Detection and measurements of indirect effects are often far from straightforward, and are mostly based on the intuition, common sense, and prior knowledge of any particular system. Abrams and co-authors described two major approaches adopted in ecological studies, namely theoretical and experimental. They stated that in practice, the theoretical and empirical approaches may be regarded as endpoints of a methodological continuum. Recently, however, we have argued that the methodological continuum to study indirect interactions is best represented by a triangle, with observational, experimental, and theoretical nodes.
Within the theoretical approach, observations (and/or carefully considered experimental data) are used together with theoretical considerations to construct a model capable of investigating interactions among the components incorporated in the model structure. This model is subsequently used to examine indirect effects between the components. There are a number of drawbacks of this approach, for example, difficulties related to obtaining sufficient details about the components represented in the models, unavoidable uncertainty as regards fluxes, parameters, initial values, etc. This uncertainty may mask the significance of the relationships studied, including indirect effects. Furthermore, as it is impossible to reproduce all the complexity of a real ecosystem, any model is a simplification of reality. Therefore, some of the potentially important interactions may be lost just by defining the model structure, while the importance of the others may be considerably altered.
Within the experimental approach, densities of individual species are manipulated (e.g., by total removal) in microcosms or experimental plots, and statistical analysis
(e.g., ANOVA, ANCOVA) are subsequently applied to estimate the magnitude of indirect effects of manipulations on densities of other species. It has been argued that this approach is best applied using a factorial design, where the densities of a number of components (e.g., species or trophic groups) are changed both alone and in combination. If implemented properly, this approach leads to a straightforward estimation of net effects. However, there is always a danger that some of the indirect interactions have not manifested owing to unavoidable time constraints of any experiment. Also, partitioning of the registered net effects may be subject to speculation. Experiments are often costly and by definition are limited by their design and the hypotheses tested. The simplicity of the experimental design may mask the significance of the relationships studied for trait-mediated effects; measurements of population abundances may need to be supplemented by behavioral observations, and/or biochemical, physiological, genetic, and other analyses. Furthermore, there is always a big question mark how applicable are the results obtained to the processes happening in the real world.
Among mathematical methods which have been used in studies of indirect effects in natural ecosystems are statistical methods (e.g., regression and correlation analysis, PCA, factor analysis, CCA, ANCOVA, ANOVA), simulation modeling (e.g., using 'what-if scenarios', sensitivity and elasticity analysis), and methods of network analysis. In particular, indirect interactions have often been analyzed using methods of network analysis. For example, Fath and Patten used methods of network analysis to show that, in the ecosystem context, direct transactions between organisms produce integral effects more positive than a simple sum of direct effects. This was in line with the view that mutualism is an implicit consequence of indirect interactions and ecosystem organization, and that the contribution of positive relationships should increase along the course of evolution and ecological succession.
It should be noted that all the methods so far applied to investigations of indirect effects have both advantages and limitations. Many of these have been previously addressed and no attempt to discuss the benefits and disadvantages of the techniques used to investigate indirect interactions has been done in this article. Neither was it intended to address any controversy and related discussion resulting from specific applications (and/or implications of such applications) of any particular method. It should be noted, however, that the methodological framework of 'comparative theoretical ecosystem analysis' (CTEA) (see below) suggests that the mathematical techniques may be best used in concert, thus allowing a detailed complementary insight into complex patterns of mechanisms underpinning dynamics of natural ecosystems.
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