perturbation experiments in community ecology: theory and practice

14
Perturbation Experiments in Community Ecology: Theory and Practice Author(s): Edward A. Bender, Ted J. Case and Michael E. Gilpin Source: Ecology, Vol. 65, No. 1 (Feb., 1984), pp. 1-13 Published by: Ecological Society of America Stable URL: http://www.jstor.org/stable/1939452 . Accessed: 23/09/2013 14:43 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology. http://www.jstor.org This content downloaded from 192.236.36.29 on Mon, 23 Sep 2013 14:43:20 PM All use subject to JSTOR Terms and Conditions

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Page 1: Perturbation Experiments in Community Ecology: Theory and Practice

Perturbation Experiments in Community Ecology: Theory and PracticeAuthor(s): Edward A. Bender, Ted J. Case and Michael E. GilpinSource: Ecology, Vol. 65, No. 1 (Feb., 1984), pp. 1-13Published by: Ecological Society of AmericaStable URL: http://www.jstor.org/stable/1939452 .

Accessed: 23/09/2013 14:43

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology.

http://www.jstor.org

This content downloaded from 192.236.36.29 on Mon, 23 Sep 2013 14:43:20 PMAll use subject to JSTOR Terms and Conditions

Page 2: Perturbation Experiments in Community Ecology: Theory and Practice

Ecology. 65(1), 1984, pp. 1-13 ? 1984 by the Ecological Society of Amenica

PERTURBATION EXPERIMENTS IN COMMUNITY ECOLOGY: THEORY AND PRACTICE'

EDWARD A. BENDER2 Department of Mathematics, C-012

TED J. CASE AND MICHAEL E. GILPIN Department of Biology, C-016, University of California at San Diego,

La Jolla, California 92093 USA

Abstract. We analyze perturbation experiments performed on real and idealized ecological com- munities. A community may be considered as a black box in the sense that the individual species grow and interact in complicated ways that are difficult to discern. Yet, by observing the response (output) of the system to natural or human-induced disturbances (inputs), information can be gained regarding the character and strengths of species interactions. We define a perturbation as selective alteration of the density of one or more members of the community, and we distinguish two quite different kinds of perturbations. A PULSE perturbation is a relatively instantaneous alteration of species numbers, after which the system is studied as it "relaxes" back to its previous equilibrium state. A PRESS perturbation is a sustained alteration of species densities (often a complete elimination of particular species); it is maintained until the unperturbed species reach a new equilibrium. The measure of interest in PRESS perturbation is the net change in densities of the unperturbed species. There is a very important difference between these two approaches: PULSE experiments yield in- formation only on direct interactions (e.g., terms in the interaction matrix), while PRESS experiments yield information on direct interactions mixed together with the indirect effects mediated through other species in the community. We develop mathematical techniques that yield measures of ecolog- ical interaction between species from both types of experimental designs. Particular caution must be exercised in interpreting results from PRESS experiments, particularly when some species are lumped into functional categories and others are neglected altogether in the experimental design. We also suggest mathematical methods to deal with temporal and random variation during experiments. Fi- nally, we critically review techniques that rely on natural variation in numbers to estimate species interaction coefficients. The problems with such studies are formidable.

Key words: community structure; competition; disturbance; perturbation; species interaction.

INTRODUCTION

To be effective, community ecology must have an- swers to the following questions. First, which species are dynamically connected to which other species? Second, which further species, though not mechanisti- cally connected, are ultimately connected through in- teractions involving intermediate species? Third, what are the signs of these interactions? That is, for a pair of species A and B, how will a change in the density of one affect the other, and what is the relationship between immediate and ultimate effects? Fourth, in the face of interactive loops of arbitrary length going in many directions, how is o ne to delimit subsystems for study? That is, what is the effect of ignoring species and what rules can be given concerning which species to ignore? Fifth, what are the consequences of lumping species into a single dynamic entity, e.g., all ants or all granivores?

In the 1960's and 1970's, the fashionable and con- ventional approach to the study of interspecific com-

I Manuscript received 25 August 1981; revised 13 Decem- ber 1982; accepted 17 January 1983.

