Functional Response of a Small Mammalian Herbivore: The Disc Equation Revisited

Functional Response of a Small Mammalian Herbivore: The Disc Equation RevisitedAuthor(s): Per LundbergSource: Journal of Animal Ecology, Vol. 57, No. 3 (Oct., 1988), pp. 999-1006Published by: British Ecological SocietyStable URL: http://www.jstor.org/stable/5107 .

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Journal of Animal Ecology (1988), 57, 999-1006

FUNCTIONAL RESPONSE OF A SMALL MAMMALIAN HERBIVORE: THE DISC EQUATION REVISITED

BY PER LUNDBERG

Department of Wildlife Ecology, Swedish University of Agricultural Sciences, S-901 83 Umea, Sweden

SUMMARY

(1) A laboratory study of bank voles feeding on willow shoots during the winter showed a decelerating (Type II) functional response of the voles to increasing shoot density.

(2) The 'disc' equation showed the best fit to the data with respect to the number of shoots attacked.

(3) The disc equation and the 'random parasitoid' equation showed equally good fit to the data with respect to the biomass consumption.

(4) In no case could the 'random predator' equation produce realistic estimates of the handling time. It is argued that the predator mode of foraging is an inappropriate way of modelling many herbivore foraging patterns.

(5) The parasitoid mode of foraging by vertebrate herbivores and its implications for plant-herbivore dynamics is briefly discussed. In particular, the possible effects on plant mortality rates should be considered.

INTRODUCTION

The instantaneous predation rate in relation to prey density, i.e. the functional response of a predator (Holling 1959), is an important concept for the understanding of the dynamics of predator-prey systems. Analyses of functional responses have also been applied to other exploitive systems such as host-parasitoid and plant-herbivore systems (e.g. Hassell 1978; Hassell & Waage 1984; Caughley & Lawton 1981; Crawley 1983). The kind of functional response most commonly found in simple two-species systems is the Type II of Holling's (1959, 1966) original classification. The most frequently used model for this monotonically decelerating functional response is the 'disc' equation.

Na=aNT( + aNTh) (1)

where Nis the initial prey density, Na the number of prey eaten by a single predator during the total time T. a is the instantaneous search rate (attack coefficient) and Th the handling time per prey. It has been shown that the disc equation cannot properly model the functional response of the predator unless the prey density is unchanged during T. In order to cope with this problem, Royama (1971) and later Rogers (1972) suggested that the 'random predator' equation

Na = N{1 -exp [-a(T-NaTh)]} (2)

would be a more appropriate model of most real systems where the initial prey density actually is decreasing during the course of exploitation by the predator. However, eqn (2)

999



Functional response of herbivores

is an implicit equation, i.e. the dependent variable (Na) appears on both sides of the equation which cannot be solved analytically. Yet, the random predator equation has become widely used in spite of its practical problems, although sometimes inappropria- tely modified (cf. Juliano & Williams 1987 for further discussions). A further modification of eqn (1) is the 'random parasitoid' equation (Rogers 1972; Hassell 1978),

Na= NN{1 -exp[-aT/(1 + aNTh)]} (3) which takes into account the fact that hosts are not removed and that they can be revisited during T.

The functional responses of herbivores have been much less studied than predators and parasitoids. For mammalian herbivores, most of the information has been gathered for sheep (e.g. Allden & Whittaker 1970) and reindeer (Rangifer tarandus (L.) (e.g. White et al. 1981). Data from free-ranging moose (Alces alces (L.)) (Renecker & Hudson 1986) indicated a Type II functional response, whereas Batzli, Jung & Gutensperger (1981) argued that brown lemmings (Lemmus sibiricus (Kerr)) showed a linear (i.e. Type I) functional response when subjected to sod blocks of natural vegetation. However, in few of these empirical studies have the nature of the functional response been explicitly examined. Theoretical studies of plant-herbivore systems (e.g. Noy-Meir 1975; Caughley & Lawton 1981; Caughley 1982), on the other hand, have usually used simplified, one- parameter models of functional responses when analysing the population dynamics of the interactors.

In this study I have extended the analysis of functional responses to include a vertebrate herbivore, the bank vole (Clethrionomys glareolus (Schreb.)) feeding on willow shoots. By testing a commonly used set of Type II models, i.e. eqns (1)-(3), I will argue that the foraging pattern of many vertebrate herbivores should be regarded more like parasitoids than predators. In fact, Holling's (1966) original disc equation seems to be most appropriate for this kind of system.

