ratio-dependent predation: an abstraction that works1

Ecology, 7 6 ( 3 ) , 1995, pp. 995-1004 0 1995 by the Ecological Society of America

RATIO-DEPENDENT PREDATION: AN ABSTRACTION THAT WORKS1,* H. RESITAKCAKAYA Applied Biomathematics, 100 North Country Road, Setauket, Neb\ York 11733 USA

ROGERARDITI Populations et Comnzmauth, Bailment 362, Universiti Paris-Sud XI, 91405 Orsay cedex, France, and Institute of Zoology and Animal Ecology, University of Lausanne. CH-1015 Lausanne, Switzerland

LEV R. GINZBURG

Department of Ecology and Evolution, State University of New York at Stony Brook, Stony Brook, New York 11794 USA

Abstract. Recent papers opposing ratio dependence focus on four main criticisms: (1) the empirical evidence we present is insufficient or biased, (2) ratio-dependent models exhibit pathological behavior, (3) ratio dependence lacks a logical or mechanistic base, and (4) more general models incorporate both prey and ratio dependence and there is no need for either of the two simplifications. We review these arguments in the light of empirical evidence from field and experimental studies. We argue that (1) empirical evidence shows that most natural systems are closer to ratio dependence than to prey dependence, (2) "pathological" dynamics in a mathematical sense is not only realistic, but the lack of such dynamics in prey-dependent models actually makes them pathological in a biological sense, (3) the mechanistic base of ratio dependence is (direct and indirect) interference and resource sharing, and (4) although more general models (with extra parameters) can never fit natural patterns worse than either prey- or ratio-dependent models, there are theoretical, practical, and pedagogical reasons for attempting to find simpler models that can capture the essential dynamics of natural systems. Key words: predation models; prey dependence; ratio dependence.

In prey-predator models the link between the dynamics of prey and predator is described by the trophic function, g, which is called the "functional response" in the prey equation (Eq. la) and the "numerical response" in the predator equation (Eq. lb):

where N is the number of prey, P is the number of predators, e is trophic efficiency, and p is predator death rate. Different formulations of the trophic function include dependence on prey density alone (prey dependence; Lotka-Volterra-type models), on the ratio of prey and predator densities (ratio dependence; Arditi and Ginzburg 1989), and on prey and predator densities separately (Table 1). Ratio dependence and prey dependence are at opposite ends of a spectrum of functional/numerical responses that are general functions

' Manuscript received 22 February 1994; revised 4 August 1994; accepted 24 August 1994. 2 T h i s article responds to Comments on ratio-dependent predation, which appeared in Ecology 75:1834-1850.

of both prey and predator densities. Functions of this more general type include those proposed by Hassell and Varley (1969), DeAngelis et al. (1975), and Beddington (1975). In a number of studies we have suggested that natural systems are generally closer to the ratio dependence end of the spectrum than to the prey dependence end; in particular, the evidence for ratio dependence comes from purposely designed experiments measuring patterns of equilibria predicted by different functional response models (Arditi et al. 1991b, Arditi and Sai'ah 1992), and from biomass relationships among trophic levels in natural ecosystems, particularly in lakes with different nutrient loadings (Arditi and Ginzburg 1989, Arditi et al. 1991a, Arditi and Sai'ah 1992, Ginzburg and Akcakaya 1992). The critiques to which we have been invited to respond (particularly Abrams 1994) contain a large number of statements that are misunderstandings of our arguments. Rather than undertaking the tedious task, of low interest to the reader, of commenting separately on every such statement, we will focus instead on the four main criticisms raised by Abrams (1994), Sarnelle (1994), and Gleeson (1994), and by previous papers cited by these authors (Freedman and Mathsen 1993, Diehl et al. 1994): (1) the empirical evidence we present is insufficient or biased, (2) ratio-dependent models exhibit pathological behavior, (3) ratio dependence lacks a logical or mechanistic base, and (4) more gen-

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TABLE1. Examples of prey-dependent, intermediate, and ratio-dependent trophic functions, g(N, P ) Name Prey -dependent

Intermediate Ratio-dependent

*N

=

Equation*

Lotka-Volterra Holling Rosenzweig-Mac Arthur

g(N,P) = aN S(N,P) = aNI(b 8 ( N 2 P )= e(N)

