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Life cycle representation of the Usher projection matrix model, where pl is the probability for an individual to stay in class l, ql is the probability to move up from class l to l + 1, ml is the probability of dying and f is the average fecundity.
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Matrix population models are widely used to predict population dynamics, but when applied to species‐rich ecosystems with many rare species, the small population sample sizes hinder a good fit of species‐specific models. This issue can be overcome by assigning species to groups to increase the size of the calibration data sets. However, the species...
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... the change of the population by a vector, N ~ t containing the numbers N l,t of individuals in L ordered size classes (l = 1,…, L) at discrete time t. Let N t ¼ P L l¼1 N l;t be the total number of individuals at time t. Like any other matrix population model, the Usher model can be interpreted as the expectation of N t independent Markov chains (Fig. 1). The relationship between N ~ t and N ~ tþ1 is described by a L 9 L transition matrix U, called the Usher ...
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... the first to the last diameter class. The method that we developed for the mixture of Usher matrix models could straightforwardly be adapted to other types of matrix projection models, such as Leslie or Lefkovitch matrix models for age-and stage-structured populations respectively. Starting from the life cycle representation of the matrix model (Fig. 1), one simply has to translate the probabil- Table 3. Observed vital rates of groups (Obs.) and average vital rates computed from the estimated transition rates (Est.): 2-year d.b.h increment (DDBH), 2-year mortality rate, 2-year fecundity rate, upper bound of diameters (DBH95) and 2-year turnover of the five groups obtained using matrix ...
Citations
... Model-based clustering based on mixture models can be used to group higher-level entities (here, species) while accounting for lower-level variability nested within them (here, trees). Such approaches have recently been proposed in various statistical frameworks such as generalised linear regression (Dunstan, Foster, Darnell, 2011;Dunstan, Foster, Hui et al., 2013;Hui et al., 2013;Mortier, Ouédraogo et al., 2015;Ouédraogo et al., 2013) or matrix population models (Mortier, Rossi et al., 2013). Mixture models are a flexible class of models allowing the simultaneous fit of models and the classification of observations into clusters (McLachlan et al., 2019). ...
... The evaluation of classification consistency was based on the two following indices (Mortier, Rossi et al., 2013): ...
The understanding of tree growth processes is crucial for promoting sustainable forest management strategies. This is a challenging task in highly biodiverse ecosystems where many tree species are observed on very few individuals and the small sample sizes hinder a good fit of species‐specific models. We propose the use of finite mixture of random coefficient regression models with multilevel nested random effects to infer guild specific fixed and random effects while evaluating the relative importance of the nested sources of variability on goodness‐of‐fit. This approach extends finite mixture of linear mixed model used for longitudinal or single group structured data contexts. A dedicated expectation–maximisation algorithm is introduced for parameter estimation. Simulations are performed for the evaluation of the misspecification of nested‐grouping structures. This work has been motivated by data collected biennially in Central African rainforests from 1986 to 2010. We show the accuracy of the proposed approach in successfully reproducing individual growth processes and classifying tree species into well‐differentiated clusters with clear ecological interpretations. Moreover, results confirm that interindividual variability appears as the most important factor to explain tropical tree species growth process variability from Central Africa forests.
... In general, the explicit calculation of the eigenvalues of the Usher matrix is not possible. Many theoretical properties of the Usher model are known, particularly its asymptotic behavior: the evolution to the stationary state is exponential and is characterized by the growth rate and the stationary distribution [79,80]. Life cycle representation of the Usher projection matrix model, where p i is the probability for an individual to stay in class i, q i is the probability to move up from class i to i + 1, m i is the probability of dying and f is the average fertility rate supposed to be constant (after [79]). ...
... Many theoretical properties of the Usher model are known, particularly its asymptotic behavior: the evolution to the stationary state is exponential and is characterized by the growth rate and the stationary distribution [79,80]. Life cycle representation of the Usher projection matrix model, where p i is the probability for an individual to stay in class i, q i is the probability to move up from class i to i + 1, m i is the probability of dying and f is the average fertility rate supposed to be constant (after [79]). ...
