Nonlinear Food Web Models and Their Responses to

In Food Webs: Integration of Patterns and Dynamics, eds. G.A, Polis and K.O. Winemiller, pp. 122-133. Chapman and Hall, New York (1996).

11 Nonlinear Food Web Models and Their Responses to Increased Basal Productivity Roger Arditi and Jerzy Michalski

Introduction We study here general food web models that include non-Lotka-Volterra trophic interactions. We set two logical requirements that any such model must satisfy: (Cl) invariance under aggregation of identical species, and (C2) mathematical separation of disconnected subwebs. We show that these logical requirements are not always satisfied by previously published models. We give the mathematical expressions corresponding to generalizations of three nonlinear functional responses: Holling (prey-dependent), DeAngelis-Beddington, and Holling-like ratio-dependent. We then address the two questions of species aggregation and of responses to basal production. Regarding species aggregation, we show that, in prey-dependent models (e.g., Holling) with two basal species and one consumer, the qualitative response of the basal level to increased basal productivity depends crucially on whether the species are lumped or not; in ratio-dependent webs, the response is not qualitatively sensitive to aggregation. In particular, this means that applicability of the theory of food chains is restricted to perfect chains if a prey-dependent model is used; applicability is much wider with a ratio-dependent model. Regarding responses to increased basal production, we show that ratio-dependent webs present unique qualitative properties. At equilibrium, the web of realized links is often simpler than the web of potential links. This realized structure depends on all parameter values and, for given parameters, alter-

nate stable states can exist. These properties can help solve the paradox that, on one hand, naturalists tend to describe very link-rich webs while, on the other hand, theoreticians predict that stable webs should be linkpoor. In the approach we follow for the theoretical study of food webs, the dynamics of each node is described by a differential equation which is a balance between production and mortality (e.g., May (1973)). Each trophic link from node Xito node X j must translate into mathematical expressions giving the number of prey consumed Gy and the number of consumers produced H yper unit time. The vast majority of published studies (e.g., Pimm (1982), Pimm (1991) for reviews) use the Lotka-Volterra interaction term to describe both of these quantities:

In fact, we have found in the literature only two models written for arbitrary food webs that do not rest on this hypothesis (Getz, 1984, 1991). When considered in per capita form, the Lotka-Volterra interaction term (1) is linear and, for this reason, we call the Lotka-Volterra model linear. The purpose of the present chapter is to study food web models that use nonlinear expressions for the individual links. Although Lotka-Volterra equations may be acceptable to describe community dynamics near equilibrium, they are not likely to apply universally for systems brought far from equilibrium by strong perturbations. Expression (1) is simply the law of mass action

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity I 123

of chemistry. It requires an assumption of random and homogeneous mixing of prey and predators-like molecules reacting in a chemical solution. It is clear that, in general, many factors can make it necessary to use more complex models. Some well-known factors are predator saturation, predator interference, prey or predator heterogeneous spatial distribution, temporally discontinuous consumption or reproduction processes, multispecies (aggregated) prey or predator variables, etc . One way of taking these factors into account is to develop highly detailed models that describe explicitly the various mechanisms involved (e.g . , space structure, age structure, etc.). This option has been followed for systems with few species (typically two). However, it becomes impractical in a theoretical context where webs can contain an arbitrarily high number of species. In such a case, it is reasonable to constrain oneself to cases where (1) is generalized to a nonlinear expression depending on population abundances only:

where, for historical reasons, the number of consumers X, has been extracted in order to make apparent the functional response gij (number of prey i consumed per consumer j per unit time). In the two-species prey-predator context (N, P), a large number of expressions for g have been suggested as more realistic altematives to the linear Lotka-Volterra functional response g = aN. In many cases this function is, as in the Lotka-Volterra case, a function of prey abundance only: g = g(N) (Holling, 1959; Watt, 1959; Ivlev, 1961; Takahashi, 1964; Rosenzweig, 197 1; Tostowaryk, 1972; Jost et al., 1973; Real, 1977). Among these expressions, the most commonly used is Holling's disc equation

et al., 1975; Beddington, 1975; Arditi and Aksakaya, 1990). Among these, we will only consider the model of DeAngelis et al. (1975) who have modified Holling's expression to include predator dependence in a phenomenological way (although derived from behavioral considerations, the model of Beddington (1975) is formally identical):