2 Authors are listed in alphabetical order.

petition was niche analysis (Mac Arthur 1958, Levins 1968). Here one measures the ecological overlap be- tween species' niches, for example, the similarity of their diet, habitat use, or time of foraging. The as- sumption is that overlap can be equated with an in- teraction coefficient in a dynamic. model such as that of Lotka and Volterra. This method has the virtue of being rapid and convenient, but it suffers from a num- ber of defects (Abrams 1975, 1980a, b, Hurlbert 1978, Case and Casten 1979, Lawlor 1980, Thomson 1980). Two stand out. First, the resources that the consumer species share may not limit consumer population growth or density, and therefore the link between overlap and dynamical effects may not be justified. Second, even if the measured niche axes are relevant from a dynamic perspective, how does one numeri- cally convert niche overlap to dynamic impact? A number of potential formulae exist in the literature, yet all are ad hoc, and none has received any empirical validation.

Given the shortcomings of this passive, observa- tional approach to the study of interspecies competi- tion, ecologists have been turning to experimental ap- proaches, in particular to perturbation studies. Here

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Page 3: Perturbation Experiments in Community Ecology: Theory and Practice

2 EDWARD A. BENDER ET AL. Ecology, Vol. 65, No. 1

N3 N3 PULSE

PRESS

Initial Growth Vector A~N33{

N1 N1

X- d 33 / dN13

r/ N23 X t 3 Components

/ / (3 Of Initial Growth

,LN13 ~ ~ ~ ~ ~ dt~ ) Vector

N2 N2

FIG. 1. A geometrical representation of PRESS and PULSE type perturbation experiments for three species communities. Each axis represents the numerical abundance of one of three species. The large open circles (0) represent the community composition before perturbation. Perturbations are represented by a solid arrow ending in an open circle. The dotted line shows the trajectory of the community following perturbation. In both experiments only species 3 has been perturbed. In the PRESS experiment, species 3 has been removed and is maintained at zero density, while species 1 and 2 adjust numerically to a new equilibrium (large filled circle [0] at the end of the dotted trajectory). The resulting values of ANi3 correspond to those used in Eq. 8. This single PRESS experiment yields estimates for the third column of the AN matrix. After similar perturbation experiments for species I and 2, the entire AN matrix can be calculated, then inverted to yield the terms required to calculate the interaction coefficients, i.e., the A matrix (Eq. 9).

In the PULSE experiment we require estimates for the initial growth rate of all three species immediately after the one- time removal of some species 3 individuals. These rates may be represented geometrically as the magnitude of each of the three component vectors depicted here on the N1 N2 plane. The dNi3/dt correspond to the terms in Eq. 5. This single PULSE experiment directly yields estimates for the third column of the A matrix. Similar experiments involving perturbations in species I and 2 would yield the remaining two columns of A.

one experimentally alters the density of some species in a community and determines consequent changes in the densities or behaviors of other species. Such studies have been made by Connell (1961), Dayton (1973), Schroeder and Rosenzweig (1975), Seifert and Seifert (1976, 1979), Brown, Davidson and Reichman (1979), Holmes et al. (1979), Hairston (1981), Wise (1981), Smith (1981), Smith and Cooper (1982), and Tinkle (1982). The hope is that the resulting community dynamics will produce conclusive proof regarding the presence or absence and strength of present-day biotic interactions, such as competition or predation be- tween species pairs. It is recognized by most workers, however, that interactions in the past may, through coevolution, become so attenuated that present-day interactions are weak, yet the community maintains structural attributes of its past history (see, e.g., Con- nell 1980). Detecting "the ghost of interactions past" is a subject beyond the scope of this paper.

Two TYPES OF PERTURBATION:

AN OVERVIEW

In what follows we assume that we have commu- nities available for investigation that are fully open to undisturbed enumeration, that the logistics of actually manipulating the population sizes of one or more species is feasible, and that suitable replicates and

controls pose no problems. Thus, we wish to discuss irreducible limits to knowledge about such systems rather than real-world problems of observation.