This thesis is devoted to the mathematical and statistical modeling of epidemic data andIt is divided into two broad parts, which are subdivided into different sections. The modeling of infectious diseases has been a subject of interest to researchers, policy makers, andmedical practitioners, most especially during the recent global COVID-19 pandemic, whichIt has been devastating to the health infrastructure and socio-economic status of many nations.It has affected mobility and interaction among citizens due to the many daily new cases and deaths.Hence, the need to contribute to understanding the mechanisms of virulence and spread using different mathematical and statistical modeling approaches. The first part is dedicated to the mathematical modeling aspect, which consists of the deterministic and discrete approaches to epidemiology modeling, which in this case is mainly focused on the COVID-19 pandemic. The daily reproduction number of the COVID-19 outbreak calculation is approached by discretization using the idea of deconvolution and a unique biphasic pattern is observed that is more prevalent during the contagiousness period across various countries. Furthermore, a discrete model is formulated from Usher’s model in order to calculate the life span loss due to COVID-19 disease and to also explain the role of comorbidities, which are very essential in the disease spread and its dynamics at an individual level. Also, the formulation of Susceptible-Infectious-Geneanewsusceptible-Recovered (SIGR) age-dependent modelling is proposed in order to perform some mathematical analysis and present the role of different epidemiology parameters, most especially vaccination, and finally, a new technique to identify the point of inflection on the smoothed curves of the new infected pandemic cases using the Bernoulli equation is presented. This procedure is important because not all countries have reached the turning point (maximum number of daily cases) in the epidemic curve. The approach is used to calculate the transmission rate and the maximum reproduction number for various countries.The statistical modelling of the COVID-19 pandemic using various data analysis models (namely machine and deep learning models) is presented in the second part in order to understand the dynamics of the pandemic in different countries and also predict and forecast the daily new cases and deaths due to the disease alongside some socio-economic parameters. It is observed that the prediction and forecasting are consistent with the disease evolution at different waves in these countries and that there are socio-economic determinants of the disease depending on whether the country is developed or developing. Also, the study of the shapes and peaks of the COVID-19 disease is presented. The peaks of the curves of the daily new cases and deaths are identified using the spectral analysis method, which enables the weekly peak patterns to be visible. Finally, the clustering of different regions in France due to the spread of the disease is modeled using functional data analysis. The study shows clear differences between the periods when vaccination has not been introduced (but only non-pharmaceutical mitigation measures) and when it was introduced. The results presented in this thesis are useful to better understand the modeling of a viral disease, the COVID-19 virus.
... 44. F. Mortier et al. (2013). « Population dynamics of species-rich ecosystems : The mixture of matrix population models approach ». ...
Dans le contexte de changements actuel, clarifier la réponse des forêts aux perturbations est indispensable pour préserver les biens et services qu’elles rendent. Le fonctionnement et le maintien des forêts dépend largement de la diversité des communautés d'arbres qui devient un enjeu majeur, en particulier dans les régions tropicales où les forêts sont les plus menacées et où les enjeux économiques, sociaux et environnementaux sont les plus importants. Cette thèse étudie la réponse aux perturbations de la diversité taxonomique et fonctionnelle d’une communauté en forêt Néotropicale. Nous analysons les trajectoires de diversité sur le long terme pour déterminer les processus écologiques sous-jacents la réponse des communautés aux perturbations, à expliciter les aspects taxonomiques et fonctionnels de la restauration, et enfin à discuter de perspectives de gestion et de modélisation de la dynamique forestière. Le dispositif expérimental de Paracou en Guyane Française a permis de suivre la réponse des communautés d'arbres sur 30 années après un gradient de perturbation. Dans un premier temps, nous avons établi et validé un estimateur de diversité fiable, pour pallier les incertitudes de botaniques des inventaires forestiers et des bases de données fonctionnelles. L'estimateur propage les incertitudes taxonomiques aux mesures de diversité via les probabilités d'associations entre noms vernaculaires et noms botaniques. L’estimateur de diversité, employé dans l'ensemble de la thèse, a été calibré pour optimiser l'estimation la précision de l'estimation en fonction des données disponibles, puis testé avec des inventaires forestiers pré-exploitation pour proposer un protocole d'inventaire optimisant le coût et la précision de ces inventaires. Dans un deuxième temps, en combinant les inventaires botaniques à un large jeu de données fonctionnel comprenant des traits des feuilles, du bois et des traits d'histoire de vie, nous avons analysé les trajectoires de diversité, de composition, et de redondance taxonomique et fonctionnelle des communautés après perturbation . Enfin, nous avons spécifiquement étudié les trajectoires de diversité et de composition des communautés recrutées. Notre étude a montré l'émergence de processus déterministes après perturbation déterminant la réponse taxonomique et fonctionnelle des communautés en favorisant le recrutement d'un pool restreint de pionnières. Nous avons montré la restauration progressive des communautés pré-perturbation et de processus stochastiques tels qu'observés en l'absence de perturbation. Les perturbations ont augmenté la richesse et l'équitabilité taxonomiques des communautés jusqu'à un certain seuil, au delà duquel la dominance de quelques pionnières diminue la richesse taxonomique, conformément à la théorie des perturbations intermédiaires. Les trajectoires fonctionnelles en revanche ont montré une augmentation de la diversité quelle que soit la perturbation et une convergences fonctionnelle entre les communautés: ce découplage entre trajectoires taxonomiques et fonctionnelles s'est expliqué par la redondance fonctionnelle des communautés, atténuant l’impact fonctionnel des perturbations. Nos résultats ont montré une restauration taxonomique et fonctionnelle tangible des communautés mais encore inachevée. A la lumière de ces résultats nous proposons une discussion sur la possibilité d'une exploitation durable des forêts et de nouvelles perspectives de modélisation de la diversité.