Note that, while Holling's model (3) has one more parameter than the basic LotkaVolterra expression (I), models that include predator interference like (4) add still another parameter. Although this would not be a serious problem when dealing with a simple predator-prey system, it is essential, in food webs containing many species, to keep the number of parameters as small as possible. With this purpose in mind, Arditi and Ginzburg (1989) have suggested that functional responses g(N, P) can be viewed as ranging along a continuous spectrum with prey dependence g(N) at one end and ratio dependence g(N1P) at the other end,

one possible ratio-dependent expression being

This function does not contain more parameters than Holling's expression (3). Therefore, it is an economical way of incorporating strong predator interference without adding new dimensions to the model. It is easy to check that the intermediate expression (4) include the forms (3) and (6) as opposite limiting cases. Clearly, such intermediate expression is biologically more "correct" since, being more versatile, it is able to apply to a broader class of situations. However, this is obtained at the cost of an extra parameter. It is therefore useful to study Other models include explicit dependence and compare the simpler models (3) and (6), on predator density: g = g (N,P) (Watt, 1959; under the assumption that the properties of Hassell and Varley, 1969; Hassell and Rog- an intermediate model such as (4) are interers, 1972; Strebel and Goel, 1973; Gomatam, mediate between those of the two extremes 1974; Rogers and Hassell, 1974; DeAngelis (3) and (6).

124 I Roger Arditi and Jerzy Michalski

Although, as already mentioned, almost all general food web models use the linear Lotka-Volterra functional response, some studies of simpler communities include the nonlinear Holling prey-dependent expression (3): in strict food chains (Oksanen et al., 1981; Hastings and Powell, 1991), in systems of two prey and one predator (Kretzschmar et al., 1993; Hastings, this volume), and in systems of two prey and two competing predators (Leon and Tumpson, 1975; Vandermeer, 1993). The effects of functional responses that include predator dependence like (4) or (6) have hardly been studied in systems containing three species or more. The only references we are aware of are Getz (1984, 1991) for food chains and webs and Arditi and Ginzburg (1989) for food chains. Indeed, generalizing nonlinear functional responses to arbitrary food webs is no easy task, as the rest of this chapter will show. In the next section, we give some necessary conditions that food web equations must fulfill for simple logical consistency. Later we propose generalizations of the nonlinear functional responses (3), (4), and (6) to the multispecies case, in keeping with the logical requirements given in the next section. Once equipped with this tool, we use it to discuss how reasonable it is to aggregate several species into a single variable, as this is done in food chain models. We then present first results obtained in food webs with ratio-dependent links, regarding response to increased primary productivity, and close the chapter with some concluding remarks.

Conditions for Logical Consistency Internal consistency of a general theory of trophic interactions requires that the equations obey some logical conditions. Two necessary conditions follow. 1. The equations must be CRITERION invariant under identification of identical species.

This criterion can be illustrated in the simple case of Figure 11.1. If equations are written, say, for one predator feeding on two prey species (three equations), and if the two prey species happen to be identical (i.e., identical

Figure 11 . l . If N , and N2 are two identical species, Criterion 1 requires that the equation calculated for N , + Niin system (a) should be identical with the equation written directly for N in system (b).

values of all pairs of parameters), then a system of two equations can be easily calculated for N = N l +N2 and for P . This reduced system must be strictly identical with the system of two equations that would be written directly for a two-species predator-prey system. Similarly, the equations should also be invariant under identification of identical predators. This property of invariance must be fulfilled by any model used to describe food web dynamics. Otherwise, the dynamics would depend on purely formal distinctions between species (e.g., artificial splitting of a species into two subspecies by marking individuals) and this is unacceptable. Thus, the criterion is required for logical consistency. Some of its biological implications can be illustrated by two simple examples. The first example is the situation of one predator P feeding on two prey species N 1 and N2. As explained mathematically in the Appendix, Criterion 1 implies that the functional response of the predator to each of the prey species must be a function of both variables N l and N2. In other terms, the total consumption of the predator in presence of the two prey species cannot be the simple addition of the two consumption terms that the predator would have in the presence of each prey separately. Otherwise, the predator would not saturate in the proper way with increasing prey availability. The second example is that of two predators Pi and P2 competing for one prey N . In such case, a consequence of Criterion 1 is that, if there is intraspecific competition within each predator population, then interspecific competition must be allowed between them.

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity I 125

(a) If the interaction terms represented by the oblique arrows become zero, Criterion 2 requires that the system of equations written for system (1) becomes identical with the system written directly for system (2). (b) Similar if species N-, becomes extinct.

Figure 11.2.

