upper asymptote), provide incompatible growth‐rate coefficients. One of the two forms of the U‐Richards model makes it possible to fix not only the upper asymptote (adult value), but also the intersection with the y‐axis (hatching value). Fitting the new model forms to data validates the usefulness of interpreting the inflection placement in addition to the growth rate. It also illustrated the advantages and limitations of constraining the upper asymptote (adult value) and the y‐axis intersection (hatching value) to fixed values. We show that the U‐Richards model can successfully replace some of the commonly used growth models, and we advocate replacing these with the U‐Richards when modelling bird growth.     

Choosing the right sigmoid growth function using the unified‐models approach

Growth of the young is an important part of the life history in birds. However, modelling methods have paid little attention to the choice of regression model used to describe its pattern. The aim of this study was to evaluate whether a single sigmoid model with an upper asymptote could describe avian growth adequately. We compared unified versions of five growth models of the Richards family (the four‐parameter U‐Richards and the three‐parameter U‐logistic, U‐Gompertz, U‐Bertalanffy and U4‐models) for three traits (body mass, tarsus‐length and wing‐length) for 50 passerine species, including species with varied morphologies and life histories. The U‐family models exhibit a unified set of parameters for all models. The four‐parameter U‐Richards model proved a good choice for fitting growth curves to various traits – its extra d‐parameter allows for a flexible placement of the inflection point. Which of the three‐parameter U‐models was the best performing varied greatly between species and between traits, as each three‐parameter model had a different fixed relative inflection value (fraction of the upper asymptote), implying a different growth pattern. Fixing the asymptotes to averages for adult trait value generally shifted the model preference towards one with lower relative inflection values. Our results illustrate an overlooked difficulty in the analysis of organismal growth, namely, that a single traditional three‐parameter model does not suit all growth data. This is mostly due to differences in inflection placement. Moreover, some biometric traits require more attention when estimating growth rates and other growth‐curve characteristics. We recommend fitting either several three‐parameter models from the U‐family, where the parameters are comparable between models, or only the U‐Richards model.     

Richards方程的数学属性及其在兽类生长过程中的应用

通过对Richards方程数学属性的分析表明 ,该方程具有变动的拐点值 ,因而在描绘兽类多种多样的生长过程时具有良好的可塑性。依据其方程参数n取值的不同 ,Richards方程包含了Spillman ,Logistic,Gompertz以及Bertalanffy方程。为了评估Richards方程对兽类生长过程的拟合优度 ,作者引用 1 0组哺乳动物兽类生长数据 ,将它与一些经典的生长模型如Spillman ,Logistic,Gompertz以及Bertalanffy方程共同进行了拟合比较。结果表明 ,Richards方程具有良好的拟合优度 ,适于描绘多种多样的兽类生长模式。     

Modeling to Predict Cases of Hantavirus Pulmonary Syndrome in Chile

Background

Hantavirus pulmonary syndrome (HPS) is a life threatening disease transmitted by the rodent Oligoryzomys longicaudatus in Chile. Hantavirus outbreaks are typically small and geographically confined. Several studies have estimated risk based on spatial and temporal distribution of cases in relation to climate and environmental variables, but few have considered climatological modeling of HPS incidence for monitoring and forecasting purposes.

Methodology

Monthly counts of confirmed HPS cases were obtained from the Chilean Ministry of Health for 2001–2012. There were an estimated 667 confirmed HPS cases. The data suggested a seasonal trend, which appeared to correlate with changes in climatological variables such as temperature, precipitation, and humidity. We considered several Auto Regressive Integrated Moving Average (ARIMA) time-series models and regression models with ARIMA errors with one or a combination of these climate variables as covariates. We adopted an information-theoretic approach to model ranking and selection. Data from 2001–2009 were used in fitting and data from January 2010 to December 2012 were used for one-step-ahead predictions.

Results

We focused on six models. In a baseline model, future HPS cases were forecasted from previous incidence; the other models included climate variables as covariates. The baseline model had a Corrected Akaike Information Criterion (AICc) of 444.98, and the top ranked model, which included precipitation, had an AICc of 437.62. Although the AICc of the top ranked model only provided a 1.65% improvement to the baseline AICc, the empirical support was 39 times stronger relative to the baseline model.