Suppose that we have a community of n species that we perturb by changing the density of some subset s (normally a single one) of the species. We wish to stress that there are very different designs that can be applied to such an experiment. In the first type (here- after referred to as PULSE experiments) perturbation is made and then the community is allowed to respond. As it does so, we measure the initial rates of popula- tion growth of each of the n species (including the perturbed species). In the second type (hereafter re- ferred to as PRESS experiments), the densities of per- turbed species are altered and maintained at predeter- mined levels by adding or removing individuals as needed. In the face of this maintained alteration, the other n - s unmanipulated species will readjust nu- merically. After this adjustment ends and the com- munity has reached a new equilibrium, one measures the net change in density of each of the n - s species. Included in PRESS experiments are experimental de- signs in which one or more species are totally removed from experimental plots. Fig. 1 gives a graphical in- terpretation of these two different techniques.

The proper interpretation of results from PULSE and PRESS experiments are often confused. To insure

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Page 4: Perturbation Experiments in Community Ecology: Theory and Practice

February 1984 PERTURBATION EXPERIMENTS 3

a proper interpretation, one must have a clear model of community interactions. We use a mathematical model to avoid the ambiguities inherent in a verbal model. The state variables of this model are numbers (or densities) of each species, and the parameters of the model are the unknowns that we wish to deduce from our experiments. The generalized Lotka-Volterra equations provide a suitable starting point in this ex- ercise. The per capita rate of growth of species i is:

NdJ =ci Ki - E aijNj for i = 1 to n, (1)

where aii = 1. Unless there is no self-interaction of i, this assumption on a,, does not reduce the generality of Eq. 1 since a scale factor can be absorbed in ci. Depending on the signs of Ki and aij, we could de- scribe predator-prey, mutualistic, competitive, or amensal interactions between any two species i and j (Mac Arthur 1972, May 1973, Case and Casten 1979). These equations are most often used as descriptions of competitive interactions, in which case aij > 0 is the competition coefficient of species j on i, K, is the carrying capacity of species i, and ci = rilKi, where ri is the intrinsic rate of growth of species i.

We assume that a stable, feasible equilibrium solu- tion Ni to Eq. 1 exists and is found by setting growth rates equal to zero, yielding

aijNj = Ki. (2) j=i

When this equilibrium is substituted into Eq. 1 we obtain

dAj1= C, E aij(Nj - Nj), for i = 1 to n. (3) N dt 1

In the next two sections we derive protocols by which the parameters in Eq. 3 (i.e., the ci's and the aij's) may be determined by PULSE and PRESS perturba- tion experiments. Much of the mathematics underlying what we call PULSE and PRESS experiments has been discussed before in the ecological literature (May 1973, Richardson and Smouse 1975, Levine 1976, Holt 1977, Lawlor 1980, Schaffer 1981). However, heretofore, no one has extended this theory to develop a protocol for community perturbation experiments in ecology.

PULSE perturbation experiments

Consider a set of species whose dynamics follow Eq. 1 and whose pre-perturbation equilibrium densi- ties Ni are known. A perturbation is made by adding or subtracting individuals of one or more of these species, and the resulting values of dNildt and Ni are measured (for all i) immediately after perturbation. These values are then substituted into Eq. 3. This gives n equations in the ci's and aij's. If n independent per- turbation experiments are done, the resulting n2 equa- tions can be solved for all the ci's and aij's. One

straightforward set of experiments results in equations that are particularly easy to solve; in experiment only species is perturbed from its equilibrium value, and it is perturbed to a density Nf. The immediately re- sulting values of dNildt (for i = I to n) are denoted by dNjf/dt (Fig. 1). Eq. 3 becomes:

dN ~ I dt N1cja1/Nf - Nf), where i Sf, (4a)

and

d ff - Nf cNf - Nf), (4 b) dt

where we have used the fact that aff = 1. It is apparent from Eq. 4a that when species compete directly (and so aif > 0), the reduction in one species will be fol- lowed by an initial increase in the other (i.e., dNildt >

0). The stronger the direct competition, the greater the response. Notice, however, that if the response of spe- cies i exceeds the response of species k (i.e., dNi1ldt >

dN,.fldt), then we cannot conclude that the direct competitive effect of f on i exceeds the direct com- petitive effect off on k (i.e., aif > akf). This is because the interaction terms in Eq. 4a are also multiplied by the ci term, which includes the intrinsic growth rate.