Hassell-Varley DeAngelis-Goldstein-O'Neill Getz Arditi-Ginzburg

g(N,P ) = g(N,P) = g(N,P) = g(N,P) =

number of prey; P

=

+ N)

aNP-"' aN/(b N cP) aN/(cP + AT) g(N/P)

+ +

Source

... Holling 1959 Rosenzweig and MacArthur 1963 Hassel and Varley 1969 DeAngelis et al. 1975 Getz 1984 Arditi and Ginzburg 1989

number of predators; a, b, c, a , and m are constants.

and Oksanen et al. (1981) developed a prey-dependent theory that predicts that, as primary productivity increases, the number of trophic levels increases and this causes a stepwise increase in biomasses of all trophic levels. This prediction of prey-dependent models is summarized in two figures, which show changes in pairs of trophic variables. Fig. 1A (based on Kerfoot and DeAngelis 1989) shows the changes in the biomass of the first trophic level (producers) as the potential primary productivity (measured for instance by nutrient loading) increases. Fig. 2A (based on Oksanen et al. 1981) shows the changes in the biomasses of first (producer) and second (herbivore) levels as a result of an increase in potential productivity (which is not shown). Sharp changes in the relationship arise as new trophic levels are added. In Figs. 1A and 2A, the last segments (labelled "4 levels") are drawn in a different style to emphasize that they correspond to the data shown in EMPIRICAL EVIDENCE Figs. 1B and 2B, respectively (see below). Persson et Trophic abundances al. (1988) have suggested that further increases in proPrey- and ratio-dependent models show striking dif- ductivity can also cause a decline in the number of ferences in the trophic abundances in food chains of trophic levels from 4 to 3. Thus, the prediction of an increase in biomasses with increasing length in response to variations in primary prey-dependent models is contingent upon simultaproductivity (Arditi and Ginzburg 1989: Table 2). In the ratio-dependent model, all levels respond propor- neous changes in the number of levels. The problem tionately, while in the prey-dependent model the re- is that such changes are not commonly observed. Sevsponses differ depending on the trophic level and on eral studies have shown that food chain lengths do not the number of levels. Prey-dependent models generally vary much, and whatever variation there is cannot be do not predict that consecutive levels will concurrently explained by variation in productivity (Ryther 1969, increase in equilibrium abundance in response to an Pimm 1982, 1988, 1992, Beaver 1985, Briand and Coincrease in productivity. The exception to this state- hen 1987, Pimm and Pitching 1987, Cohen et al. 1990). ment is a structurally unstable model (Gatto 1991, A study by Persson et al. (1992) found that three-link Gleeson 1994) that we will discuss below (see Density systems had lower productivity than four-link systems, and therefore provided counter-evidence to the above dependence a t only the top level). We analyzed the relation between abundances of var- generalizations. However, besides the fact that the difious trophic levels in 175 lakes (Ginzburg and A k p k - ference was not significant, one should note the disincluded in aya 1992) and concluded that the patterns of abundance crepancy in the number of con~n~unities were more consistent with the ratio-dependent model these studies: while Persson et al. (1992) is based on than with the prey-dependent model. This analysis has 11, Briand and Cohen (1987), for instance, is based on been criticized (Abrams 1994, Diehl et al. 1994, Sar- 113 communities. It is also important to note that an nelle 1994) on the grounds that other (non-ratio-de- increase in the number of trophic levels is not assumed pendent) models can also predict similar patterns, and by ratio dependence, but it is not incompatible with it that some experimental evidence favors other n~odels. either. In a ratio-dependent system, such an increase We discuss each of these arguments in detail. would not result from changes in the stability properties Different number of trophic levels.-Fretwell (1977) of trophic interactions, as proposed by Oksanen et al. era1 models incorporate both prey and ratio dependence and there is no need for either of the two simplifications. In this paper, we argue that (1) empirical evidence shows that most natural systems are closer to ratio dependence than to prey dependence, (2) "pathological" dynamics in a mathematical sense is not only realistic, but the lack of such dynamics in prey-dependent models actually makes them pathological in a biological sense, (3) the mechanistic base of ratio dependence is (direct and indirect) interference and resource-sharing, and (4) although more general models (with extra parameters) can never fit natural patterns worse than either prey- or ratio-dependent models, there are theoretical, practical, and pedagogical reasons for attempting to find simpler models that can capture the essential dynamics of natural systems.