Conclusions

Instead of choosing a single model, we present a set of candidate models that can be used in modeling and forecasting confirmed HPS cases in Chile. The models can be improved by using data at the regional level and easily extended to other countries with seasonal incidence of HPS.     

Shapes and functions of bird-growth models: how to characterise chick postnatal growth

We compare four candidate models (logistic, Gompertz, von Bertalanffy, and extreme value function) for modelling the growth of birds. We fitted the models to two empirical data sets of chick growth (six biometric measurements) of African black oystercatchers Haematopus moquini from South Africa and little stints Calidris minuta from Russia, and identified the best-fitting growth curves by Akaike's information criterion. We also determine fitted and derived parameters, including the relative value (size) at hatching, the placement of inflection, the (normalised) growth rate constant, and the adult value (upper asymptote). The preferred model together with these factors describes how fast (or abruptly) the curves asymptote, and illustrates why growth is poorly characterised by the growth rate constant alone. Though the extreme value function model has not (as far as we know) been applied to chick growth data before, it appears to return the best fit for some parameters in our data sets. For example, we found that in African black oystercatchers two very different models best characterise two of the measurements: the extreme value function model and the Bertalanffy model for tarsus growth and body mass growth, respectively. In addition, we discuss the usefulness of fixing the upper asymptote to the adult value (e.g., adult body mass) and recommend a fixed upper asymptote in most cases.     

Genetic-statistical analysis of growth in selected and unselected mouse lines

The growth of males sampled from two mouse lines long-term selected for over 86 generations on body weight (DU6) or on protein amount (DU6P) was analysed from birth till 120 days of age and compared to the growth of an unselected control line (DUKs). Animals from the selected lines are already approximately 40 to 50% heavier at birth than the controls. This divergence increases to about 210 to 240% at the 120 day of age. With birth weights of 2.2 and 2.4 g and weights of 78 and 89 g at the 120 day these selection lines are the heaviest known mouse lines.

The fit of three modified non-linear growth functions (Gompertz function, Logistic function, Richards function) was compared and the effect of three different data inputs elucidated. The modification was undertaken to use parameters having a direct biological meaning, for example: A: theoretical final body weight, B: maximum weight gain, C: age at maximum weight gain, D (only Richards function): determines the position of the inflection point in relation to the final weight. All three models fit the observed data very well (r² = 0.949–0.998), with a slight advantage for the Richards function. There were no substantial effects of the data input (averages, single values, fitting a curve for every animal with subsequent averaging the parameters).

The high growth of the selected mice is connected with very substantial changes in the final weight and in the maximum weight gain, whereas the changes of the age at the point of inflection were, although partially significant, relatively small and dependent on the model used.     

Rapid electroelution of nucleic acids from agarose and acrylamide gels

The alkaline/filter DNA elution technique measures single-strand DNA breaks in mammalian cells based on the DNA molecular weight dependent retention of the macromolecule on 2-μm-pore-size filters. Described here is a modification of the technique which uses [³H]thymidine-labeled DNA of γ-irradiated cells as an internal reference. Thus, an increased precision is obtained in the assessment of this type of DNA damage at biologically significant radiation doses (i.e., where cell survival occurs). The measure of DNA damage is based on the actual initial DNA elution rate, i.e., arithmetic ratio of the elution of “test” DNA (i.e., ¹⁴C-labeled DNA) relative to the elution of “reference” DNA (i.e., ³H-labeled DNA). The repair of this damage on postirradiation incubation of the cells is detected as a decrease in the rate of “test” DNA cluted relative to “reference” DNA from unincubated cells. For Chinese hamster V79–171 cells irradiated with 5 Gy (500 rads), repair can be resolved into two first-order processes having rate constants (at 24°C) of ～0.190 and ～0.017 min^?1.     

