COMPACTNESS VERSUS INTERIOR-TO-EDGE RATIO; TWO APPROACHES FOR HABITAT’S RANKING Attila R. Imre KFKI Atomic Energy Research Institute, H-1525 Budapest POB 49, Hungary. E-mail:
[email protected] Received 22 April 2005; accepted 14 December 2005
ABSTRACT In landscape ecology spatial descriptors (or indices) can be used to characterize habitats. Some of these descriptors can be used for habitat’s ranking; this ranking is very important for conservation purposes. We would like to show that two traditional descriptors, namely the compactness and interior-to-edge ratio can give contradictory results. Being the second one is a more relevant descriptor, we would like to propose to avoid the further use the compactness in habitat’s ranking.
Key Words: landscape ecology, compactness, interior-to-edge ratio, fragmentation, habitat, ranking, Diamond’s rule.
INTRODUCTION In landscape ecology isolated habitats can be ranked by their descriptive attributes and/or by their spatial attributes (also called spatial descriptors or indices). Having isolated patches originated from a bigger primal habitat, one can expect that the descriptive attributes (for example the type and density of trees) are more or less similar. In that case the primary tool for the comparison and ranking of the habitats has to be based on the spatial attributes (O’Neill, et al., 1988; Forman, 1990). The most widely used spatial descriptors are the area (A), interior-to-edge ratio (I/E), and compactness (C) (see for example Patton, 1975; Petersen and Turner, 1994; Baskent and Jordan 1995; Davidson 1998 and Bogaert et al., 1998). Accepting the fact that due to the limited financial resources not all fragmented habitats can be saved from further deterioration, one need a good way for ranking the habitats and protect/develop the best ranked ones. This is the reason why proper ranking is quite important. While the area and compactness are genuine geometrical descriptors, interior-toedge ratio has biological meaning. The edge is the outer part of the patch, an ω-wide belt (Figure 1), shielding the interior from the external disturbances. Higher I/E ratio means relatively bigger undisturbed central area which is favourable from ecological point of view (Saunders et al., 1991). Obviously the most important descriptor is the area; it cannot be denied that a big habitat is better than a small one, even when the shape of the big one is not as good as for the small one, except some extreme case when the shape is very much distorted, irregular or the habitat is very narrow (i.e. the whole area is in the edge zone). But when the studied habitats have approximately similar areas, one should choose an other descriptor for ranking. Diamond’s rule (Diamond, 1975) (which ranks the most rounded, i.e. most compact habitat as the best one) is a traditional method for habitat-ranking (Patton, 1975). For long Acta Biotheoretica (2006) 54: 21–26 DOI: 10.1007/s10441-006-5909-0
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Springer 2006
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ATTILA R. IMRE Figure 1. Schematic representation of the the interior (white), edge (grey) and edge thickness (arrow) on a circular patch
time it was believed that higher compactness means higher I/E value, i.e. the Diamond’s rule is correct – Diamond’s paper has been cited 385 times since 1975 (ISI Web of Science)-, but as it as been shown earlier (Imre, 2001, Bogaert, 2001) these two quantities are not strongly correlated, i.e. they can be changed independently. While compactness is a size independent quantity, I/E ratio depends not only on the shape of the habitat but also on the area, therefore ranking a set of habitats with different shape and with different area one might obtain contradictory results (a particular habitat might rank very good using compactness but worse by using I/E value) and vice versa (Imre, 2001). In landscape ecology, the proper or inproper use of the spatial descriptors is a current issue (Li and Wu, 2004). Previously it has been shown that compactness and I/E ranking might have contradictory results applied on a set of patches with different areas (Imre, 2001). In this short paper we would like to demonstrate that the two traditional rankingdescriptors, namely the compactness and I/E value do not have strong correlation, not even for patches with same area. Therefore it is possible that comparing habitats with more or less equal area, compactness and I/E value would give contradictory ranking. Because we believe that the I/E value (together with the habitat’s area) is more important descriptor for the habitat’s quality than the compactness, therefore we would like to propose to avoid the use of the Diamond’s rule whenever it is possible.