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RATIO-DEPENDENT PREDATION

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(1981), but from minimum population sizes required for persistence, as a function of total production. This view is more in line with the "productivity space" hypothesis of Schoener (1989): although more productive lakes do not in general have more trophic levels, a 1-m2 lake will not support a population of top carnivores. Persson et al. (1992) also found other differences between three-link and four-link systems compatible with prey dependence, but the sample sizes in these analyses were even lower, with only 5 or 6 data points for each regression. In contrast, Ginzburg and A k p k aya (1992) compiled data sets from 175 lake ecosystems, with sample sizes of 11 to 119 for each regression. All of these regressions showed increased bion~assesat all trophic levels as a function of increased nutrient loading, but none of the studies in which these data were collected mentioned an increase in the number of trophic levels with productivity. A study similar to those compiled by Ginzburg and

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log Chlorophyll a FIG.2. Relationship between phytobiomass and herbivore biomass. (A) Prey-dependent model prediction (different segments correspond to the assumed increases in the number of trophic levels). (B) Ratio-dependent model prediction, superimposed with data from Mills and Schiavone (1982). The data displayed in (B) deal with four-level lakes and should be compared only with the last portion of the prey-dependent prediction in (A).

Akgakaya (1992) was performed by Mills and Schiavone (1982) on a series of 13 lakes. This study differed from the others in its detailed identification of species in each of the lakes, which made it possible to know that each lake had the same number of trophic levels. In these lakes, zooplankton density increased as a function of phytoplankton density (Fig. 2B), which in turn increased as a function of the nutrient concentration (total phosphorus; Fig. 1B). Log-log regressions for both relationships give positive slopes that are significantly greater than zero (P < 0.01). Since all of the lakes had four trophic levels, and an increase in the number of trophic levels is impossible, the prey-dependent theory would predict a decreasing function for both relationships (the last "segments" of the relationships in Figs. 1A and 2A, labelled "4 levels"), whereas the ratio-dependent theory correctly predicts the increasing relationships. Clearly, ratio dependence is a more realistic description of these data than prey dependence. Abrams (1994) cites Diehl et al. (1994), who claim

H. RESIT AKCAKAYA ET A L .

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Ecology, Vol. 76, No. 3

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FIG.3 . Residuals from log-log regressions among nutrient concentrations (total phosphorus, measured in mg/m3), phytoplankton density (chlorophyll a density, measured in mg/m3), zooplankton biomass (measured in mg/m3), and fish biomass (measured in kg/ha) in lakes. (A) Nutrient concentration and phytoplankton density (data from Mills and Schiavone 1982); (B) nutrient concentration and phytoplankton density (data base compiled from studies listed in Ginzburg and A k p k a y a 1992: Table 2); (C) nutrient concentration and zooplankton biomass (data from Hanson and Peters 1984); (D) phytoplankton density and zooplankton biomass (data from Pace 1984); (E) nutrient concentration and fish biomass (data from Jones and Hoyer 1982); (F) phytoplankton density and fish biomass (data from Jones and Hoyer 1982).

that the residuals from the Daphnia-algae regression in our earlier paper (Arditi et al. 199la) is consistent with a stepwise pattern. We do not see any such pattern. Since Diehl et al. do not give any statistical evidence for this claim, we invite readers to make their own judgments by inspecting their figure (Diehl et al. 1994: Fig. 2), and similar plots of residuals in Fig. 3. These graphs show the residuals from the regression in Fig. 1B (on data of Mills and Schiavone 1982), as well as from regressions in the trophic biomass relations compiled by Ginzburg and A k p k a y a (1992). We suggest (1) that none of these relationships shows a stepwise pattern, and (2) that there is no extrinsic evidence for an increase in the number of trophic levels in any of these studies (there is strong evidence to the contrary in the data of Mills and Schiavone 1982).