Trans-Theta Logistics: A New Family of Population Growth Sigmoid Functions

Sigmoid functions have been applied in many areas to model self limited population growth. The most popular functions; General Logistic (GL), General von Bertalanffy (GV), and Gompertz (G), comprise a family of functions called Theta Logistic ([Formula: see text] L). Previously, we introduced a simple model of tumor cell population dynamics which provided a unifying foundation for these functions. In the model the total population (N) is divided into reproducing (P) and non-reproducing/quiescent (Q) sub-populations. The modes of the rate of change of ratio P/N was shown to produce GL, GV or G growth. We now generalize the population dynamics model and extend the possible modes of the P/N rate of change. We produce a new family of sigmoid growth functions, Trans-General Logistic (TGL), Trans-General von Bertalanffy (TGV) and Trans-Gompertz (TG)), which as a group we have named Trans-Theta Logistic (T [Formula: see text] L) since they exist when the [Formula: see text] L are translated from a two parameter into a three parameter phase space. Additionally, the model produces a new trigonometric based sigmoid (TS). The [Formula: see text] L sigmoids have an inflection point size fixed by a single parameter and an inflection age fixed by both of the defining parameters. T [Formula: see text] L and TS sigmoids have an inflection point size defined by two parameters in bounding relationships and inflection point age defined by three parameters (two bounded). While the Theta Logistic sigmoids provided flexibility in defining the inflection point size, the Trans-Theta Logistic sigmoids provide flexibility in defining the inflection point size and age. By matching the slopes at the inflection points we compare the range of values of inflection point age for T [Formula: see text] L versus [Formula: see text] L for model growth curves.     

Differences in Physical and Biochemical Properties of Thermus scotoductus SA-01 Cultured with Dielectric or Convection Heating

A thermophile, Thermus scotoductus SA-01, was cultured within a constant-temperature (65°C) microwave (MW) digester to determine if MW-specific effects influenced the growth and physiology of the organism. As a control, T. scotoductus cells were also cultured using convection heating at the same temperature as the MW studies. Cell growth was analyzed by optical density (OD) measurements, and cell morphologies were characterized using electron microscopy imaging (scanning electron microscopy [SEM] and transmission electron microscopy [TEM]), dynamic light scattering (DLS), and atomic force microscopy (AFM). Biophysical properties (i.e., turgor pressure) were also calculated with AFM, and biochemical compositions (i.e., proteins, nucleic acids, fatty acids) were analyzed by attenuated total reflectance-Fourier transform infrared (ATR-FTIR) spectroscopy. Gas chromatography-mass spectrometry (GC-MS) was used to analyze the fatty acid methyl esters extracted from cell membranes. Here we report successful cultivation of a thermophile with only dielectric heating. Under the MW conditions for growth, cell walls remained intact and there were no indications of membrane damage or cell leakage. Results from these studies also demonstrated that T. scotoductus cells grown with MW heating exhibited accelerated growth rates in addition to altered cell morphologies and biochemical compositions compared with oven-grown cells.     

Association between Traffic-Related Air Pollution in Schools and Cognitive Development in Primary School Children: A Prospective Cohort Study

Background

Air pollution is a suspected developmental neurotoxicant. Many schools are located in close proximity to busy roads, and traffic air pollution peaks when children are at school. We aimed to assess whether exposure of children in primary school to traffic-related air pollutants is associated with impaired cognitive development.

Methods and Findings

We conducted a prospective study of children (n = 2,715, aged 7 to 10 y) from 39 schools in Barcelona (Catalonia, Spain) exposed to high and low traffic-related air pollution, paired by school socioeconomic index; children were tested four times (i.e., to assess the 12-mo developmental trajectories) via computerized tests (n = 10,112). Chronic traffic air pollution (elemental carbon [EC], nitrogen dioxide [NO2], and ultrafine particle number [UFP; 10–700 nm]) was measured twice during 1-wk campaigns both in the courtyard (outdoor) and inside the classroom (indoor) simultaneously in each school pair. Cognitive development was assessed with the n-back and the attentional network tests, in particular, working memory (two-back detectability), superior working memory (three-back detectability), and inattentiveness (hit reaction time standard error). Linear mixed effects models were adjusted for age, sex, maternal education, socioeconomic status, and air pollution exposure at home.Children from highly polluted schools had a smaller growth in cognitive development than children from the paired lowly polluted schools, both in crude and adjusted models (e.g., 7.4% [95% CI 5.6%–8.8%] versus 11.5% [95% CI 8.9%–12.5%] improvement in working memory, p = 0.0024). Cogently, children attending schools with higher levels of EC, NO2, and UFP both indoors and outdoors experienced substantially smaller growth in all the cognitive measurements; for example, a change from the first to the fourth quartile in indoor EC reduced the gain in working memory by 13.0% (95% CI 4.2%–23.1%). Residual confounding for social class could not be discarded completely; however, the associations remained in stratified analyses (e.g., for type of school or high-/low-polluted area) and after additional adjustments (e.g., for commuting, educational quality, or smoking at home), contradicting a potential residual confounding explanation.

