The Promise and Limitations of Spatial Models in ...

The Promise and Limitations of Spatial Models in Conservation Biology Uno Wennergren; Mary Ruckelshaus; Peter Kareiva Oikos, Vol. 74, No. 3. (Dec., 1995), pp. 349-356. Stable URL: http://links.jstor.org/sici?sici=0030-1299%28199512%2974%3A3%3C349%3ATPALOS%3E2.0.CO%3B2-X Oikos is currently published by Nordic Society Oikos.

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OIKOS 74: 349-356. Copenhagen 1995

The promise and limitations of spatial models in conservation biology Uno Wennergren, Mary Ruckelshaus and Peter Kareiva

Wennergren, U., Ruckelshaus, M. and Kareiva, P. 1995. The promise and limitations of spatial models in conservation biology. - Oikos 74: 349-356. We review the application of spatially explicit models to conservation biology, and discuss several problems regarding the use of these models. First, it is unclear whether increasing the complexity of spatial models to include age structure enhances our ability to predict population growth in temporally varying environments. Second, if simulations of individual behavior are used to identify options for landscape management, predictions about the fate of dispersing organisms are likely to be hugely in error unless dispersal attributes are known to a far greater degree of accuracy than is reasonable to expect. Third, the compelling metaphor of extinction debts resulting from habitat destruction in competitive communities stands firm as a cautionary tale even when the metapopulation models include multiple trophic levels - but the question remains of how widespread and tight are the tradeoffs between dispersal capacity and competitive superiority. Given the shakiness of spatial models as a foundation for specific conservation recommendations, we conclude they may be more useful as a tool for exploring the design of spatially-structured monitoring schemes, so that management mistakes might be detected before they become irreversible. U . Wennergren, Dept of Biology, Linkoping Unit:, S-581 83 Linkoping, Sweden. - M . Ruckelshaus, Dept of Biological Science, Florida State Univ., Tallahassee, FL 323062043 USA. - P.Kareiva, Dept of Zoology, Univ. of Washington,Seattle, WA 98195, USA.

Until recently, models of spatially structured populations were largely of interest to only a handful of theoretical ecologists. However, habitat fragmentation and human disturbance of landscapes have been so vividly portrayed as major ecological perturbations, that spatial models concerning the consequences of landscape change now represent an extremely vigorous area of research (Kareiva and Wennergren 1995). Unfortunately, the popularity of spatially explicit models is no guarantee that this research effort proceeds in fruitful directions. In particular, when spatial models enter the arena of conservation biology, they face two dangers: (1) they are interpreted too literally in situations where they represent metaphors more than predictive models, or (2) in the interest of making the models realistic, researchers render them so unwieldy and "data hungry" that they are impractical to use. In this paper we review several ave-

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Accepted 29 August 1995 Copyright O OIKOS 1995 ISSN 0030-1299 Printed in Denmark - all rights reserved OIKOS 74:3 (1995)

nues of spatial modeling that have major implications for conservation biology. Our purpose is to raise questions about the use of these different modeling approaches in conservation biology, keeping in mind the competing demands of simplicity (for the sake of minimal data requirements) and realism (for the sake of particular applications).

When do we need spatial structure in simple linear models of population growth? Linear matrix projection models (i.e., models with no density dependence) are often used to gain a quick assay