Another study that is brought up by Abrams (1994) as empirical evidence against ratio dependence is by Hansson (1992), who has plotted the relationship between zooplankton and phytoplankton for lakes with two and three trophic levels. The results look like Fig. 4. Note that the higher trophic level is plotted as the horizontal axis, and all error is assumed to be in chlorophyll (phytoplankton). We consider Hansson's (1992) results to be strong evidence for ratio dependence for two reasons. First, ratio dependence predicts that all levels should change proportionally, whereas prey-dependence predicts that there should be no change in phytoplankton biomass in "2-level" systems. Hansson's study shows that there is an increase in phytoplankton biomass. contrary to prey-dependent theory. Second, a ratio-dependent model with three levels (e.g.,

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Herbivore Biomass FIG. 4. Relationship between phytoplankton (chlorophyll a density) and herbivores (zooplankton biomass) in lakes with two vs. three trophic levels, based on Hansson (1992) Fig. 4 . Note that the axes are reversed compared to Fig. 2.

Ginzburg and Akgakaya 1992: Eq. 2) predicts that the slope of the relationship between zooplankton and phytoplankton biomasses should decrease with added levels. Hansson's (1992) study shows exactly this (since it shows an increased slope for the inverse relationship with the higher trophic level as the independent variable), and therefore is consistent with ratio dependence. Abrams (1994) also repeats the criticism of Diehl et al. (1994) about our statistical analysis in which we assumed that the regression goes through the origin. Diehl et al. repeat the regression with an intercept parameter. The x-intercept value is very small and Diehl et al. do not provide a test to prove that it is statistically different from zero. Note that most studies on relationships among trophic levels use log-log regressions, which implicitly assume that the relationships have zero intercepts in the original variables. We do not think that this is an important issue, considering the amount of evidence consistent with concurrent increases in consecutive trophic levels (even when the number of trophic levels remains the same), a pattern not predicted by the prey-dependent model. Even if the regression intercepts the abscissa, this would be at a very low algal concentration, an extreme situation in which we recognize that ratio-dependence may not be applicable. Actually, we do agree that it is at the extremes of low and high densities that strict ratio dependence may not be valid. When the prey-topredator ratio is very high, predators are not likely to encounter effects of interference, and their rate of predation will be mostly determined by the density of their prey. As we have argued before, ratio dependence and prey dependence are at opposite ends of a spectrum of functionallnumerical responses that are general functions of both prey and predator densities:

999

where g are three types of trophic function (prey-dependent, intermediate, and ratio-dependent) in the general prey-predator model (Eq. 1; see Table 1). We do not claim that ratio dependence is universally applicable; what we claim is that within the range of densities that are usually found in natural systems, the observations of trophic biomass patterns are more consistent with ratio dependence than with prey dependence. We suggest that in the absence of detailed information and data for a more general function, the simplest assumption about the form of predation is ratio dependence. Models of trophic interactions with ratiodependent functional response will make more accurate predictions than models that assume prey dependence. Interference a t only the top level.-Sarnelle (1994) shows that the relationship among trophic-level abundances (Ginzburg and Akcakaya 1992) can also be predicted by food chain models with partial interference only at the top level and prey dependence in the other levels. His model predicts the slopes of log-log regressionsbetween two consecutive trophic levels in a four-level system as follows: inl(1 - m) between producers and herbivores, (1 - m)/m between herbivores and carnivores, and m between carnivores and top-carnivores, where in is the coefficient of mutual interference for the top carnivores (see Hassell-Varley equation in Table 1). This is an interesting suggestion that mixes partial interference at one level with prey dependence in other levels. We agree that this is a possible mechanism; however, there are two difficulties with this approach. First, there are no data on the relationship between carnivores and top carnivores, and Sarnelle's argument depends heavily on this unobserved relationship. In contrast, ratio dependence does not predict different slopes for different levels, and therefore does not rely on this relationship any more than relationships among other trophic levels. The second difficulty relates to compatibility with data. The observed slopes between various trophic levels are mostly around 1 (Ginzburg and Akgakaya 1992, McCarthy et al. 1995). If both ml(\ - in) and (1 - m)lm are to be equal to 1, m must be equal to 0.5. In fact the observed slopes are not exactly 1, but vary between 0.6 to 1.8 (crudely estimated from the reciprocals of the slopes in Ginzburg and A k p k a y a 1992). This means that for Sarnelle's suggestion to be true, in cannot be much different from 0.5. Even m = 0.6 or 0.4 will be at the limit of compatibility with observed relations. This makes Sarnelle's suggestion very restrictive: not only should there be interference only at the top level. but it should have a coefficient m not very different from 0.5. This crude calculation has been refined by McCarthy et al. (1995). As a more general argument, one might say that there are many combinations of interference coefficients that will be more-or-less compatible with the observed slopes (of log-log regressions of trophic biomasses). Ginzburg and Akqakaya (1992) suggested