Conclusions

Children attending schools with higher traffic-related air pollution had a smaller improvement in cognitive development.     

Characterization of acyl chain position in unsaturated phosphatidylcholines using differential mobility-mass spectrometry

Glycerophospholipids (GPs) that differ in the relative position of the two fatty acyl chains on the glycerol backbone (i.e., sn-positional isomers) can have distinct physicochemical properties. The unambiguous assignment of acyl chain position to an individual GP represents a significant analytical challenge. Here we describe a workflow where phosphatidylcholines (PCs) are subjected to ESI for characterization by a combination of differential mobility spectrometry and MS (DMS-MS). When infused as a mixture, ions formed from silver adduction of each phospholipid isomer {e.g., [PC (16:0/18:1) + Ag]⁺ and [PC (18:1/16:0) + Ag]⁺} are transmitted through the DMS device at discrete compensation voltages. Varying their relative amounts allows facile and unambiguous assignment of the sn-positions of the fatty acyl chains for each isomer. Integration of the well-resolved ion populations provides a rapid method (< 3 min) for relative quantification of these lipid isomers. The DMS-MS results show excellent agreement with established, but time-consuming, enzymatic approaches and also provide superior accuracy to methods that rely on MS alone. The advantages of this DMS-MS method in identification and quantification of GP isomer populations is demonstrated by direct analysis of complex biological extracts without any prior fractionation.     

Blood lactate accumulation and muscle deoxygenation during incremental exercise.

Near-infrared spectroscopy (NIRS) could allow insights into controversial issues related to blood lactate concentration ([La](b)) increases at submaximal workloads (). We combined, on five well-trained subjects [mountain climbers; peak O(2) consumption (VO(2peak)), 51.0 +/- 4.2 (SD) ml. kg(-1). min(-1)] performing incremental exercise on a cycle ergometer (30 W added every 4 min up to voluntary exhaustion), measurements of pulmonary gas exchange and earlobe [La](b) with determinations of concentration changes of oxygenated Hb (Delta[O(2)Hb]) and deoxygenated Hb (Delta[HHb]) in the vastus lateralis muscle, by continuous-wave NIRS. A "point of inflection" of [La](b) vs. was arbitrarily identified at the lowest [La](b) value which was >0.5 mM lower than that obtained at the following. Total Hb volume (Delta[O(2)Hb + HHb]) in the muscle region of interest increased as a function of up to 60-65% of VO(2 peak), after which it remained unchanged. The oxygenation index (Delta[O(2)Hb - HHb]) showed an accelerated decrease from 60- 65% of VO(2 peak). In the presence of a constant total Hb volume, the observed Delta[O(2)Hb - HHb] decrease indicates muscle deoxygenation (i.e., mainly capillary-venular Hb desaturation). The onset of muscle deoxygenation was significantly correlated (r(2) = 0.95; P < 0.01) with the point of inflection of [La](b) vs., i.e., with the onset of blood lactate accumulation. Previous studies showed relatively constant femoral venous PO(2) levels at higher than approximately 60% of maximal O(2) consumption. Thus muscle deoxygenation observed in the present study from 60-65% of VO(2 peak) could be attributed to capillary-venular Hb desaturation in the presence of relatively constant capillary-venular PO(2) levels, as a consequence of a rightward shift of the O(2)Hb dissociation curve determined by the onset of lactic acidosis.     