CRITERION 2. The system of equations for the food web must separate into independent mathematical subsystems i f the community splits into disconnected subwebs. For example, the four-dimensional system of two predators P i , P2 feeding on two prey Nl, N2 in Figure 11.2a should separate into two two-dimensional systems when preferences are such that PI eats Nl only and P2 eats N2 only. Similarly, the system of five equations of Figure 11.2b should separate into two independent subsystems if species N2 is brought to extinction. Thus, if species P2appears in the dynamic equation of species PI (e.g., as a competitor for species N2), it should disappear from this equation when species N2 disappears. The customary community-level LotkaVolterra model

fulfills both criteria. This is due to the linearity of the interactions appearing in the per capita growth rates (the quantity in the parentheses). When the interactions are not linear, satisfying the logical criteria becomes more difficult. For example, the Appendix shows that one of the systems that is often written for the situation of Figure 11.1 violates Criterion 1. In the next section we present several food web dynamic models that contain nonlinear interactions and that fulfill the criteria.

Generalizations of Some Trophic Equations The majority of predator-prey models considered in the ecological literature have the following general form:

where f(N) is the resource growth function (e.g., logistic); e is the conversion efficiency; p. is the consumer mortality; and g(N,P) is the functional response. The form of (8) assumes that the consumer's growth rate is proportional to the rate at which resources are consumed. Although not all models make this hypothesis (Getz, 1991; Berryman, l992), we will keep it here and consider generalizations of (8) for different forms of g(N,P). Generalizations of models that cannot be written in this form are also possible (Berryman et al., 1995). The general equations presented below (9, 10, and 11) fulfill the logical criteria of the previous section. These equations are general enough to describe food webs that include omnivory, loops, and cannibalism. Note that each generalization suggested here is not necessarily unique: starting from a given functional response, there may exist several possible extensions to the multispecies case. We present those that appeared to be most natu-

126 I Roger Arditi and Jerzy Michalski

ral. Because of space limitations in the present chapter, we must refer the reader to another paper where the generalization rationale is more fully explained (Appendix 1 in Michalski and Arditi, 1995). Rolling Functional Response In Holling's functional response (3), a is the searching efficiency of predator P for prey N (number of prey encountered per unit of searching time) and b=aTh where Th is the handling time per prey caught. A possible generalization of (8) resting on this functional response is the following:

where R(i) is the set of all resources of species i; C(i) is the set of all consumers of species i; Xbasaiis the vector of abundances of all basal species (those that do not have resources); hÃis the relative preference of predator Xj for prey Xi (i.e., the weight that predator Xj will give to prey Xi in comparison with alternate prey species); ajhà is the searching efficiency of predator Xj for species Xi; bij is a, multiplied by the handling time of predator Xj for prey Xi; ev is the conversion efficiency of prey Xi into biomass of Xj; IJL;is the mortality of a nonbasal species (for basal species it is supposed to be included in the growth function f,). The Lotka-Volterra competition model can be used as the function/;' (Xbasai) for basal species. It has the necessary property of collapsing into the simple logistic if all basal species are identical (see Appendix). For nonbasal species, f, (Xbasal)is zero.

DeAngelis-Beddington Functional Response The functional response (4) was introduced by DeAngelis et al. (1975) as a phenomenological expression, the authors stating that they were not suggesting any explicit mechanism. Beddington (1975) built almost the same model, with the idea that one should start from Rolling's concept of a handling time and also take into account the time wasted by predators on random encounters with other predators (i.e., direct interactions like fighting). If we now make the additional assumption that each predator individual can encounter all other individuals belonging to all species that feed on the same prey, then a generalization of (8) can be

where cik takes account of the time wasted by an individual predator Xi on an encounter with an individual predatorXk; and other symbols are as for (9). C (R(i)) is the set of all consumer species that share a common resource with consumer Xi. Ratio-Dependent Functional Response As mentioned in the Introduction, Arditi and Ginzburg (1989) have suggested that, in situations characterized by strong space and time heterogeneities, the functional response can be approximated by a function of the preyto-predator ratio (5). This function should be concave and monotonic, characterized by an initial slope and an upper asymptote. We will use here Equation (6), which is just expres-