It is important here to distinguish between direct and indirect interactions. The interpretation of the aij in Eqs. 2 and 3 depends on whether we allow the N's to represent only species on a single trophic level or on mixed trophic levels. Consumer species directly affect the resource species that they eat and vice ver- sa, so the aij's between consumer i and its prey j are always nonzero, but two consumers or two prey will only have nonzero interaction terms between them if they directly interfere with one another. In a single trophic level description, like the typical Lotka-Vol- terra competition equations, resource dynamics are not explicitly considered, and competition for resources, as well as interference competition, is parameterized into the ai's. Thus, in this parameterization, even pure exploitation competitors without interference will have nonzero aij's between them. Whether or not PULSE experiments will detect such exploitation competition depends critically on whether or not resource dynam- ics occur on a much faster time scale than consumer dynamics. We defer discussion of this problem (see Overlooked Species).

We now solve Eq. 4a, b for the unknown aij's and the ci's. Setting = j in Eq. 4a and f = i in Eq. 4b and combining, we obtain:

d''NiN(Ni - N1) a dtan 5)

dN.- , ad 5 d Ni(Nj - Nj)

dN11 dt dt (5b)

Ni (N - Ni)

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4 EDWARD A. BENDER ET AL. Ecology, Vol. 65, No. 1

Every term on the right-hand side is experimentally measurable. A problem arises, however, if species i is not self-limiting (e.g., a prey species limited only by predators), then aii = 0 and dN/ildt = 0, and alir for j #/ i is not well defined; however, ciai~j is well defined. In this case we use

Ci~li dt (5)0 N(NJ - Nj)

instead of Eq. 5a, b. Notice that by performing a single experiment, we

can obtain a column of ciaij terms, and thus after n such experiments, each perturbing a different species, we obtain an entire matrix containing all n2 of the ciaij terms. At this stage we may solve for all the aij terms themselves (unweighted by ci) because aii = 1. Let A

denote the matrix of the aij's. It may be of interest to determine only some of the entries in A by performing less than n experiments. Determining either a single row or column of A requires the full set of n experi- ments. Two experiments are required in practice even to determine a single aiUj. A more important case is the following: suppose we are only interested in a subset of s species. We can determine cla for i, I I to s by doing experiments for all f = Ito s and measuring dNiJ/ dt. This result is important because we may be uncer- tain if other species are involved in the interactions or be unable to measure or perturb those species.

It is also possible to perform n experiments, each involving the simultaneous perturbation of q species (q < n); however, one must insure that all n pertur- bations are linearly independent. As before, at least n experiments have to be performed before one can solve for the entire A matrix.

In practice, of course, there are always unknown error terms associated with the measurement of the Ni's and dNildt's. Replicate experiments will be re- quired to assess these errors, increasing many fold the number of needed experiments. Regression methods may be adopted to find the best estimates of the aij's. Seifert and Seifert (1976, 1979) adopt just this PULSE approach in their experimental analysis of the insect communities living in Heliconia plants. Their regres- sion technique results in estimates of ciaii terms (but not ci or ,ij alone) and should not be confused with attempts by others to apply regression to natural (i.e., unmanipulated) variation in numbers (e.g., Schoener 1974, Hallett and Pimm 1979). These later studies are discussed below (see Natural Variation Techniques).

To achieve the greatest accuracy in Eq. 5, large per- turbations are desirable since they will make dNijA1/t, dNij/dt, Nj - Ni and N1 - Nj large and thus less sen- sitive to errors in measurements and random fluctua- tions. On the other hand, Ni should not be too small since it is a factor in the ratios of Eq. 5. A reasonable time interval is needed to estimate dNi/dt, but the Ni's

must not change so much that dNildt changes appre- ciably over the measurement period. We cannot re- move a species completely in PULSE perturbation de- signs because its per capita growth rate ([dNj1/dt]/Nj) would then be impossible to determine.