COMPARISON In general, the perimeter and the area of an object embedded into two-dimension can be described as: P = CP L
(1)
A = CA L2
(2)
and where P, A, L, C P and C A are the perimeter, area, a characteristic linear size (like radius, diameter, side-length, etc) and the two corresponding form-factors, respectively. For example for a circle (when L is the radius) C P = 2π and C A = π , while for a square (when L is the side-length) C P = 4 and C A = 1. Obviously other characteristic linear sizes (diameter instead of radius, diagonal instead of side-length) could be chosen; this choice would influence the value of the form factors but would not influence the conclusions of this paper (see the Appendix).
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From these two equations one can define a size-independent compactness as: A CA C= 2 = 2 P CP
(3)
Because in this paper we would like to compare objects with same area, therefore it is advisable to use area instead of linear size (from equation (2)): A L= (4) CA The edge and the interior with ω edge depth can be calculated as: 2 A E = C A L 2 − C A (L − ω)2 = A − C A −ω CA and
I = C A (L − ω) = C A 2
A −ω CA
(5)
2 (6)
therefore for a general two-dimensional object the I/E value can be written as: I A = −1 (7) E A 2C A ω − C A ω2 CA Comparing equation (3) and equation (7), one can see that (even in case of constant area) the compactness depends on C A and C P , while I/E depends on C A and ω. The edge depth (ω) is not only a geometrical quantity; it depends greatly on the type and amount of external disturbances (Gehlhausen et al., 2000). It is usually a few tens of meters but some times it can be as high as 400 meter (Chen, 1991). Habitat patches can be originated from one big habitat by fragmentation, therefore it can be assumed that all patches have more or less the similar surroundings, i.e. they have the same type and amount of external disturbances. Therefore one might fix ω, keeping only the C A dependence for I/E. In that case it can be seen that while the compactness depends on both form-factors (C A and C P ), the I/E value depends only on C A (one can change compactness by changing C P only while keeping A and C A – and therefore I/E - on constant value). Therefore one might conclude that I/E and C can be changed independently, as long as C P and C A are independent. After this conclusion, the only remaining step is to prove that the two formfactors are independent; the easiest way to do that is to show a simple example. Having an equilateral triangle and an arrow-head (Figure 2) and choosing the side-length (in this case the side length is equal with the distance of the farthest points) as characteristic linear size one can see that while C P are equal for the two shapes, C A s can be different (depends on gamma), showing that these two quantities are independent. To demonstrate the applicability of the conclusion that I/E and C are not correlated and therefore they can give contradictory ranking results, two pine forest fragments (Pinus sylvestris L.) from the Belgian Kempen region (Bogaert et al., 1999; Imre and Bogaert 2004, Imre and Bogaert 2006) was analyzed. The black-and-white map of the two fragments can be seen on Figure 3. It should be mentioned here that instead of artificially scaling two different-shaped objects to the exactly same area for our purpose,
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Figure 2. Two objects to demonstrate the independence of form factors. The perimeter form-factors are the same for both objects (C P = 3), while the area form-factor (C A ) is 0.866 for the triangle while its can be any value (depending on the value of gamma) between zero and 0.866.
Figure 3. Two pine forest patches (Bogaert et al., 1999; Imre and Bogaert, 2004) with approximately the same are, where the ranking by compactness and ranking by I/E values give opposite result. The edge depth has been chosen for 10 meters. a: A = 23.8 hectare, C = 0.02065, I /E = 4.05; b: A = 23.6 hectare, C = 0.02083, I /E = 3.68. In compactness ranking, patch “b” is better than patch “a”, while in I/E ranking, patch “a” is better than patch “b”.
real (although not exactly equal area) samples was chosen for this demonstration. The areas of these fragments are approximately the same (a little bit below 24 hectare), therefore for ranking one should use compactness or I/E value. Using compactness ranking, patch “b” is better than patch “a”, while in I/E ranking, patch “a” is better than patch “b” (the corresponding values can be seen in the figure legend). Although the difference is small, this contradiction demonstrates the problem of using compactnessranking in real habitats.