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that coefficients might be around 1, distributed over all the levels, since this is one of the combinations that will be compatible with the observations. Thus we did not ignore the restriction of adjacency or imbeddedness in our analysis as Sarnelle (1994: 1837) claims; we only suggested that if all interference coefficients are around 1, all slopes (including those between non-adjacent levels) will also be close to 1. Sarnelle's suggestion is another set of interference coefficients, namely, (0, 0, 0.5), for the 2nd, 3rd, and 4th levels, respectively, that will also be compatible with the observed patterns. A new analysis of the data confirms our initial assumption of distributed interference, and shows that although Sarnelle's suggestion is a possibility, in the face of available data, it represents a very unlikely combination of interference coefficients. It demonstrates that most of the plausible combination of interferences are clustered around the point that represents (1, 1, l), rather than (0, 0, 0.5) (McCarthy et al. 1995). Sarnelle (1994) also offers results of statistical relations for two-trophic-level systems as evidence against ratio dependence. However these results are due to the bias introduced by Model I regression (Sarnelle 1994). Reanalysis of data by Model I1 regression shows that both two-level and all-level systems are compatible with ratio dependence (McCarthy et al. 1995). Density dependence a t only the top level.-Another way to make a prey-dependent model predict the proportionality of equilibria is to introduce a quadratic mortality for the top trophic level ( ~ l e e s o i1994; dNs/dt = asN4Nj - a&). This idea is not new; it has been proposed by Gatto (1 99 1) and discussed by Ginzburg and A k p k a y a (1992). As Gleeson (1994) demonstrates, this mechanism does predict the proportionality of equilibria, but under very restrictive conditions. First, only the top level must have density-dependent mortality; all other levels should have no self-limitation, and no source of mortality other than predation. To see how unrealistic this assumption is, set the abundance of food (one lower level) and predator (one upper level) to zero for any single level except for the first and last ones. The population will have a growth rate of zero; i.e., it will remain unchanged in the absence of any resources! Adding any additional (density-dependent or independent) mortality term to make the model more realistic will prevent the proportionality of equilibria. Second, this mechanism produces the proportionality of equilibria only if all functional responses are linear relationships (i.e., type I). When nonlinear functional responses (arising from handling time, or a limit to the maximum rate of consumption) are introduced, the equilibria are no longer proportional. Third, the mortality at the top level should be precisely quadratic; if it is a combination of linear and quadratic terms, the proportionality, again, does not follow. In summary, Gleeson's model is structurally unstable; any small modification in the direction of making it more

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general or realistic would eliminate its prediction of the proportionality of trophic equilibria. Finally, this mechanism does not explain the observed proportionality in experimental systems where the mortality is controlled, and thus is known to be linear, not quadratic (Arditi et al. 1991b, Arditi and Sai'ah 1992). Gleeson (1994) also objects to infinite ratio of prey to predators. As explained by A. A. Berryman, A. P. Gutierrez, and R. Arditi (unpublished manuscript), this does not produce any unrealistic value in the variables that are actually observed (e.g., the number of prey consumed), if the functional responses are non-linear and asymptotic. Gleeson's model, on the other hand, makes the unrealistic assumption of unlimited rate of p e r capita consumption in proportion to available food. Split trophic levels.-Abrams (1994) states that predictions of ratio-dependent models in terms of reaction of trophic-level biomasses to increases in primary productivity are also predicted by prey-dependent models when there are two or more species on each trophic level. We believe this is a deficiency of prey-dependent models, since it means that (1) arbitrary splitting of species will change the predictions of the model and (2) prey-dependent models are not suitable for modeling food chains, since lumping species into trophic levels will change the predictions. In other words, even if a food web has prey-dependent dynamics for each separate interaction, the dynamics of the food chain (after aggregating non-identical species into one trophic level) would no longer behave as prey dependent (Arditi and Michalski 1995). Both prey-dependent and ratio-dependent models of trophic interactions have made the implicit assumption that the same models can be used to model both communities (in terms of biomass in trophic levels) and prey-predator interactions (in terms of abundances of the two populations). While we believe that this assumption deserves much scrutiny, we would like to point out that communities are never entirely described at the level of the single species; some aggregation (e.g., into functional groups) is always done. Models that are not very sensitive to the degree of lumping several species into common variables are therefore preferable. Ratio-dependent models have been used to describe dynamic properties at the population level (Akqakaya 1992), as well as equilibria! properties at both the trophic level and the population level (Arditi et al. 1991a, b, Arditi and Sai'ah 1992, Ginzburg and Akqakaya 1992). In other words, both the populationlevel and the trophic-level interpretation of ratio-dependent equations are compatible with the evidence, whereas applying prey-dependent models to aggregated trophic levels leads to inconsistencies. More complicated models.-Abrams (1994) also lists a number of more complicated models that make predictions of trophic abundances similar to those of ratio-dependent models. These are mostly models that explicitly model factors (various heterogeneities) that