of the viability of endangered populations. Although these models surely are too simplistic to offer genuine predictions, they provide a valuable connection between field data on age-structured or stage-structured demographic rates and a population's prospects for persistence. These linear models can also be used to explore the effectiveness of different management efforts that are designed to rescue a species from its decline (Crouse et al. 1987, Doak et al. 1994). Conspicuously lacking from published age-structured viability analysis is a consideration of spatial structure and population subdivision, even though conservation biologists give great attention to concepts of metapopulations and habitat fragmentation. Of course, there are numerous spatial models or metapopulation models in conservation, but they typically neglect age-structure (e.g., Lindenmayer and Lacy 1995). The dilemma is that including both spatial subdivision and age-structure in the same model inevitably makes a model very unwieldy. The competing complexities of age versus spatial structure suggests an interesting theoretical question: when can projection matrices with spatial and age-structure be safely collapsed into simpler models? Matrix theory tells us that age and spatially structured populations (see Appendix A for the appropriate matrix representation) with constant dispersal and demographic rates will converge to a stable spatial and age strucure and will eventually grow or shrink at a constant rate (the population's intrinsic rate of change or dominant eigenvlaue of the matrix; Caswell 1989). Even if the environment varies modestly, one can use stochastic theory to show that there is an expected asymptotic rate of population change with a stable probability distribution describing variation about this asymptotic rate (Cohen 1979, Tuljapurkar 1990). In contrast to smallscale environmental changes, none of the existing theory strictly applies for greater magnitudes of environmental variation, yet many organisms experience large fluctuations in birth and death rates. For example, as a result of a 10°C-change in temperature, insect reproduction may vary tenfold (Bieri et al. 1983); and as a result of random droughts, amphibian reproduction is often episodic with total reproductive failure in drought years, and tenfold variation even among the years in which reproduction successfully occurs (Pechmann et al. 1991). We decided to model such dramatic variation by introducing it to a spatially and age-structured matrix (see Appendix A), and then to generate population trajectories. Our question was whether the random population trajectories generated by the full spatially and age-structured model could be approximated by either a spatial model without age-structure (Appendix B), or a scalar model with neither age- nor spatial structure (Appendix C). In lieu of an exhaustive analysis, for illustrative purposes we report here only a subset of our studies. Specifically, we considered two types of life cycles: one represented by an owl and an aphid (Fig. 1). We used cyclic variation drawn from a common probability distribution that was synchronized among patches but resulted in differing re-

Spotted owl type

juveniles

adults

--in optimal

environment

Aphid type

juveniles

adults

Fig. 1. Life cycles of spotted owl and aphids, illustrating qualitative differences as well as the full range of variation in fecundity or survival allowed in the simulation model with environmental variability. The demographic data used to generate these life cycles came from McKelvey et al. 1993 and from Bieri et al. 1983.

alized population trajectories in each patch. We imagined demographic rates to vary with "patch quality", with patch quality varying from 0 to 1 and having the effect on demography depicted in Fig. 2. Then patch quality itself was simulated as varying through time in a cyclic fashion, at a rate that would prohibit using a Taylor expansion to estimate sensitivity of population growth rate on changes in particular demographic parameters. This simulates large-scale climatic fluctuations driving demographic variations, but exerting unequal effects on different subpopulations. The magnitude of the variation in each "patch" or subpopulation ranged from four-fold to seven-fold within any cycle of variation (depending on the particular sequence of random draws). Using this approach, one can ask how much error is entailed in predicting long run stochastic growth rate if the full model (i.e. data from a full age-structured population broken into different patches connected by dispersal) is collapsed so that age-structure is neglected, or such that both age and spatial structure are neglected. Since the answer may depend on where in the life cycle variation enters (through reproduction or survival) and on what

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Fig. 2. The effect of patch quality on population growth rates as a result of changes in either juvenile survivorship, adult survivorship, or fecundity. These changes in growth rate result from changes in specific demographic entires ranging between the default maximum value and 0.1 of that maximum value. This variation in matrix entries is so large that any analytical treatment of the sensitivity of population growth rates is not appropriate (because the perturbations are not small).

Table 1. The set of simulations run for the life history trpes of spotted owl and pea aphid. The simulations were made in two steps: first an initial 300 timesteps with the full matrix model so that the age and stage structure stabilized as much as they were going to, and then for another 300 timesteps when the three models were run in parallel. We ran 5 replicates of each combination of dispersal class and source of demographic variability as depicted in this table (i.e., five replicates of all nine combinations). Effect of environmental variation on: Juvenile survivorship

Adult survivorship

Fecundity

n =5 5 5

5 5 5

5 5 5

Newborns Subadults Adults

Dispersing age class:

life stage disperses, we considered several different life history scenarios (see Table 1). In Fig. 3, we see that in some cases even though temporal variation is large, the behavior of the complicated model can be well-described by s simple scalar model. In fact, for both the spotted owl and the aphid life cycles, a scalar model predicts long-run growth rate within 10% of the "true" value if demographic variation enters through changes in fecundity as opposed to survival. Empirical studies suggest that natural variation in fecundity will often be greater than variation in survivorship for annual plant species (Schaal 1984, Silvertown et al. 1993); this result has also been observed for long-lived animals such as the spotted owl (McKelvey et al. 1993). Not only can the magnitude of fecundity variation be greater than the