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity I 127

into a single "trophic species." For example, the earliest natural food web ever published (the web associated with the cotton plant and the boll weevil, as reported by Pirnrn et al. (1991)), contains nodes consisting of a single species as well as others made up of "29 parasites," "52 other weevils," etc. An extreme case is given by the ecological theory of food chains where all species on the same trophic level are lumped into a single variable. Such an operation is performed under the assumption that, in the given context, aggregation has relatively little influence on the dynamic properties of the food web. This conjecture, which has never been closely investigated theoretically, will be examined on the simple example of Figure 11.3. Aggregawhere the parameters a; and bà no longer tion can be justified only if the properties have the mechanistic (microscopic) interpre- of the detailed web of Figure 11.3a are not tation of Equations (9 and 10). Two auxiliary qualitatively different from those of the agvariables are used: gregated web of Figure 11.3b. o n e key property that we will look at is the response of equilibrium abundance to variation in basal productivity (i.e., production at the prey level). In the case of twospecies predator-prey systems, it was shown by Arditi and Ginzburg (1989) that very different properties appear depending on whether the functional response is prey-dependent or ratio-dependent. Because in preywhere, analogously with h y ,the relative pref- dependent systems the predator isocline is a erence of consumer X j for resource Xi (among vertical line, prey equilibrium abundance is other resource species); the parameter ft is independent of its own productivity: inthe relative competition efficiency of con- creased basal productivity generates a resumer X, (among other predator species) for sponse of predators only. In ratio-dependent resource Xi. Note that the variables X? and systems (and in some intermediate models), xfi)are defined in a nested way. They cannot both species respond positively because the be written as simple explicit functions of den- predator isocline is a slanted line. sities, except in a few cases. Grossly speakIn the case of a three-species system like ing, X? is the part of species X i that is currently being accessed as a resource by species xfi)is the part of species X, that is currently acting as a consumer on species Xi. Equations (12-13) actually describe a competition model, in which predators compete by sharing their prey interspecifically; this is the multispecies extension of the sharing rationale that underlies the single-species ratiodependent functional response. Figure 11.3. N l is the preferred and Nt the less

sion (3) where the argument N has been replaced by NIP. A possible generalization of this response to an arbitrary food web is

The Effect of Species Aggregation When building food webs, ecologists very frequently pool different biological species

preferred prey. Lumping the two species into a single variable is only justified if the properties of the aggregate N , + N2 in system (a) are not qualitatively different from those of the single species N in system (b).

128 I Roger Arditi and Jerzy Michalski Table 11.1. Responses of equilibria to an increase of basal productivity in the detailed model of Figure 11.3a means that equilibrium increases, + means that it stays and the aggregated model of Figure 11.3b. constant. Aggregation preserves the response of total prey when assuming ratio dependence but not with prey dependence. Compare the responses of N\+N: and of N*. Aggregated model

Detailed model N; Prey dependence Ratio dependence

t

or+

T

N,

N;+N;

P*

T T

T ?'

? T

Figure 11.3a, the response to basal production in prey-dependent Holling-type models has been investigated by Kretzschmar et al. (1993). They have shown that, following an increase in productivity of species Nl and N2, the equilibrium value Nl may either not respond or respond positively depending on the specific model used. The values N2, Ni N2, and P* all respond positively. With ratio-dependent interactions, it is easy to check that all equilibria respond positively. Table 11.1 compares these responses in the three-species system of Figure 11.3a and those in the two-species system of Figure 11.3b (as given by Arditi and Ginzburg (1989)). Thus, depending on whether they are described as separate variables or as a single lumped variable, the response of total prey greatly differs in the prey-dependent case. In other terms, even if a prey-dependent interaction were acceptable at a detailed level (say, because the assumption of homogeneous mixing were satisfied for each prey species), it would no longer be acceptable after aggregation of nonidentical prey. In particular, this means that approximating an ecological community as a food chain with a single "species" per trophic level can be a highly questionable operation when working with the prey-dependent model. The ratiodependent model is much more compatible with species aggregation, an operation that is frequently necessary in field studies.

+

P*

N*

T T

Ñ

T

of a given basal species can induce almost any pattern of changes in the equilibria of the various species in the web, and this depends on the values of model parameters, not only on food web structure. We observed the same indeterminacy in computer simulations of nonlinear systems described by (9) and (11). Two things, however, distinguish responses of ratio-dependent webs (11) from prey-dependent webs (7) or (9): simplification of the web structure and structural changes. This can best be shown by examples. Consider a community represented by Figure 11.4a, with dynamics governed by (1 1). Assume that growths of Xi and X2 are described by the same separate logistic functions and that X4 prefers X2 to Xl but is more efficient than X3 in competition for Xi (i.e., n24 > hI4 and P14 > pi3). Assume also that the difference between the relative preferences is greater than the difference between the relative efficiencies (i.e., (hXhl4)/ (h24+ h14) > (p14-(313)/(Pl4+(3 13)). Then, if the carrying capacities of Xi and X2 are the same, one finds at equilibrium x F x ~ , X P= 0 x4c(1) = 0 x4^ = x4, i e X3 feeds exclusively Xl on X4 and feeds exclusively on X2. The link from XI to X4 is not realized: X4 is still ready to eat Xi, but it doesn't; it is outcompeted by X3, due to the 3