RATIO-DEPENDENT PREDATION

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form the mechanistic basis of ratio dependence (see The mechanistic basis . . . below). We believe these results are not evidence against ratio dependence; on the contrary, they support our basic statement that ratio-dependent summarization is a better abstraction of trophic dynamics than prey dependence. We will return to this issue in the Conclusion section. Spatial heterogeneity A set of experiments has demonstrated the effect of spatial distribution on the type of predation in two species of cladocerans (Arditi and Sai'ah 1992; see also Arditi et al. 1991b). As expected, the species with a homogeneous distribution (Daphnia magna) followed the prey-dependent model, and the species with a heterogeneous distribution (Simocephalus vetulus) followed the ratio-dependent model. Further, Arditi and Sai'ah (1992) demonstrated that it was possible to change the predation pattern of both species by artificially modifying their spatial behavior: S. vetulus followed the prey-dependent model when forced to a homogeneous distribution, and D. inagna followed the ratio-dependent model when forced to a heterogeneous distribution. Thus our experiments demonstrate that at the appropriate temporal and spatial scales, the preydependent view is preferable to ratio dependence. Abrams (1994) repeats the arguments about these experiments raised by Ruxton and Gurney (1992). Obviously, measuring algal concentrations and the functional response directly would be a more accurate test, as Arditi et al. (1991) wrote. However, this was technically impossible, making it necessary to use the indirect method based on counting the predators. The other arguments repeated by Abrams were addressed in a previous reply by Arditi et al. (1992). Mutual interference Our analysis of mutual interference in predator-prey and parasitoid-host experiments (Arditi and Akqakaya 1990) was a study dealing with the functional response at the behavioral time scale (i.e., the traditional conception of the functional response) and based on experimental data found in the literature. Using the intermediate model g ( N I P ) , we quantified the position of the functional response along the spectrum of Eq. 2. We found that prey dependence g ( w (i.e., in = 0) was rejected in 15 out of 15 cases (m always significantly >0), whereas ratio dependence g(N1P) was rejected in only 3 out of 15 cases (in significantly < I ) . This result clearly suggests that the systems we analyzed were closer to ratio dependence than to prey dependence. Abrams' (1994) claim that our selection of data sets was biased is unfounded. Our criteria for selecting studies were described in Arditi and Akqakaya (1990). We made an exhaustive search of suitable studies, i.e., measurements of the number of prey eaten with variations in both prey density and predator density. We removed all data points that were meaningless

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because prey had been almost exhausted (>90% consumption) or had been hardly attacked ( < l o % consumption). Such situations generate huge uncertainties about the estimates of the searching efficiencies, making estimation of in impossible. We ended with the 15 studies mentioned. Abrams (1994) also criticizes our analysis of data of Katz (1985), which consisted of field observations and field experiments. Abrams notes that Katz did not detect predator dependence in the field; lack of detection was due to the fact that field densities were much lower and much less variable than experimental densities. For all prey and predator densities observed in the field, predators were saturated by superabundant food. Katz's figure shows that, in field conditions, there was hardly any response to prey density either. In other words, the functional response was constant, which is equally consistent with ratio dependence as with prey dependence. The lack of dependence on prey abundance made field densities inappropriate for studying the functional response, and Katz had to manipulate both prey and predator densities experimentally. We obtained his data from his Fig. 8 (Katz 1985) and performed the analysis reported in our paper (i.e., exact measurement of the coefficient of mutual interference, m) and found m = 0.87  0.35 (mean  1 SE). Any reader who would look at Katz's figure would see very clearly that both factors affect the number of prey eaten per predator: predator-dependence is very clear.