range in survival probabilities, but population growth rates have been found to be most strongly affected by variation in fecundity in a number of plant species (van Groenendael 1985, Caswell 1989, Horvitz and Schemske 1995). Theoretically, it remains to be seen whether variation in fecundity is generally more likely to allow collapsing of spatial and age-structured dynamics down to a scalar model. Our simulations certainly make it clear that under some circumstances subdivided populations with age structure can be adequately described by simple scalar models. This points out an important area for future theoretical work: it would be valuable to know in advance what patterns of demographic and environmental variation lend themselves to simplified models, and conversely what patterns require detailed spatiallystructured and age-structured models.

Uncertain estimation of model parameters and the limitations of landscape models with individual behavior juv

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Fig. 3. The errors that occur when one approximates a complex age and spatial model (i.e., Appendix A) with a simple scalar model or patch model. The errors depend on what part of the demography varies over time and what age classes disperse. The italicized juv, sub and ad refer to which age classes disperse (juvenile, subadult or adult). OIKOS 74:3 (1995)

Enthusiasm for simulating the dispersal behavior of individuals in fragmented landscapes is widespread among conservation biologists (Pulliam and Dunning 1995). The point of such an exercise is to use the collective success of a population of individuals who disperse and reproduce to predict the merits of alternative land management schemes (Turner et al. 1995). While the goals of these individual behavior models are admirable, it is questionable whether adequate data exist to support those goals. In particular, virtually all spatially explicit behavior models require a parameter that describes the mortality risk while dispersing, as well as the dispersal capacity of individuals. It is extraordinarily difficult to estimate these dispersal parameters for any species, much less an endangered species. Thus, models of this genre often rely on observations from only a handful of organisms, scattered among different sites and different years, perhaps even studies via different methods (e.g., Verner et al. 1992). We believe that one should expect large errors in estimating dispersal attributes from the scant field data typical of threatened or endangered spe35 1

when one includes population dynamics that include some fashion of density-dependence. Clearly we need to think more carefully about the application of spatially explicit population models that are based on simulations of individual behavior - the "realism" of these models is no guarantee of their usefulness. ERROR TYPE 0 DISPERSAL

LANDSCAPE

010 ERROR IN MODEL PARAMETER

Fig. 4. Prediction error as a function of input error in dispersal mortality and landscape parameters. Prediction and input errors are plotted for organisms dispersing in landscapes with 16% suitable habitat. Results shown include landscape errors (2, 8 and 16% of landscape quality is misclassified) and dispersal mortality errors (2, 8, 16, 24 and 32% overestimation of mortality during dispersal) resulting from organisms dispersing through landscapes filled with square and linear habitat patches 9 and 16 cells in size.

cies. Consequently, we became interested in the translation of error in estimating attributes of dispersal behavior into error in predicting what fraction of dispersers successfully locate habitat patches. To address this question we simulated individuals searching for habitat patches via a random walk in different landscapes. Obviously the answer depends on what proportion of the landscape is suitable habitat - if this proportion is either very high or very low, even an error-riddled model will successfully predict that virtually all dispersers succeed (when most of the landscape is suitable) or virtually all of the dispersers fail (when essentially none of the habitat is suitable). But if 2% to 25% of the landscape is suitable, modest errors in estimating attributes of dispersal behavior propagate into huge errors in predicting the success of individuals in a variety of habitat patch configurations (Fig. 4). In contrast, errors in categorizing and mapping habitat patches in general matter much less than do errors in dispersal behavior (Ruckelshaus', Hartway and Kareiva, unpubl.). These results do not depend qualitatively on the way searching movement is modeled, and hold up with correlated random walk movement as well as with a capacity of long-distance orientation towards suitable patches. The grim message emerging from this analysis is that, however appealing may be spatially explicit models of individual behavior, they are likely to be highly unreliable given our uncertainties about dispersal behavior and rates. The only solution may be to seek alternative, less "data-hungry" modeling approaches or to collect better information on dispersal behavior (while worrying a bit less about the details of landscape maps). Alternatively, there is some hope that errors in predicting dispersal success are dampened