Shrub cover expressed as an ‘arthropod island’ in xeric environments

Based on the vague importance of shrub cover, an attempt was made to create a theoretical concept framework known as an ‘arthropod-island’ analytical model. These models were based on the multi-bond correlation between shrubs, soil properties, and above- and below-ground biotic communities. By utilizing published datasets on (i.e., above- and below-ground) arthropod communities related to shrub species and age, the proposed models for an ‘arthropod island’ were applied in order to determine their fitness for xeric ecosystems. It was found that the ‘arthropod-island’ concept could be the result of statistical differences in ecologically adaptive (i.e., preferable) redistribution of arthropods among the microhabitats beneath the shrub canopy and in the open spaces. Taxon density, relative to the richness and Shannon indices, was found to be more sensitive to the selected models. The relative interaction intensity index [RII = (A ? B)/(A + B), A = shrub canopy value, B = intershrub value] was found to be more suitable for the ‘arthropod island’ at the community level. The relative neighbor effect [RNE = (B ? A)/max(A, B)] and RII were found to be suitable at the population level, while the fitted model heavily depended on the variety of arthropod taxon. It was suggested that there were consistent ‘arthropod island’–shrub relationships between shrub species and between shrub ages in terms of arthropod density at the community level. The arthropod taxon was found to indicate an inconsistent ‘arthropod island’–shrub relationship between shrub species that differed from shrub ages at the population level.     

Growth temperature can alter the temperature dependent stimulation of photosynthesis by elevated carbon dioxide in Albutilon theophrasti

Stimulation of photosynthesis in response to elevated carbon dioxide concentration [CO2] in the short-term (min) should be highly temperature dependent at high photon flux. However, it is unclear if long-term (days, weeks) adaptation to a given growth temperature alters the temperature-dependent stimulation of photosynthesis to [CO2]. In velveltleaf (Albutilon theophrasti), the response of photosynthesis, determined as CO2 assimilation, was measured over a range of internal CO2 concentrations at 7 short-term measurement (12, 16, 20, 24, 28, 32, 36 degrees C) temperatures for each of 4 long-term growth (16, 20, 28 and 32 degrees C) temperatures. In vivo estimates of VCmax, the maximum RuBP saturated rate of carboxylation, and Jmax, the light-saturated rate of potential electron transport, were determined from gas exchange measurements for each temperature combination. Overall, previous exposure to a given growth temperature adjusted the optimal temperatures of Jmax and VCmax with subsequently greater enhancement of photosynthesis at elevated [CO2] (i.e., a greater enhancement of photosynthesis at elevated [CO2] was observed at low measurement temperatures for A. theophrasti grown at low growth temperatures compared with higher growth temperatures, and vice versa for plants grown and measured at high temperatures). Previous biochemical based models used to predict the interaction between rising [CO2] and temperature on photosynthesis have generally assumed no growth temperature effect on carboxylation kinetics or no limitation by Jmax. In the current study, these models over predicted the temperature dependence of the photosynthetic response to elevated [CO2] at temperatures above 24 degrees C. If these models are modified to include long-term adjustments of Jmax and VCmax to growth temperature, then greater agreement between observed and predicted values was obtained.     

Analyses of spatial patterns and population processes of clonal plants

The nonrandom spatial structure of terrestrial plants is formed by ecological interactions and reproduction with a limited dispersal range, and in turn this may strongly affect population dynamics and population genetics. The traditional method of modelling in population ecology is either to neglect spatial pattern (e.g. in transition matrix models) or to do straightforward computer simulation. We review here three analytical mothods to deal with plant populations in a lattice-structured habitat, which propagate both by seeds that scatter over the whole habitat and by vegetative reproduction (producing runners, rhizomes, etc.) to neighboring vacant sites. [1]Dynamics of global and local densities: Dynamical equations of population density considering nearest-neighbor correlation (spatial clumping) are developed as the joint dynamics of global average density and local density (comparable to mean crowding) based onpair approximation. If there is a linear trade-off between seed production and vegetative reproduction, the equilibrium abundance of the population may be maximized by engaging both means of reproduction. This result is accurately predicted by the pair approximation method, but not by mean-field approximation (neglect of spatial structure). [2]Cluster size distributions: Using global and local densities obtained by pair approximation, we predicted cluster size distribution, i.e. the number of clusters of occupied sites of various sizes. [3]Clonal identity probability decreasing with distance: Multi-locus measurement of allozymes or other neutral molecular markers tells us whether or not a given pair of individuals belong to the same clone. From the pattern of clonal identity probability decreasing with the distance between ramets, we can estimate the relative importance of two modes of reproduction: vegetative propagation and sexual seed production.     

