Clustering species using a model of population dynamics and aggregation theory
Introduction
The high species diversity of some terrestrial or sea ecosystems such as tropical rain forests or coral reefs has raised many questions about their functioning (Hubbell and Foster, 1986, Hubbell, 1997, Whitmore, 1998). Ecologists have tried to simplify this diversity by assigning species to functional groups, i.e. groups of species that have the same functions in the ecosystem (Díaz and Cabido, 1997, Köhler et al., 2000, Fonseca and Ganade, 2001, Baker et al., 2003, Mcgill et al., 2006). Even if marked patterns such as the dichotomy between pioneers and climax species in tropical rain forests have been identified (Swaine and Whitmore, 1988, Baker et al., 2003), the definition of functional groups has remained an inaccessible Holy Grail, the distribution of species along functional gradients always being continuous rather than discrete. To build functional groups, ecologists typically grouped species on the basis of their similarity with respect to ecological characteristic or functional traits (Gourlet-Fleury et al., 2005). The methods used to group species were mainly cluster analysis, when they were not simply an educated guess.
People interested in the modelling of the dynamics of species-rich ecosystem have also paid attention to the grouping of species. The motivation of modellers was basically not to find functional groups, but rather to compensate for the scarcity of data for the less abundant species, that are also the most numerous. The scarcity of data for these rare species prevented from estimating the parameters of the models of population dynamics with enough precision. By pooling species, more sizeable data sets could be formed and reliable parameter estimates could be obtained. Despite this motivation, modellers have mainly stuck to the paradigm of functional groups, i.e. the grouping of species was made on the basis of their similarity with respect to their characteristics (Köhler and Huth, 1998, Köhler et al., 2000). Often the groups of species were built independently from the model of population dynamics (e.g. Favrichon, 1998). Sometimes the building of the groups of species was linked to the model of population dynamics, the grouping being based on the residuals of the model (Vanclay, 1991a, Vanclay, 1992, Gourlet-Fleury and Houllier, 2000).
When pooling species into a group, the number of available observations increases and thus the variance of the estimators of model parameters decreases. But at the same time, an estimation bias is introduced since the values of the parameters for a given species are confounded with those of the group. The wider the group is, the larger the bias is and the smaller the variance is. The bias vanishes when each group is a singleton restricted to a single species, but the variance is then maximum. To assess the interest of a species grouping from the modeller’s point of view, it is thus necessary to compute the quadratic error that results from the groups, where the quadratic error is the square bias plus variance.
This study aims at assessing the interest of groups of species from the modeller’s point of view, i.e. on the basis of the quadratic error on model’s predictions that it brings. The null grouping is when there are as many groups as species and each group identifies with a species (in other words, no effective grouping is made). A grouping of species will be considered as justified if it brings a lower quadratic error than the null grouping. The quadratic error will be interpreted as a disaggregation error in the context of aggregation theory. Aggregation theory deals with the error implied when shifting the level of description of a system from a detailed level to an aggregated less-detailed level (Iwasa et al., 1987, Iwasa et al., 1989, Ritchie and Hann, 1997). In the present case, the aggregation consists in replacing s species with g groups of species. Once the disaggregation error is defined, a method for defining groups of species follows by searching, for a given number g of groups, the grouping that minimizes the disaggregation error.
In this study, we presented a general framework useable with any model of population dynamics. We then applied the grouping strategy to 94 well represented tree species of a tropical rain forest in French Guiana. We chose to use a matrix model for size-structured populations to model population dynamics, and we addressed three questions: (i) How to build the disaggregation error? (ii) Is there a statistical interest to build groups, compared to null grouping? (iii) What happens if the groups are built according to a different strategy, either using the same model of population dynamics (groups of Favrichon, 1994, resulting from a cluster analysis), or using a different model (groups of Gourlet-Fleury and Houllier, 2000)?
Section snippets
Aggregation diagram
Let s be the number of species. For each species , observations , …, are available. Each observation is considered as a random variable drawn from a distribution that depends on unknown parameters . These parameters are those of the model of population dynamics. Expectations and variances will refer to the distributions . Parameters are estimated from observations using an estimator . The model of population dynamics is here considered as an application
Species characteristics
Fig. 3 shows the correlation circle of the PCA of the table giving the vital rates (, , ) for each species. The recruitment rate is positively correlated with the mortality rate, and together these two rates define the turnover rate. The turnover rate explains the first axis of the PCA. The upgrowth transition rate is almost independent from the turnover rate and explains the second axis of the PCA. The mortality rate is actually close to the recruitment rate for all species, so that
Clustering method
On the basis of the Usher matrix models and for the 94 species studied at Paracou, the choice of modellers to build groups of species is justified: for reasonably well chosen groupings, the gain in variance that results from data pooling over-compensates in terms of quadratic error for the bias that results from the groups. The positive balance in terms of quadratic error is obtained for a large range of number of groups ( in the present case) and for different grouping methods. Only when
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