Extending the metapopulation concept to communities suffering from habitat destruction Recently, Tilman et al. (1994) extended the application of single-species metapopulation models to communities of interacting competitors, with interesting implications for biodiversity preservation in the face of habitat destruction. Tilman et al's approach is the latest in a long series of similar models in which the idea of "patches in a spatial world" is used for inspiration, but the model itself is nonspatial (in the sense that no patches or organisms are assigned to any real position, and all patches are equally accessible to one another). Nonetheless, these nonspatial metaphors are valuable because they focus our attention on the importance of dispersal. Metapopulation models are especially useful for illustrating that if dispersal rates are low, the destruction of habitat can cause species extinctions because the colonization of empty suitable patches is impaired by permanent removal of habitat from the landscape. Tilman et al. emphasize the additional importance of tradeoffs between dispersal ability and competitive ability - pointing out that when such tradeoffs exist, habitat destruction can threaten even common species (because these competitively superior species are comparatively poor dispersers), and that the loss of these common species will lag behind the immediate removal of habitat, producing an "extinction debt". This is a powerful metaphor, which should motivate field ecologists to think more carefully about dispersal of all species, not just the rare species that already are listed as endangered and threatened. Because Tilman et al's model is restricted to competitive interactions within one trophic level, it is interesting to ask whether the "extinction debt" generalizes to multitrophic-level interactions. By adding a second trophic level to Tilman et al's basic metapopulation competition model, we learned that extinction debts may be exacerbated when an additional trophic level is added (Fig. 5). It also appears that in a two-trophic level model, the higher trophic level is more vulnerable to an extinction debt (Fig. 5). Clearly, the presence of an extinction debt hinges on a tradeoff between dispersal and the ability to succeed in a patch after arrival. In addition to seeking experimental confirmation of such a fundamental tradeoff, it would pay to examine models in which the tradeoff had been modestly relaxed - thereby asking whether an extinction debt still occurs with only a loose OIKOS 74:3 (1995)

- fraction of habitats occupied by herbivore species i, p, fraction of habitats occupied by predator species i, c, -dispersal rate of species i, m! - extinction probability for species i, D fraction of habitats removed from the community.

v,

species lost in a *yStetem without predators

4

herbivore species

o predator yrecies

0%

50% habitat destruction

1O O X

Fig. 5. Species extinctions in a metapopulation due to habitat destruction, assuming the following metapopulation model with two trophic levels:

correspondence between dispersal and competitive abilities.

Given the limitations of Can We use theory to suggest monitoring schemes for predicting population change? If spatial models will not be able to provide much more than general "alerts" regarding the dangers of landscape fragmentation, it may be useful to pay greater attention to spatially explicit monitoring schemes designed to predict population trajectories. As part of an experimental study of habitat fragmentation in the devastated zone surrounding Mount St. Helens (see Turchin and Kareiva

The terms in the herbivore equation are as follows: number of new patches occupied next time step: the bracketed term is the fraction of potentially occupiable patches, depending on the amount of habitat destruction, the fraction of patches with superior or equal competitors, and the fraction of patches with predator species. The loss in occupancy comes from mortality or the presence of superior competitors or predators. The predator equation follows the same scheme; with the exception that the fraction of potentially occupiable patches is a function of patches occupied by herbivores minus the fraction already occupied by superior or equal predator species. The basic dynamic is that a species compensates for lower competitive ability by a high dispersal rate, allowing it to occupy vacant patches. Overall species diversity is then totally dependent on how much dispersal and extinction rates vary among species. For example, a system with 30 plausible species is characterized by 60 parameters that determine the final diversity. The extinction rates mainly reduce the fraction of patches occupied and are of minor importance. We tried different kinds of trade-offs such as more or less convex functions. The qualitative result was robust and hence we only present the results from linear trade-offs in a system with 20 herbivores and 10 predators. We examined sixteen different combinations of dispersal rates and degrees of trade-off to produce the graphs depicted. For each trophic level, either high or low mean dispersal was combined with either high or low differentiation in dispersal rates - with the lowest dispersal rate starting at either 0.1 or 0.2, and increasing as species decline in competitive rank by either 0.015 or 0.02. For example, predators with both high differentiation in dispersal rates and high dispersal rates overall were combined in a system with herbivores characterized by low differentiation in dispersal rates but high dispersal rates overall. Note that almost all species go extinct after 75% habitat destruction and all the predators go extinct even earlier.