CONCLUSIONS In this short paper we demonstrated that the compactness and the I/E values are not necessarily correlated; it is possible to find a set of shapes with the same area where the ranking by compactness (i.e. using Diamond’s rule) and the ranking by I/E values would give the contradictory result. This contradiction is caused by the difference of the perimeter form factors (C P ) of the patches. This problem – together with the one described earlier (Imre, 2001) where the difference in the area of the shapes can cause contradiction in the rankings by compactness and by I/E value – shows that compactness
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is not a proper tool of habitat’s ranking. Because I/E values has ecological significance (see “edge” effect (Forman, 1995)) while compactness is a purely geometrical descriptor, therefore we would like to propose to avoid the further use of compactness as a ranking tool to avoid contradiction in the ranking.
APPENDIX The choice of L (characteristic linear size) has some minor influence on equations (5)–(7). For example in case of a circular habitat, choosing the radius as L would give as equations (5)–(7) in this present form but choosing L as diameter would change them. In that case L-ω should be replaced with L-2ω and equation (7) would turn to: I A −1 (8) = E A 2 4C A ω − 4C A ω CA This difference would not change the conclusion that I/E depends only on C A while compactness depends on both form factors. Similarly, any other choice would change only some constants in equation (7) but they would not influence the final conclusion.
REFERENCES Baskent, E.Z. and Jordan, G.A. (1995). Characterizing spatial structure of forest landscapes. Canadian Journal of Forest Research 25: 1830–1849. Bogaert, J., P. Van Hecke, D. Salvador-Van Eysenrode, and I. Impen. (1998). Quantifying habitat edge for nature reserve design. Coenoses 13: 131–136. Bogaert, J., P. Van Hecke, and I. Impens, (1999). A reference value for the inerior-to-ege ratio of isolated habitats. Acta Biotheoretica 47: 67–77. Bogaert, J. (2001). Size dependence of interior-to-edge ratios: size predominates shape. Acta Biotheoretica 49: 121–123. Chen, J. (1991). Edge effects: microclimatic pattern and biological responses in old-growth Douglas-fir forests, Ph.D. dissertation, Univ. of Washington, Seattle, USA. Davidson, C. (1998). Issues of measuring landscape fragmentation. Wildlife Society Bulletin 26: 32–37. Diamond, J.M. (1975). The island dilemma: lessons of modern biogeographic studies for the design of natural reserves. Biological Conservation 7: 129–146. Forman, R.T.T. (1990). Ecologically sustainable landscapes: the role of spatial configuration. Changing landscapes: an ecological perspective, I.S. Zonneveld, R.T.T. Forman (eds.), Springer-Verlag, New York, pp. 261–278. Forman, R.T.T. (1995). Land mosaics: the ecology of landscapes and regions. Cambridge University Press, Cambridge, p. 632. Gehlhausen, S.M., M.W. Schwartz and C.K. Augspurger (2000). Vegetation and microclimatic edge effects in two mixed mesophytic forest fragments. Plant Ecology 147: 21–35. Imre. A.R. (2001). About the ranking of isolated habitats with different shapes: and interior-to-edge ratio study. Acta Biotheoretica 49: 115–120. Imre, A.R. and J. Bogaert (2004). The fractal dimension as a measure of the quality of habitats. Acta Biotheoretica 52: 41–56. Imre, A.R. and J. Bogaert (2006). The Minkowski-Bouligand dimension and the interior-to-edge ratio of habitats, Fractals 14: 49–53.
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ISI Web of Science, http://wos01.isiknowledge.com. Li, H. and J. Wu (2004). Use and misuse of landscape indices. Landscape Ecology 19: 389–399. Peterson, G.W. and R.E. Turner (1994). The Value of Salt Marsh Edge vs. Interior as a Habitat for Fish and Decapod Crustaceans in a Louisiana Tidal Marsh. Estuaries 17: 235–262. Patton, D.R. (1975). A diversity index for quantifying habitat “edge”. Wildlife Society Bulletin 3: 171–173. Saunders, D.A., R.J. Hobbs and C.R. Magules (1991). Biological consequences of ecosystem fragmentation: a review. Conservation Biology 5: 18–32.