... To address this problem, modellers usually cluster species into groups using a variety of methods (Swaine and Whitmore, 1988; Steneck and Dethier, 1994; Favrichon, 1994; Bellwood and Wainwright, 2001; Gitay and Noble, 1997). Mixture models that cluster based on similar species responses rather than similar species traits have been proposed in the framework of generalized linear models (GLM) (Dunstan et al., 2011; Dunstan et al., 2013; Hui et al., 2013; Ouédraogo et al., 2013) and more recently in the context of homogeneous matrix population models (Mortier et al., 2013). In this paper, we propose a new class of mixture of inhomogeneous matrix population models that allows the simultaneous clustering of species based on vital rate processes (recruitment, growth and mortality) and selection of group-specific explicative environmental variables. ...
... We are able to identify the correct number of underlying clusters for all the different processes with correlations as high as 0.9 between consecutive repeated measures (Figure 2). We use two matching indices, I 1 and I 2 (Mortier et al., 2013), to assess the clustering performance and compare each species group allocation based on the maximum a posteriori estimate to the true group membership. These indices are based on the K O K contingency ...
Understanding how environmental factors could impact population dynamics is of primary importance for species conservation. Matrix population models are widely used to predict population dynamics. However, in species-rich ecosystems with many rare species, the small population sizes hinder a good fit of species-specific models. In addition, classical matrix models do not take into account environmental variability. We propose a mixture of regression models with variable selection allowing the simultaneous clustering of species into groups according to vital rate information (recruitment, growth and mortality) and the identification of group-specific explicative environmental variables. We develop an inference method coupling the R packages flexmix and glmnet. We first highlight the effectiveness of the method on simulated datasets. Next, we apply it to data from a tropical rain forest in the Central African Republic. We demonstrate the accuracy of the inhomogeneous mixture matrix model in successfully reproducing stand dynamics and classifying tree species into well-differentiated groups with clear ecological interpretations. Copyright © 2014 John Wiley & Sons, Ltd.
Impacts of climate change on the future dynamics of Central African forests are still largely unknown, despite the acuteness of the expected changes and the extent of these forests. The high diversity of species and the potentially equivalent diversity of responses to climate modifications are major difficulties encountered when using predictive models to evaluate these impacts.
In this study, we applied a mixture of inhomogeneous matrix models to a long-term experimental site located in M'Baïki forests, in the Central African Republic. This model allows the clustering of tree species into processes-based groups while simultaneously selecting explanatory climate and stand variables at the group-level. Using downscaled outputs of 10 general circulation models (GCM), we projected the future forest dynamics up to the end of the century, under constant climate and Representative Concentration Pathways 4.5 and 8.5. Through comparative analyses across Gcm versions, we identified tree species meta-groups, which are more adapted than ecological guilds to describe the diversity of tree species dynamics and their responses
to climate change.
Projections under constant climate were consistent with a forest ageing phenomenon, with a slowdown in tree growth and a reduction of the relative abundance of short-lived pioneers. Projections under climate change showed a general increase in growth, mortality and recruitment. This acceleration in forest dynamics led to a strong natural thinning effect, with different magnitudes across species. These differences caused a compositional shift in favour of long-lived pioneers, at the detriment of shade-bearers.
Consistent with other field studies and projections, our results show the importance of elucidating the diversity of tree species responses when considering the general sensitivity of Central African forests dynamics to climate change.
HOw to balance the compenting demands of logging companies, conservation NGOs, local communities, mining companies and governments in the landscapes of central Africa? We explored this question through participatory modelling and scenario development. With these tools we helped the FSC Program for the Congo Basin secure an agreement between all parties regarding the management of Intact Forest Landscapes in FSC certified concessions.
Understanding how drought affects annual tree growth in tropical forests is of crucial importance to predict their response to climate change. Previous studies, mainly led in the N eotropics and in S outheast A sia, have yielded contradictory results which might be explained by differences in species studied, in the tree development stages considered, or by differences in other environmental factors than water availability.
Here, we described the growth responses of functional groups of tree species to drought in a C entral A frican semi‐deciduous moist forest. Species groups were automatically defined using a finite mixture model, which grouped species according to their growth model parameters. The growth model considered the variation in species response to drought, and the effect of competition for resources and of tree development stage on growth. Groups were further characterized by species functional traits. Nine species groups were identified. They differed in their ability to acquire, use and conserve resources, as suggested by their differences in maximum growth rate, regeneration guild, maximum dbh, wood density and leaf habit. The species were organized along a light requirement gradient that here closely matched a broader continuum of plant strategies for resource use, from slow‐growing shade‐tolerant conservative species to fast‐growing pioneer acquisitive species.
Tree growth decreased with drought intensity, and species drought tolerance was found to be related to resource use strategy: slow‐growing species using a conservative strategy were the least sensitive to variations in water availability, while fast‐growing species using an acquisitive strategy were the most sensitive.
Synthesis . Shade‐tolerant species, characterized by a low potential growth rate and thus a conservative strategy of resource use, were found to be the least sensitive to drought. This supports the hypothesis of a single axis summarizing multiple traits that represents a general trade‐off between the conservation and rapid acquisition of resources.