Food Web Responses to Increased Primary Productivity Abrams (1993) has shown that, in LotkaVolterra food webs (Equation 7), the response to increased nutrient input was essentially indeterminate: Increasing productivity

Figure 11.4. Change of the effective food web structure due to a rapid increase of the carrying capacity of XI. Dotted arrows indicate possible, but effectively nonexisting links.

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity I 129

interplay of competition efficiencies and food preferences. Thus, the number of links effectively present is less than the number of potential links: the food web structure is simplified. This can happen because the strengths of the links are dynamic variables themselves. Now, if this equilibrium is disturbed by a sudden increase of Kl, then the link from X1 to X4 reappears (x4'() is no longer zero) and, eventually, a new equilibrium is reached at which X3 becomes extinct (Figure 11.4b). The new equilibrium will only be attained if the increase of Kl is sufficiently large and rapid (how large and rapid depends on all parameter values). Thus, it depends on the history of the system. This also means that alternate equilibrium states can exist for the same set of parameters. Changes of the effective structure are due to the coexistence of food preferences and of consumption competition in the model. This allows changes of diet in response to abundances of all species in the web. Simplifications and changes of web structure can be further illustrated by a more complicated example. We considered the food web represented on Figure 11.5. For the growth functions of the basal species we chose

i.e., we assumed no competition between plants. Using computer simulations to solve Equation (1 1) for this web, we found many reasonable parameter sets for which all 11 species coexisted in a stable equilibrium. However, only a fraction of all possible links effectively existed at equilibrium. One of the effective equilibrium structures produced by a given set of parameters is presented on Figure 11.6a. We then perturbed this parameter set by increasing K2 by a factor of 3. The system found a new equilibrium with a different effective structure: One link disappeared and another one appeared (Figure 11.6b). Note that, if we superimpose the webs of Figures 11.6a and 11.6b, we obtain a web with more links than each of the composing webs. When we also varied other parameters (carrying capacities of other plants, competition efficiencies, and food preferences), we obtained many different structures (not shown here). If all realized structures are superimposed, a much more complicated picture is obtained. This can explain why naturalists tend to describe webs with many more links than theoreticians would like to see (Pimm, 1982). As is well known, naturalists tend to build webs by accumulating knowledge obtained in different seasons and at different places (i.e., with different parameter values).

Conclusions

Figure 11.5. Food web used in the simulation. Species 1-4 are plants, species 5-7 are herbivores, and species 8-1 1 are carnivores. Each of the carnivores can eat all herbivores and all the other carnivores with the exception of the top carnivore (species l l ) which is free from predation. The arrows indicate potential links.

We have argued in this chapter for the need of general food web models that include nonlinear (i.e., non-Lotka-Volterra, see the Introduction) trophic interactions. We have explained the two logical requirements of (Cl) invariance under aggregation of identical species, and (C2) mathematical separation of disconnected subwebs. Then, we have shown that it is indeed possible to build such models and we have given mathematical expressions corresponding to generalizations of three nonlinear functional responses: (1) Holling; (2) DeAngelis-Beddington; and (3) Hollinglike ratio-dependent. Finally, we have used these expressions, particularly those for model (3) which is the most different from the usual Lotka-Volterra food webs, to address the question of species aggregation and the question of web responses to basal pro-

130 I Roger Arditi and Jerzy Michalski

Figure 11 6. Effective equilibrium realizations of the food web of Figure 11.5 for different values of the carrying capacity of species 2. (a) K 2 = 0.1; (b) K2 = 0.3. The arrows indicate effectively existing links.