Comparison of growth curves of mice selected and unselected for postweaning gain

Summary Mice were sampled from a line selected for increased postweaning weight gain from three to six weeks and from a randombred control line originating from the same base population. Body weights were recorded at each of 14 ages from day 5 to day 98. The Richards and logistic growth functions were fitted to the growth trajectories of each individual mouse by a generalized non-linear least squares procedure. Estimated growth parameters (asymptotic weight, rate, shape of curve, age and weight at inflection, mean absolute growth rate and mean relative growth rate) were computed for each individual. The effects of line, litter within line, sex and line × sex interactions on these estimated parameters were then studied.Both the Richards and logistic functions fitted the data equally well and the plotted trajectories coincided over most of the growth curve. There was excellent agreement between the estimates of asymptotic weight and both age and weight at inflection based on the different functions. However, both functions apparently underestimated the asymptotic weight.Analyses of the line differences showed that selection for postweaning gain increased the mean absolute growth rate over the entire curve but had no effect on the relative growth rate or the shape of the growth curve. Full-sib analyses suggested the presence of considerable genetic variation and some high genetic correlations among the estimated growth parameters.Paper number 2942 of the Journal Series of the North Carolina State University Agricultural Experiment Station, Raleigh, North Carolina. This research was supported in part by Public Health Service Research Grant GM 11546. Computing services were supported by NIH Grant FR-00011.     

Estimating Initial Epidemic Growth Rates

The initial exponential growth rate of an epidemic is an important measure of disease spread, and is commonly used to infer the basic reproduction number $\mathcal{R}_{0}$ . While modern techniques (e.g., MCMC and particle filtering) for parameter estimation of mechanistic models have gained popularity, maximum likelihood fitting of phenomenological models remains important due to its simplicity, to the difficulty of using modern methods in the context of limited data, and to the fact that there is not always enough information available to choose an appropriate mechanistic model. However, it is often not clear which phenomenological model is appropriate for a given dataset. We compare the performance of four commonly used phenomenological models (exponential, Richards, logistic, and delayed logistic) in estimating initial epidemic growth rates by maximum likelihood, by fitting them to simulated epidemics with known parameters. For incidence data, both the logistic model and the Richards model yield accurate point estimates for fitting windows up to the epidemic peak. When observation errors are small, the Richards model yields confidence intervals with better coverage. For mortality data, the Richards model and the delayed logistic model yield the best growth rate estimates. We also investigate the width and coverage of the confidence intervals corresponding to these fits.     

Current and future distribution of the deciduous shrub Hydrangea macrophylla in China estimated by MaxEnt

Climate change has a significant impact on the growth and distribution of vegetation worldwide. Hydrangea macrophylla is widely distributed and considered a model species for studying the distribution and responses of shrub plants under climate change. These results can inform decision‐making regarding shrub plant protection, management, and introduction of germplasm resources, and are of great importance for formulating ecological countermeasures to climate change in the future. We used the maximum entropy model to predict the change, scope expansion/reduction, centroid movement, and dominant climate factors that restrict the growth and distribution of H. macrophylla in China under current and future climate change scenarios. It was found that both precipitation and temperature affect the distribution of suitable habitat for H. macrophylla. Akaike information criterion (AICc) was used to select the feature combination (FC) and the regularization multiplier (RM). After the establishment of the optimal model (FC = QP, RM = 0.5), the complexity and over‐fitting degree of the model were low (delta AICc = 0, omission rate = 0.026, difference between training and testing area under the curve values = 0.0009), indicating that it had high accuracy in predicting the potential geographical distribution of H. macrophylla (area under the curve = 0.979). Overall, from the current period to future, the potential suitable habitat of this species in China expanded to the north. The greenhouse effect caused by an increase in CO₂ emissions would not only increase the area of high‐suitability habitat in Central China, but also expand the area of total suitable habitat in the north. Under the maximum greenhouse gas emission scenario (RCP8.5), the migration distance of the centroid was the longest (e.g., By 2070s, the centroids of total and highly suitable areas have shifted 186.15 km and 89.84 km, respectively).     

Inflection, canards and excitability threshold in neuronal models

A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh–Nagumo, reduced Hodgkin–Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.