One argument against ratio dependence involves a so-called "pathological" behavior: under certain conditions, a ratio-dependent model can predict that both variables (prey and predator abundances) get "arbitrarily close to the axes" (Freedman and Mathsen 1993), which is pathological in a mathematical sense. In biological terms this "pathological" dynamics will be manifested by both species increasing initially, then predators consuming all the prey and both species going extinct. For biologists this is a very reasonable outcome, especially in the absence of refugia, as demonstrated by the famous experiments of Gause (1934) with Paramecium and Didiniuin. The prey-dependent Lotka-Volterra model, however, cannot predict such an outcome (assuming that prey increase in the absence of the predator, and predators decrease in the absence of prey). Thus, in a biological sense, it is the preydependent model that shows pathological behavior. A realistic model of prey-predator interactions should be able to predict the whole range of dynamics observed in such systems in nature. A ratio-dependent model can have stable equilibria, limit cycles, and the extinction of both species as a result of overexploitation. For example, a lynx-hare model by Akcakaya (1992) shows limit cycles that, for some parameter combinations, lead to extinction. Modifying the model by adding a small prey refuge changes this dynamics

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to limit cycles with period close to the observed period of lynx-hare cycles in Canada, with parameters estimated from field studies, mostly independently of the time series being predicted. It is interesting to note that this type of prediction was never made by a LotkaVolterra model for the lynx-hare system. Another claimed pathology of the ratio-dependent model is the fact that predators can persist on an arbitrarily small prey abundance. We wish to recall here that the whole theoretical framework, in which we and our critics work, rests on the assumption that populations are large enough that N and P (the number of prey and predators, respectively) can be accepted as continuous variables. When densities are very low, other factors dominate the dynamics and the whole approach in terms of differential equations becomes inappropriate, for the ratio-dependent model as well as for the prey-dependent model.

The difference between the prey-dependent models and the ratio-dependent models that we propose is in the formulation of the functional and numerical responses. The question is whether the rate of predation (functional response) and the rate of predator reproduction (numerical response) are better modeled as functions of the prey density N or as functions of the ratio NIP. Modeling the functional response with a ratio (i.e., with "available resources per consumer") is a result of a direct sharing mechanism. Of course, there is also some kind of sharing in prey-dependent models as well, but this sharing is indirect, operating through the predator equation. The difference arises in the way we perceive the time scale at which the biological processes of reproduction and predation occur. In ecological systems, these two time scales are usually very different: the behavioral time scale is normally much faster than the population dynamics time scale. Arditi and Ginzburg (1989) argued that in order for a preypredator model to be internally consistent, the functional response must be measured at the same (slower) time scale as that of population dynamics. Making the two equations in a prey-predator model consistent by bringing both time scales to that of reproduction has been proposed by Arditi and Ginzburg (1989) as a motivation for such a simplification. Just like a molecular motion image for the Lotka-Volterra model, our time-scale-based motivation is just that, a motivation or a metaphor and not a proof of validity (as Abrams [I9941 seems to suggest); it was never meant as evidence for ratio dependence. Only by comparing actual predictions of the two views with data can we select the one that does not fail the test of consistency (Ginzburg and A k ~ a k a y a1992). Both Abrams (1994) and Oksanen et al. (1992) give the impression that they treat continuous-time models based on differential equations as some sort of "true"

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models. We believe that population processes are intrinsically discrete: populations vary by whole individuals; fractions do not exist. But this does not mean that difference equations are any better; they also describe the population variable as continuous (hence the approximation of large populations), but treat time as discrete. Population dynamics processes are neither purely discrete nor purely continuous in time, and both differential and discrete equations are approximations. This does not mean that differential equations have no place in ecology. In fact, differential equations serve in population ecology the role of easily analyzable approximations to underlying discrete processes. The choice of time scale in selecting such a continuous approximation was our main motivation for revising the coupling terms in prey-predator model. In other words, ratio-dependence recognizes, and attempts to correct, one of the most fundamental inadequacies of differential equations (that of different time scales of consumption and reproduction) in modeling population dynamics, whereas a prey-dependent model simply ignores this inadequacy.

