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. 2016 Jul;19(7):729-42.
doi: 10.1111/ele.12617.

N-dimensional hypervolumes to study stability of complex ecosystems

Ceres Barros  1   2 , Wilfried Thuiller  1   2 , Damien Georges  1   2 , Isabelle Boulangeat  3 , Tamara Münkemüller  1   2
Affiliations

N-dimensional hypervolumes to study stability of complex ecosystems

Ceres Barros et al. Ecol Lett. 2016 Jul.

Abstract

Although our knowledge on the stabilising role of biodiversity and on how it is affected by perturbations has greatly improved, we still lack a comprehensive view on ecosystem stability that is transversal to different habitats and perturbations. Hence, we propose a framework that takes advantage of the multiplicity of components of an ecosystem and their contribution to stability. Ecosystem components can range from species or functional groups, to different functional traits, or even the cover of different habitats in a landscape mosaic. We make use of n-dimensional hypervolumes to define ecosystem states and assess how much they shift after environmental changes have occurred. We demonstrate the value of this framework with a study case on the effects of environmental change on Alpine ecosystems. Our results highlight the importance of a multidimensional approach when studying ecosystem stability and show that our framework is flexible enough to be applied to different types of ecosystem components, which can have important implications for the study of ecosystem stability and transient dynamics.

Keywords: Climate change; ecosystem stability; land-use changes; n-dimensional hypervolumes; perturbations.

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Figures

Figure 1
Figure 1
The utility of phase portraits for studying stability. A system of a) two species can be represented by b) a classical two-dimensional phase portrait. The system’s state at equilibrium is represented by a circling behaviour in b) that corresponds to oscillations of species abundances in a). This concept can be extended to higher dimensions, where the c) dynamics of a three species community are represented by d) a three-dimensional phase portrait. In multidimensional space, states at equilibrium become clouds of points in d), which can be represented by n-dimensional hypervolumes (schematic cubes). Comparisons between hypervolumes can be related to the ball-and-cup analogy of resilience, as they indicate departures from the initial state that can happen e) within the same basin of attraction, f) when the system shifts to an alternative stable state, or g) when the equilibrium is displaced (see Beisner et al. 2003; Horan et al. 2011).
Figure 2
Figure 2
Framework scheme. Several types of time series data can be used (Step 1). In our study case, we used simulated plant functional groups’ (PFG) abundances and community weighted mean (CWM) trait values per habitat-land-use combination, under a given scenario of land-use and/or climate changes. Variables used for hypervolume calculations should be scaled and uncorrelated (Step 2), which was ensured by selecting axes extracted from Principal Components Analyses (PCAs) on scaled time series of PFG abundances and of CWM trait values. Pre- and post-perturbation hypervolumes are then calculated using, in this example, the PCAs factor scores referring to control (scenario 1) and post-perturbation data (remaining scenarios), and then compared (Step 3). Comparisons between hypervolumes can be complemented using other metrics (Step 4) for a further analysis of community changes. In Step 3, ‘POC’ stands for ‘proof-of-concept’ hypervolumes (see methods section ‘Step 3. Comparing hypervolumes to analyse community changes’).
Figure 3
Figure 3
Full system trajectories under different scenarios and land-use practices. The full trajectories of thickets and scrubland vegetation are shown for three scenarios of climate and/or land-use changes, under three types of land-use practices. The first 500 years correspond to the control scenario (in orange), followed by another 500 years of climate and/or land-use changes: land-use abandonment without and with climate change in blue and red (scenarios 2 and 4, respectively) and land-use intensification in purple (scenario 3). Since we are graphically constrained to three dimensions, we plotted the trajectories using relative abundances of chamaephyte (full lines), herbaceous (dashed lines) and phanerophyte (dotted lines) plant functional groups (by adding up separate group’s abundances per life form type). The three-dimensional plot in b) corresponds to trajectories in non-disturbed areas – first two panels in a) – whereas in c) it corresponds to trajectories in intensified grazed areas – last panel in a).
Figure 4
Figure 4
Overlap in disturbed and non-disturbed areas. Proportion overlap between control and post-perturbation hypervolumes of a,c) PFG raw abundances – a) and c) only differ in the y-axis scale – and b) CWM trait values. The proportion of overlap (overlap) was calculated as the ratio between the intersection volume and the total volume occupied by the two hypervolumes (standard errors shown as error bars). Observed mean overlaps are shown by scenario, across all habitat types and grouped by disturbed areas (areas under present grazing or mowing regimes and areas that will become grazed on mown under scenarios of land-use intensification) and non-disturbed areas (all areas that are not currently grazed or mown and those that will remain so, under land-use intensification scenarios). Standard errors are shown as error bars. Comparisons between proof-of-concept (‘POC’) and control scenario hypervolumes are shown in a) and b), but not in c), so that overlap values obtained in other scenario comparisons can be seen.
Figure 5
Figure 5
Mean distances and changes in size, in disturbed and non-disturbed areas. Mean centroid distances between control and post-perturbation hypervolumes and differences in their sizes (post-perturbation minus pre-perturbation; Δsize) are shown for a,c) PFG raw abundances and b,d) CWM trait values. Negative Δsize values indicate that the post-perturbation hypervolume was smaller than the pre-perturbation hypervolume, and vice-versa for positive Δsize values. Both metrics are shown by scenario, across all habitat types and grouped by disturbed areas (areas under present grazing or mowing regimes and areas that will become grazed on mown under scenarios of land-use intensification) and non-disturbed areas (all areas that are not currently grazed or mown and those that will remain so, under land-use intensification scenarios). Standard errors are shown as error bars. Comparisons between proof-of-concept (‘POC’) and control scenario hypervolumes are also shown.
Figure 6
Figure 6
Temporal stability measured by hypervolume overlap. Temporal stability was analysed by modelling the temporal response of the proportion of overlap (overlap) under different habitat-land-use combinations, using generalised additive models (GAMs) with a Gaussian smoother fitted for each habitat-land-use combination. Each coloured point corresponds to the comparison between a hypervolume at a given time slice and the first hypervolume, with colours referring to land-use (the first year of each 15 year time slice is indicated in the x-axis). Dashed vertical lines indicate the start and end of simulated climate changes.
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