duction. Although many problems remain to be investigated, the present analysis permits the statement of three points that we believe important. First, conditions C l and C2 are required for any general-purpose model of food webs. They are not necessarily required for every model of every specific community. For example, we have shown in the Appendix that a commonly used model for the case of one predator feeding on two prey does not fulfill condition C l . However, this model may be acceptable in some conditions, such as if the two prey species are known to be different, not substitutable, and not separately limiting. Indeed, Vandermeer (1993) mentioned that he was aware that his model could not apply to all situations. Second, on the question of aggregation, we have shown that, in prey-dependent models with two basal species and one consumer, the qualitative response of the basal level to increased basal productivity depends crucially on whether the species are lumped or not; in ratio-dependent webs, the response does not change with aggregation. This finding is particularly relevant to the theory of food chains which describes ecosystems with a single lumped species at each level. For example, prey-dependent food chain models predict that herbivores do not respond to basal productivity in chains with three levels (plants-herbivores-camivores) . They respond only in chains with two or four levels (Hairston et al., 1960; Slobodkin et al., 1967). Empirical data collected along gradients of productivity in terrestrial and aquatic ecosystems show that herbivore biomass increases with productivity and that herbivore

and plant biomasses are correlated (e.g., McCauley et al. (1987), McNaughton et al. (1989), Moen and Oksanen (1991), and Cyr and Pace (1993)). Prey-dependent food chain theories can conform to these observations only with the additional ingredient that the number of trophic levels varies with productivity (Oksanen et al., 1981; Persson et al., 1988). However, applicability of this theory is restricted to perfect chains because of the nonconservation of response patterns through aggregation. As stated by Pirnm (1992), "the one-species-per-trophic-level models work when the systems are indeed that simple." For their part, ratio-dependent models do not encounter these difficulties since we have shown here that they are qualitatively insensitive to aggregation and since they predict positive response to basal productivity at all trophic levels (Arditi and Ginzburg, 1989; Arditi et al., 199 1; Ginzburg and Akcakaya, 1992). Third, our first simulation analyses of ratio-dependent food webs confirm that detailed web structure matters when studying responses to basal productivity. This has been shown both experimentally (Leibold and Wilbur, 1992) and theoretically in prey-dependent webs (Abrams (1993) and, indirectly, Yodzis (1988)). Indeed, as food webs become more complex, indeterminate and counterintuitive responses become more frequent, particularly because of intraguild predation (Polis and Holt (1992) and references therein). Sufficiently complex ratio-dependent webs present qualitative properties that have not been observed in prey-dependent webs. First, at equilibrium, the web of realized links may be simpler than the initial web

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity I 131

of potential links, i.e., species may abstain from consuming some resources that they are able to consume. Second, the realized structure depends on all parameter values and, as these values change, the structure can change, i.e., species may switch resources. Both of these phenomena are known to occur in nature. It remains to be seen whether they occur at conditions comparable to those of the theoretical models of foraging theory and whether they represent optimal diet strategies. Third, for given parameter values, alternative stable states can exist. Different structures will be realized depending on the food web history and shifts from one structure to another can be induced by strong perturbations. Taken together, these three properties can explain why empirical webs built from accumulated observations made at different places and different times tend to be very link-rich although they are unfeasible from the theoretical point of view.

If N l and N 2 are identical species, then r 1 r2 = r, yl2 = y2 = 1 , a l = a2 = a , K l = K2 = K , bl = b2 = b. If we consider the joint variable N = N l N2, we can write =

+

Acknowledgments Our concern for logical consistency of food web models was aroused by a letter from Egbert G. Leigh, Jr. We thank Resit Akgakaya, Mark Burgman, Wayne Getz, Lev Ginzburg, Bob Holt, and Stuart Pimm for stimulating remarks. This work was supported by the Swiss Priority Program on the Environment (grant 5001-034810) and by the French Programme 'Environnement, Vie et Sociites' (GDR 1107).

Appendix We give here an example of a system of trophic equations that does not satisfy the logical criteria given in this chapter. Consider a system of two competing prey and one predator. It might seem a natural thing to do to write functional responses of the predator for each prey as separate Holling functions, as this was done by Leon and Tumpson (1975), Vandermeer (1993), Hastings (1995):

The basal production terms collapse nicely but, in the consumption term, the variables N l and N2 persist as independent variables. This is absurd since the two species are supposed to be identical. This shows that the system (Al) does not satisfy Criterion 1. Another way of writing the dynamic equations is to consider the predator-functional response as a function of a weighted sum of prey abundances: All prey species combine to saturate the predator. It can be checked that the following system satisfies Criterion 1.

132 I Roger Arditi and Jerzy Michalski

,

d p - a(elhlN1 + e2h2N2) dt 1 + b,hiNl + b2h2N2

(A,&l

- I J ~ P. This system, which is the application of Equation (9) to the three-species web considered, is somewhat overparameterized and can be simplified with appropriate changes of parameters.

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