In sum, the mechanism of prey dependence is based on an analogy with molecular mass action, whereas the mechanism of ratio dependence is based on resource sharing and (pseudo-) interference, originating from differences in time scales of feeding and reproduction, and from spatial heterogeneities. It would, of course, be possible to model the mechanisms of temporal heterogeneity, spatial structure, resource-sharing, and interference explicitly. We suggest (and we have growing evidence for this) that such detailed models will give, on a large scale, long-term average population equilibria whose patterns of abundance will match approximately the predictions of the ratio-dependent model. In general, models with more details give better descriptions of natural dynamics. To see this one does not even have to look at complicated models: by just adding one more parameter, models that are intermediate between prey and ratio dependence (e.g., DeAngelis et al. 1975) can cover the whole spectrum from pure prey dependence to pure ratio dependence. Our argument is not against these more complicated models; it is against prey-dependent, and in favor of ratiodependent models, of similar detail and complexity. We believe that there are situations in which models with different levels of complexity and detail are most useful. The next question that arises is this: if more complicated (so-called "mechanistic") models can describe natural dynamics correctly, what is the use or contribution of ratio-dependent models (which are called "phenomenological")'? There are three distinct answers to this question: one theoretical, one practical, and one pedagogical. Pimm (1992) summarizes one of the answers by writing: "One scientist's mechanism is another's phenom-

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enon." Obviously, ratio dependence is a mechanism from the point of view of food-web or community dynamics, and it is a phenomenon from the point of view of physiology and behavior, A similar statement is true for any type of model, including so-called mechanistic models of predation. One can always look for more mechanisms at lower levels of organization (e.g., physiology and behavior), but this does not preclude the usefulness of models at higher levels (e.g., food-web or community dynamics). Laws of planetary motion were criticized 300 yr ago for lack of mechanism. Today we still do not have a universally accepted theory of gravity, but this did not prevent space travel. The appropriateness of an abstraction depends on the problem to be studied, and whether ratio dependence is a useful abstraction or not should be judged by its predictions (for example of large-scale patterns in ecosystems) compared to the predictions of prey-dependent models. On this basis, the ratio-dependent theory, we believe, has so far fared better than its alternative. Thus, the theoretical contribution of ratio-dependent theory so far has been to show that natural systems usually seem to be closer to ratio dependence than to prey dependence. We believe that this is a more meaningful and interesting statement than simply saying that, overall, natural systems sometimes show prey dependence, sometimes ratio dependence, and sometimes something in the middle. The second use of a ratio-dependent abstraction is practical. In both applied and theoretical ecology there are times where simpler models that can capture essential dynamics of a natural system are preferred to more detailed models. In applied ecology this need usually arises from a lack of data, whereas in theoretical ecology it arises when simple models are necessary as building blocks of models of larger systems, for example food webs. The third use of ratio-dependent models is pedagogical. Before reaching hasty verdicts about who is setting "predator-prey theory back by decades," it is necessary to evaluate what has been achieved by "the vast majority of predator-prey modelling during the twentieth century" (Abrams 1994:1842, 1849). Hanski (1991) summarized this development as a platform of models that was "constructed by adding all sorts of conceivable, and often quite a few inconceivable, modifications to previous, simpler and in some sense more fundamental assumptions", and suggested that a similar development was possible with-a ratio-dependent basis, instead of the prey-dependent basis that was used. Even though everyone agrees now that simple prey-dependent models (such as Lotka-Volterra models) are inadequate, they still form the basis of teaching prey-predator interactions in many textbooks, just as they formed the basis of developing the theory. If it is necessary to start from a simple model, either as a theoretical or a pedagogical basis, does it not make

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sense to start from an equally simple model that fits patterns observed in natural communities? We thank Robert Armstrong, Donald DeAngelis, and an anonymous reviewer for helpful comments. This is contribution number 915 from Ecology and Evolution, State University of New York at Stony Brook. R. Arditi acknowledges the support of the Swiss Priority Programme on the Environment (grant 5001-348 10) and Programme Environnement, Vie et Societes of the French CNRS (GDR 1107).

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