MATERIALS AND METHODS

Organisms Bank voles were live-trapped in September-October 1986 in abandoned pastures close

to Umea, North Sweden (63?46'N,20?18'E). The animals were transferred to individual cages in the laboratory and provided commercial rabbit/guinea-pig feed, hay and water ad lib. The thirty-two voles were kept in natural daylight conditions and at approximately 10?C throughout the experimental period.

Willow shoots used in the experiments were male Salix myrsinifolia Salisb.-phylicifolia L. long-shoots from approximately ten clones close to Umeea. Only shoots from the middle-upper parts of the 2-3 m high shrubs were used, and were collected for experiments from the field within 2 h prior to the trials.

Procedure

Each experiment consisted of six simultaneous trials with shoot densities and different individual voles of both sexes. Altogether eight such experiments were performed from 11 February to 13 March 1987. Each trial was conducted in a 1 x 1 m arena with 0-75 m high plate walls and with the floor covered with gravel. The shoots were placed in the arena in a regular pattern at a distance of 14-5 cm from each other. The densities used were 2, 4, 9, 16, 25 and 36 shoots m-2. The shoots were selected to be as equally large as possible within

1000



TABLE 1. Summary of the experimental set-up for the six different willow shoot densities used in the functional response experiments with bank voles. Eight trials

were performed for each shoot density

Shoot Mean shoot Mean total biomass density (m-2) mass (g) S.E. available (g) S.E.

2 1.28 0.03 2.55 0.06 4 1.31 0.03 5.23 0.13 9 1.29 0.03 11-59 0.25

16 1.29 0.02 20-72 0.44 25 1.28 0.03 31-92 0.77 36 1.24 0.01 44-61 0.33

and between all trials, and were clipped and weighed to the nearest 0-01 g before being placed in the experimental arena. The position of each shoot was noted so that the biomass removed could be determined for each individual shoot. The shoot characteristics are summarized in Table 1.

Individual voles were tested in the arenas for 3 h around noon under natural daylight conditions and at 5-10 ?C. They were provided with water and a small hide throughout the trial. After the termination of the experiments each shoot was determined as attacked or not and again weighed. Biomass loss was recorded for each shoot. The average water loss per shoot during the 3-h trial was determined from the unattacked shoots. The shoots lost 0-2-7% (mean 0-6%) of their initial weight due to desiccation and all biomass losses were corrected accordingly.

Analysis The three different kinds of Type II functional response models (eqns (1)-(3)) were

tested to fit the experimental data with non-linear least-square procedures (SAS Institute Inc. 1984, 1985). Individual data points were used in the analyses to avoid underestimates of the S.E.s of parameter estimates (Juliano & Williams 1987).

RESULTS

Since herbivores are often able to attack a large number of 'prey' but commonly only partially consume them, it is not unambiguous in which way 'predation' rate should be measured. In this study practically all shoots were attacked in each trial almost irrespective of shoot density (Fig. 1). Not surprisingly, bank voles were able to exploit the complete arena during the 3-h trials, although shoots were left untouched more often at the highest densities (Fig. 1). The consumption rate, on the other hand, showed a more

pronounced decelerating pattern with increasing shoot density (or rather total biomass available which was highly correlated to shoot density; rs = 0-997, P < 0-001, N= 48) (Fig. 2). Although the random predator equation showed an adequate fit to both the attack rate and consumption rate data, it could in no case produce realistic estimates of Th (i.e. Th always negative and/or greater than total time irrespective of iteration method used). However, an even better fit, as judged from inspections of the S.E.s and C.V.s of the parameter estimates (cf. Juliano & Williams 1987), could be achieved with both the random parasitoid and disc equations. Table 2 summarizes the parameter estimates and their accuracy for both the attack and consumption cases. The random parasitoid and disc equations produced almost identical parameter estimates for the consumption rate

P. LUNDBERG 1001



1002 Functional response of herbivores

30-

a) /

0

O /l w 20- 0 o /

0

.- 10- E

z

X , I I I I I I

0 10 20 30

Number of shoots available

FIG. 1. Functional response of bank voles feeding on willow shoots with respect to number of shoots attacked. Data points are means (? S.E.) of eight trials. The curve is the best non-linear

least-square fit of the disc equation using data from individual trials.

3-0-

o 20-

0

0

E i 1.0

CD I " I l I l I l I l I

0 10 20 30 40 50

Biomass available (g) FIG. 2. Functional response of bank voles feeding on willow shoots with respect to biomass removed. Data points are means (?+ S.E.) of eight trials. The curve is the best non-linear least-

square fit of the disc and random parasitoid equations using data from individual trials.

but they differed for the rate of shoot attack data. Here, the random parasitoid equation gave much higher estimates of both Th and a, and also higher C.V.s of the estimated parameter values (Table 2). Thus, for the rate of biomass consumption as a function of shoot density both the random parasitoid model and the disc equation gave equally good descriptions of the data. For the shoot attack rate, however, the disc equation produced parameter estimates with less variability than did the random parasitoid equation.