1989, and Kareiva 1991), we have been manipulating the number of fireweed (Epilobium nrtgusrifolium) patches within 100 m2 plots to four different levels: 2, 3, 5, and 10 patches per 100 m2 (for all treaments, fireweed must be cut down and ~ u l l e dur, to reduce the freciuencv . . of patches to these uncharacteristically low densities). These fireweed patches represent the only local habitat for ladybird beetles and their aphid prey. Our experiment thus provides a model for the sort of habitat destruction that has prompted spatially explicit landscape simulations of population dynamics. By tracking the proportion of fireweed patches occupied by ladybird beetles as a function of the number of left standing in each plot, one can see a dramatic effect of habitat loss on ladybird beetle populations. However, this effect is not seen until three to four weeks after the fireweed has been experimentally removed. The challenge is to design spatially explicit monitoring schemes that foreshadow the

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Fig. 6. The average fraction of fireweed patches occupied by ladybugs over a one month period (August) and its relation to the colonization intensity observed in plots that have been fragmented to varying degrees as described in the text. Each point is an average of 5 to 10 experimental plots; plots included are those for which there were at least four censuses in which occupancy is recorded (and hence at least three records of colonization events).

ultimate collapse of a population due to landscape modifications, before it is too late to do anything about the problem. Our study of the fireweed system suggests that the probability that a vacant habitat patch is colonized per time interval may be a good indicator of the longterm fate of a population (Fig. 6). In particular, by sampling all patches weekly, one can use as an index of colonization rate the number of vacant patches that are colonized divided by the product of the number of vacant and occupied patches within the plot. This measure of colonization intensity is a good predictor of the average habitat occupancy for ladybird beetles, after the effects of the fireweed destruction have been allowed to reveal themselves (Fig. 6). Any regular monitoring scheme that recorded presence and absence with respect to specified locations could be used to estimate such a colonization frequency. We expect that a detailed spatial model of dispersal and local extinction would indicate that such monitoring data in general could provide advance warning of population collapses due to habitat loss. Certainly, it would be useful to apply different monitoring schemes to data generated by models of spatially-structured populations, with the idea of testing the effectiveness with which monitoring programs can anticipate population trends generated by landscape change. Although spatial models may not be the panacea ecologists are looking for, it is certainly reasonable to expect that these models could help us better design the ways in which we collect spatially-structured data for use in predicting the fate of populations in fragmented landscapes.

The perils and promise of spatially explicit models There is no question that examinations of landscape patterns and the effects of habitat fragmentation on dispersal behavior and success are crucial to conservation of species. It is not, however, clear whether spatial models will contribute much to these efforts. To a large ex-

tent, spatial models seem to represent either loose metaphors with little justification for their complexity, or highly specific descriptions that could never be adequately parameterized. This does not mean that the pursuit of spatially explicit models is futile, but it does suggest that we need to examine carefully what we hope to accomplish with spatial theory. If we intend to move beyond the use of theory for simply demonstrating possibilities, we need to be a great deal more skeptical about the merits of spatially explicit models. One of the more underrated uses of spatially explicit theory in conservation biology is as a tool for assessing the merits of different monitoring schemes aimed at detecting threats to populations as a result of landscape change. Acknowledgements - We were supported by grants from the National Science Foundation, USDA and EPA (to Peter Kareiva), by the Swedish Institute and Swedish NFR (to Uno Wennergren), and finally by Florida State University's support of Mary Ruckelshaus.

Reference Bieri, M., Baumgartner, J., Bianchi, G., Delucchi, V. and von Atx, R. 1983. Development and fecundity of pea aphids as affected by constant temperatures and pea varieties. - Mitt. Schweiz. Entomol. Ges. 56: 163-171. Caswell, H. 1989. Matrix population models. - Sinauer, Sunderland, MA. Cohen, J. 1979. Long-run growth rates of discrete multiplicative processes in Markovian environments. - J. Math. Anal. Appl. 69: 243-25 1. Crouse, D., Crowder, L. and Caswell, H. 1987. A stage-based population model for loggerhead sea turtles and implications for conservation. - Ecology 68: 1412-1423. Doak, D., Kareiva, P. and Klepetka, B. 1994. Modelling population viablility for the desert tortoise in the Western Mojave desert. - Ecol. Appl. 4: 446460. Horvitz, C. C. and Schemske, D. W. 1995. Spatiotemporal variation in demographic transitions of a tropical understory herb: projection matrix analysis. - Ecol. Monogr. 65: 155192. Kareiva, P. 1991. Population dynamics in spatially complex environments: theory and data. - Philos. Trans. R. Soc. Lond. B 330: 175-190. - and Wennergren, U. 1995. Connecting landscape patterns to ecosystem and population processes. - Nature 373: 299302 .

