Desirable mathematical properties of indicators for biodiversity change

Numerous indicator approaches are found in the scientific literature to describe changes in biodiversity. It is however far from clear which indicators are most appropriate and which are less suitable to summarize trends in biodiversity. One reason for this lack of clarity is that so far the mathematical properties of indicator approaches have had little attention. In this paper, we derive a number of desirable mathematical properties of indicators from economic price-index theory and apply these in the form of tests to 10 metrics to summarize changes in biodiversity. The metrics species richness, Simpson index, Shannon index, Buckland's modified Shannon index and Sorensen's similarity coefficient violate the monotonicity and proportionality test. The percentage of increasing minus declining species also fails the proportionality test, and in the case where trends are assessed relative to the preceding year, this metric also violates the identity test. Most of these indicators are sensitive to spatial scale. The arithmetic and geometric mean of population indices and the mean abundance have better mathematical performance, but the first two are sensitive to appearing and disappearing species in the system surveyed. The metric mean abundance however can only be applied under particular conditions and has some undesirable properties. Unlike the arithmetic mean, the geometric mean is not sensitive to the base year chosen and has the most favourable mathematical properties of the indicators evaluated. The geometric mean can be straightforwardly extended to take into account unequal values of species if desired.

[1]  F. Vivien,et al.  The convention on biological diversity: A conventionalist approach , 2005 .

[2]  Georgina M. Mace,et al.  A Framework for Improved Monitoring of Biodiversity: Responses to the World Summit on Sustainable Development , 2005 .

[3]  J. Biesmeijer,et al.  Biodiversity change is scale-dependent: an example from Dutch and UK hoverflies (Diptera, Syrphidae) , 2011 .

[4]  D. Noble,et al.  Developing indicators for European birds , 2005, Philosophical Transactions of the Royal Society B: Biological Sciences.

[5]  Robert K. Colwell.  Beta diversity: synthesis and a guide for the perplexed , 2010 .

[6]  Barry R. Noon,et al.  Utility and limitations of species richness metrics for conservation planning , 2006 .

[7]  E. Bayne,et al.  A new method to estimate species and biodiversity intactness using empirically derived reference conditions , 2007 .

[8]  William J. Sutherland,et al.  Ecological census techniques: A handbook: Edited by William J. Sutherland Cambridge University Press, 1996. £75.00/$125.00 hbk, £27.95/$44.95 pbk (xv + 336 pages) ISBN 0 521 66135 5 / 367921 1 , 1997 .

[9]  Ian F. Spellerberg,et al.  Monitoring Ecological Change: Index , 1991 .

[10]  S. T. Buckland,et al.  Monitoring change in biodiversity through composite indices , 2005, Philosophical Transactions of the Royal Society B: Biological Sciences.

[11]  William J. Sutherland,et al.  Ecological Census Techniques: Contents , 1996 .

[12]  M. Diekmann.  Species indicator values as an important tool in applied plant ecology – a review , 2003 .

[13]  V. Volterra.  Variations and Fluctuations of the Number of Individuals in Animal Species living together , 1928 .

[14]  F. Jiguet,et al.  Birds are tracking climate warming, but not fast enough , 2008, Proceedings of the Royal Society B: Biological Sciences.

[15]  Rudolf de Groot,et al.  A conceptual framework for selecting environmental indicator sets , 2008 .

[16]  M. Hisschemöller,et al.  Ecological Indicators: between the two fires of science and policy , 2007 .

[17]  S. E. Nielsena,et al.  A new method to estimate species and biodiversity intactness using empirically derived reference conditions , 2007 .

[18]  D. Waller,et al.  Biotic Impoverishment and Homogenization in Unfragmented Forest Understory Communities , 2004 .

[19]  R. Biggs,et al.  A biodiversity intactness index , 2005, Nature.

[20]  S. Quader,et al.  Improvements to the Red List Index , 2007, PloS one.

[21]  D. Noble,et al.  The generation and use of bird population indicators in Europe , 2008, Bird Conservation International.

[22]  J. Lockwood,et al.  Biotic homogenization: a few winners replacing many losers in the next mass extinction. , 1999, Trends in ecology & evolution.

[23]  S. Gaines,et al.  Species diversity: from global decreases to local increases , 2003 .

[24]  J. Olden,et al.  Biotic homogenization and conservation prioritization , 2007 .

[25]  Arco J. van Strien,et al.  Wild Bird Indicators: Using Composite Population Trends of Birds as Measures of Environmental Health , 2010 .

[26]  J. Randers,et al.  The Living Planet Index: using species population time series to track trends in biodiversity , 2005, Philosophical Transactions of the Royal Society B: Biological Sciences.

[27]  Bert M. Balk,et al.  Price and Quantity Index Numbers: Models for Measuring Aggregate Change and Difference , 2012 .

[28]  H. Van Dyck,et al.  Declines in Common, Widespread Butterflies in a Landscape under Intense Human Use , 2009, Conservation biology : the journal of the Society for Conservation Biology.

[29]  A typology of indicators of biodiversity change as a tool to make better indicators , 2009 .

[30]  Sutherland.  Ecological Census Techniques , 2006 .

[31]  Jim Schieck,et al.  Indices for monitoring biodiversity change: Are some more effective than others? , 2009 .

[32]  M. Heinrich.  Handbook of the Convention on Biological Diversity , 2002 .

[33]  J. Lamarque,et al.  Global Biodiversity: Indicators of Recent Declines , 2010, Science.