Although the consumption rate in each trial increased with increasing total biomass available, the consumption per shoot clearly decreased (Fig. 3a). The same pattern was



P. LUNDBERG

TABLE 2. Parameter estimates of the functional response functions for bank voles feeding on willow shoots, where a = the attack coefficient and Th = handling time

Number of shoots attacked Disc equation

Random parasitoid equation

Random predator equation

Biomass consumption Disc equation

Random parasitoid equation

Random predator equation

Parameter estimate

a 0326 Th 0-008 a 1-115 Th 0-012 a 0350 Th -0-150

a 0-050 Th 0-912 a 0-054 Th 0-913 a 0005 Th -3-557

Asymptotic S.E.

0-029 0.009 0-610 0-016 0-141 0-118

0-013 0-192 0-016 0-192 0-001 0-986

C.V. (%) 61-6

780-0 379-1 924-0 319-5 545-2

180-2 145-9 205-3 145-7 138-6 192-1

F

874-7

873-7

1001-5

139-7

139-7

120-1

R2

(%) 97.4

97.4

97-0

86-9

86-9

84-2

(a)

Ib) S 0 I'

a) 0 >0 0.0

LQ) ci) -0

-ii)

(- C

f ~~~~0 0

f ~~~~~~0. 0-

0.

10 20 30

0-2

0*I I {

I I I I I

10 20 30 0

Shoot density

FIG. 3. The biomass removed per shoot (a) and proportion of biomass removed per shoot (b) by bank voles feeding on willow shoots at different densities (number of shoots per m2).

found for the proportional consumption per shoot (Fig. 3b) since all shoots used were

virtually of the same initial mass. The fact that consumption per shoot decreased with increasing shoot availability indicate that the voles' exploitation of shoots follow, at least qualitatively, the marginal value theorem (Charnov 1976; Stephens & Krebs 1986), i.e. the shoots can be regarded as patches themselves and that time spent handling each shoot decreases with increasing intake rate experienced in the environment. As intake rate decelerates at the highest shoot densities, the change in proportional consumption per shoot should also decrease more slowly towards higher densities. This seemed in fact to be the case (Fig. 3). Since the initial shoot mass varied so little, it was impossible to detect any differences in consumption of individual shoots within each trial.

1003

-o a)

u 0.3 0,

0

0 0 .0

(n

0.2 a) Q.

V

w

0 -

S U, 0~ 0

co0 I I

- f I -



Functional response of herbivores

Na

Nax -

N

100- 0 0"

o

o Pcrit--- -

?E 0 Ncrit N max N

FIG. 4. The possible effect of a Type II functional response (upper graph, where N= shoot density and Na = number of shoots attacked) combined with partial consumption of ramets (cf. Fig. 3b) on herbivore-plant dynamics. If there is a critical level of proportional consumption of individual shoots (Pcrit) below which biomass losses due to herbivory have no effects on survival and future reproduction (lower graph), the death rate of the plant population will be disproportionately low

at population densities above Ncrit.

DISCUSSION

The random predator equation was developed from the disc equation in order to properly cope with the fact that the prey density usually is significantly changed during the exploitation of the prey population (Royama 1971; Rogers 1972; Hassell 1978). This is true for most predators but not necessarily so for many herbivores which often only consume parts of the plants. If the consumption per plant or shoot is relatively small, and if not only a small proportion of the plant is available forage to the animal, it is quite possible that the 'prey' density in fact remains virtually constant, or nearly so, during an experimental period. In fact, there seem to be good reasons to regard the foraging pattern of many herbivores as more parasitoid than predatory; the individual plants can be reattacked and they are not necessarily killed after initial herbivory (but might be so after repeated grazing or due to secondary infections). The results from the present study support this view. The random parasitoid equation and the disc equation produced the best fit to the data, both in terms of shoots attacked and biomass consumption. The random predator equation, usually thought of as an improvement of the disc equation for most experimental situations, was apparently unsuitable for this plant-herbivore system.

The marginal value theorem (Charnov 1976), or more recent modifications of it (e.g. McNair 1983; Stephens et al. 1986), has been proved to be a useful tool for the understanding of a variety of foraging patterns. Recently, it has also been argued that the concept of 'patches as prey' (Stephens & Krebs 1986) is appropriate when diet choice and patch use is a simultaneous and equivalent problem to the forager. In the present study, the problem of 'diet choice' was reduced since all shoots were as similar as possible, both with respect to origin and size. Rather, the exploitation of shoots in this situation seems to be reduced to a straightforward patch problem to the voles, i.e. the shoots might be regarded as patches themselves and exploited accordingly. Other recent studies have indeed successfully applied the 'plant as patch' concept to herbivores foraging on discrete

1004



individual plants (or rather ramets) (e.g. Vivas & Saether 1987). Moreover, the rate at which such patches are exploited should, according to the present results, not primarily be analysed as a predator-prey problem.