Lindenmayer, D. B. and Lacy, R. C. 1995. Metapopulation viability of leadbeater's possum, Gymnobelideus leadbeateri, in fragmented old-growth forests. - Ecol. Appl. 51: 164182. ~ c ~ e hK., e Noon, ~ , B. and Lamberson, R. 1993. Conservation planning for species occupying fragmented landscapes: the case of the northern spotted owl. - In Kareiva, P., Kingsolver, J. and Huey, R. (eds), Biotic interactions and global change. Sinauer, Sunderland, MA, pp. 424-450. Pulliam, R. and Dunning, J. 1995. Spatially explicit population models. - Ecol. Appl. 5: 2. Schaal, B. A. 1984. Life-history variation, natural selection and maternal effects in plant populations. - In: Dino, R. and Sarukhan, J. (eds), Perspectives on plant population ecology. Sinauer, Sunderland, MA, pp. 188-206 Silvertown, J., Franco, M., Pisanty, I. and Mendoza, A. 1993. Comparative plant demography-relative importance of lifecycle components to the finite rate of increase in woody and

herbaceous perennials. - J. Ecol. 81: 465-476. Tilman, D., May, R., Lehman, C. and Nowak, M. 1994. Habitat destruction and the extinction debt. -Nature 371: 6 5 4 6 . Tuljarpurkar, S. 1990. Population dynamics in variable environments. - Springer, New York. Turchin, P. and Kareiva, P. 1989. Aggregation in Aphis varians: an effective strategy for reducing predation risk. - Ecology 70: 1008-1016. Turner, M., Arthaud, G. J., Engstrom, R. T., Hejl, S. J., Liu, J., Loeb, S. and McKelvey, K. 1995. Usefulness of spatiallyexplicit population models in land management. - Ecol. Appl. 5: 12-16. Van Groenendael, J . 1985. Differences in life histories between two ecotypes of Plantago lanceolata L. - In: White, J. (ed.), Studies on plant demography. Academic Press. New York, pp. 5 1 4 7 . Verner, J., McKelvey, K. S., Noon, B. R., Gutierrez, R. I., Gould. G. I.,Jr, and Beck, T. W. 1992. The California spotted owl: a technical assessment of its current status. - USDA Forest Service Gen. Technical Report PSW-GTW-133, Berkelcy. CA.

Appendix The three models that are run in parallel. B and C are simplifications of the full matrix model A. All parameters in the models are derived from the same demographic data as in the full matrix model. We transform age specific data to population specific data by calculating eigenvalues and eigenvectors of matrices, i. e. the population growth rates and stable age distributions. In the full matrix model the parameters are both age and patch specific, in the patch model they are only patch specific and in the scalar model there is only one parameter for the whole population. When these three models ran in parallel with constant demography, they all produced the same long-run growth rate.

A. The full matrix model This model includes both age- and space- specific rates (demography). Each individual is characterized by both its age class and in what patch it exists; this determines its fecundity, survivorship and dispersal rate. Example with 4 patches and 6 age classes, for more details of these kinds of models, see Caswell (1989):

nf - number of individuals in age class 1 and patch i, Fi - fecundity of individuals in age class 1 and patch i, P/probability of individuals in age class 1 surviving and staying in patch i, d/]-probability of individuals in age class 1 surviving and dispersing from patch j to patch i.

B. The patch rnodel Population growth rates, h-values, are specific for each patch. The h-values are the leading eigenvalue of Leslie matrices for the patches and these matrices then vary over time according to environmental variation. The hvalues comprise both fecundity and survivorship. The patch specific dispersal rates, d,, comprise both dispersal and survivorship. The patch specific dispersal rate is the age specifc dispersal rate, dl,, in the full model times the stable age structure for the specific patch, i.e., the age structure that is coupled with the h-value.

C. The scalar model One single growth rate for the whole population which is the actual leading eigenvalue of the full matrix:

,

n , , = h,n,

OIKOS 74:3 (1995)