PRESS perturbation experiments

Again imagine a set of n species obeying Eq. 1 and at a stable equilibrium. These equilibrium densities are given in matrix form as

N = A-1K, (6)

where A l is the inverse matrix of A; the latter contains the direct interaction terms aii. N is the column vector of equilibrium densities, and K is the column vector of Ki's. From Eq. 6 it is apparent that if K1j is varied and

t " K's for i ?j are fixed, then

AN= (7)

Thus the selective alteration of different species car- rying capacities allows for the direct solution of the a1ij-'- terms (i.e., the elements of the inverse matrix). By performing n such experiments, one could solve for the entire A-1 matrix and by inversion find A. For most communities, however, an investigator has no prior knowledge of the K s, and selective manipulation of these variables is impossible. It is easier and more practical to manipulate species numbers directly, even though the derivation of the meaning of the results is more circuitous.

Suppose species/is experimentally maintained at a level NAtJf and that the remaining n - I species are allowed to reach new equilibrium levels denoted as NIf. Then at this new equilibrium dNildt 0 0 for i #A.f and so Eq. 3 reduces to

It

Y al jAN j = 0, for i #Jf (8) Ji=l

where the matrix AN1, is the observable change in den- sity of species j following a PRESS perturbation of spe- cies f (i.e., AN.,, N - Nj,). If this experiment is repeated for all ni species, we obtain n(n - 1) equa- tions that can be solved for A (using the fact that aii I for all i) as shown in Appendix 1. Two equivalent formulae are derived:

aij = ANj- 'V/ANj' 1), and (9)

ANjf_ -z _I -1 AA JJ = f- jf V1. ( 10)

where 3ij ) and ANjj'-1' represent elements from the inverse of the A matrix and the AN matrix, respective- ly. To recapitulate, to measure the interaction coeffi- cients aij we do n experiments that yield the observ- able quantities AN1j. AN-' is the inverse of the matrix. Eq. 9 yields 0Hir, and Eq. 10 shows that the experi-

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February 1984 PERTURBATION EXPERIMENTS 5

mental responses in density are directly related to the elements of the inverse of A, not A itself (Fig. 1).

Although we have implied that Nff is arbitrary, this is not quite true. A species must never be held at such a level that another species goes extinct. For example, if Nff were such that Nif = 0, then Eq. 8 would not necessarily hold. (In this case the analysis breaks down because the mathematically relevant equilibrium value for the ith species is negative.) Unlike PULSE exper- iments, N1. = 0 is permissible (provided no other species goes extinct).

How can we achieve the greatest accuracy in Eq. 9? This is done by making the perturbations as large as possible with the caveat that a species must never be reduced to such a level that another species goes extinct. This caveat can present problems since we usually cannot predict nonextinction. Analysis of the effect of experimental error in Eq. 9 is much harder than in Eq. 5 because AVj'1' and AN1ii-1' result from inverting the matrix AN of measurements. If AN iS "ill conditioned," then even minor changes in its entries can cause major changes in the entries of AN-1. A dis- cussion of ill conditioning can be found in Fadeev and Fadeeva (1963), and Pomerantz (1981) illustrates the problem with ecological data. Generally speaking, a matrix with a small determinant is likely to be ill con- ditioned, and this might arise when two or more species interact in a similar fashion with other members of the community (i.e., are nearly equivalent members of the same guild).