The functional response of herbivores has been paid comparatively little attention in the literature (but see Crawley 1983), seemingly due to the fact that plant-herbivore dynamics is still a relatively unexplored field. If considered in this context (e.g. by Noy- Meir 1975; Caughley & Lawton 1981; Crawley 1983), the functional response has usually been assumed to take the form of a Type II curve, although rarely related to any explicit models of the foraging behaviour. Other (i.e. empirical) studies have indeed found curvilinear relations between resource density and intake rate indicating a Type II functional response (e.g. Arnold & Dudzinski 1967; Allden & Whittaker 1970; Renecker & Hudson 1986). In none of these cases, however, have the data been tested to fit any functional response model. There are also examples of linear functional responses from at least two vertebrate herbivores, i.e. reindeer (White et al. 1981) and brown lemmings (Batzli, Junge & Gutensperger 1981). Common to these two species is that they inhabit a relatively resource-poor environment in the arctic or subarctic, and there are reasons to believe that most vertebrate herbivores in such low-productive areas are food-limited (e.g. Fretwell 1977; Oksanen & Ericson 1987). A linear functional response is indeed what we would expect if this is the case (cf. also Crawley 1983). In contrast, herbivores in more productive habitats where food is not the sole limiting factor would respond differently, i.e. showing a decelerating (Type II) functional response curve, at least in cases of a single food resource when no 'switching' (Holling 1965; Murdoch & Oaten 1975) potentially can occur. Consequently, the suggestion by Batzli, Jung & Gutensperger (1981) that the crucial difference in response to available forage is to be found between ruminants (e.g. sheep, reindeer) and non-ruminants (e.g. voles) seems to be doubtful.

Results from functional response tests in an oversimplified experimental situation should of course be treated with caution when applied to real world systems (cf. Abrams 1982). Therefore, the implications of the findings from the present study for plant- herbivore dynamics cannot be other than speculative. However, the foraging pattern showed by bank voles in these experiments lead to some remarks on the interaction between herbivore foraging and plant population structure. Let us assume that herbivores indeed can influence the mortality rate in plant populations (e.g. Ericson & Oksanen 1987; but see also Crawley 1983 for review). The situation depicted in Fig. 4 would then lead to a disproportionately low plant (or shoot) mortality rate at sufficiently high plant densities. Strictly, this would require a more or less homogenous shoot stand without too large differences in individual qualities of the shoots. If there is a larger set of plant species and the functional response to the different species in isolation differ substantially, the outcome of the dynamics would become more equivocal (cf. Colton 1987). Nevertheless, the fact that individual shoots are not necessarily killed after a herbivore attack does indeed influence the herbivore-plant dynamics to become rather unlike predator-prey systems. Future analyses of the interactions between plant populations and their herbivores should take such facts into consideration.

ACKNOWLEDGMENTS

Marten Astrom skilfully assisted at various stages of the study and he and Kjell Danell commented on the manuscript. Financial support was received from the Swedish Natural Science Research Council.

P. LUNDBERG 1005



1006 Functional response of herbivores

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Arnold, G. W. & Dudzinski, M. L. (1967). Studies on the diet of the grazing animal. III. The effect of pasture species and pasture structure on the herbage intake of sheep. Australian Journal of Agricultural Research, 18, 657-666.

Batzli, G. O., Jung, H.-J. G. & Gutensperger, G. (1981). Nutritional ecology of microtine rodents: linear foraging-rate curves for brown lemmings. Oikos, 37, 112-116.

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Charnov, E. L. (1976). Optimal foraging: the marginal value theorem. Theoretical Population Biology, 9, 129- 136.

Colton, T. F. (1987). Extending functional response models to include a second prey type: an experimental test. Ecology, 68, 900-912.

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Ericson, L. & Oksanen, L. (1987). The impact of controlled grazing by Clethrionomys rufocanus on experimental guilds of boreal forest floor herbs. Oikos, 50, 403-416.

Fretwell, S. D. (1977). The regulation of plant communities by food chains exploiting them. Perspectives in Biology and Medicine, 20, 169-185.

Hassell, M. P. (1978). The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton.

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(Received 9 February 1988)



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Functional Response of a Small Mammalian Herbivore: The Disc Equation Revisited