Contrasts between PULSE and PRESS experiments

Fundamental differences.-A fundamental differ- ence between PULSE and PRESS experiments is the relationship between the aij's and the strength of a species' response. For PULSE perturbations, re- sponse strength, at least around equilibrium, is mea- sured by dNif/dt, which is related to aif in a simple way by Eq. 4a. For PRESS perturbations response strength is measured by Ni - Nif, which is directly coupled to the elements of the inverse matrix, the

sij(-D's as visible in Eq. 10. That is, the response of species i following a PRESS perturbation in species, depends on terms connecting species i with f in the inverse of the A matrix, not the A matrix itself. Even for a pure competition community where all the aij terms are greater than or equal to zero, the off-diag- onal terms in A-' may take on any sign. Each of the aij'-1' terms is a function of all the elements in the A

matrix not just the aij term. These aii"l) terms may be interpreted as the total effect that species has on the equilibrium density of species i and thus includes terms for the direct competitive effect via aij, plus terms for effects which are mediated through paths (the "loops" of Levins [1975]) involving additional species in the community (see Lawlor 1979). For example, speciesj may affect k, which in turn affects 1, which finally

affects i; this would be one possible path of length 3. The fact that Eq. 10 involves A-1 rather than A has been a source of considerable confusion. We devote the remainder of this section to that subject.

Consider, for example, the classical niche-theoretic example of three species whose niches are aligned symmetrically along a single niche axis. Each species has a Gaussian-shaped niche with overlap a < 1.0 with its nearest neighbor. The two end species have overlap of -a4 with one another (May 1975). For example, if a = 0.5, the A matrix is:

1 0.5 0.0625 A = 0.5 1 0.5

0.0625 0.5 1

and the inverse of A is:

1.4"2 -0.889 0.356 A- = -0.889 1.889 -0.889

0.356 -0.889 1.422

From the sign of a,,'-" and a(311'- we see that the net effect of the two end species on the equilibrium den- sity of one another is positive even though these species are direct competitors. The removal of either end species from the three-species community causes an initial increase in the density of the other end species at a rate given by Eq. 4a, but soon the trajectory turns around. At the new equilibrium, there is a net decrease in the density of the remaining end species (but an increase in the middle species). In fact, if K1 = K2 =

K:3 = 1, then N1 = N3 = 0.889 and N2 = 0. I l 1, while setting N11 = 0 gives N21 = N31 = 0.667.

Such indirect or gratuitous "mutualism" easily crops up in mathematical descriptions of multispecies com- petition communities when n > 2. Holt (1977) de- scribes examples whereby alternate prey species re- duce each other's equilibrium abundance, whether or not they compete, in the presence of a food-limited predator. He calls this indirect effect "apparent com- petition." In other examples, a predator may have a net positive effect on its prey, or the prey may have a negative effect on some of its predators, within the context of all the community interactions. Colwell and Fuentes (1975) provide other biological examples of such counterintuitive interactions.

In species-rich guilds, the results of a species re- moval experiment may be complex. The particular case described in Fig. 2 is highly symmetrized, but the character of the conclusion is robust for any richly connected competitive system. We set up a system of 18 competing species following the Lotka-Volterra equations (Eq. 1). All equilibrium densities are nor- malized to unity. The character of the competition ma- trix is that studied by May and Mac Arthur (1972). Species have Gaussian niches aligned along a single niche dimension, and nearest neighbors have a = 0.8; at time t = 10, an arbitrary member of the community

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6 EDWARD A. BENDER ET AL. Ecology, Vol. 65, No. 1

no ~~~~~~~~~~~~~~~~~~. -.-..............................s

=

FIG. 2. Time trajectories of an 18-species competition de- scribed in the text. All densities are initially normalized to unity and are thus coincident for the first 10 time periods. At t= 10. a single species is removed; since the system is sym-

metrical, it makes no difference which one. The behavior for the next 140 time periods is shown. Ten additional species ultimately go extinct.

is removed. The time trajectories of the remaining 17 species are shown in the remainder of the figure. Be- cause of the symmetry, all but one of the species move as pairs. As this figure shows, half the species go ex- tinct following this perturbation experiment, negating the application of a PRESS analysis.

Just as the removal of one species may result in a population decrease in its competitor species, so might the removal of a predator result, through indirect ef- fects, in a numerical decrease in some of its prey. Porter (1976) performed experiments enclosing fresh- water algae with and without their zooplankton graz- ers. When zooplankton were excluded, certain species of algae increased dramatically in abundance. Other species, however, showed significant declines. Porter observed that the types of algae that increased in the presence of grazers had durable cell walls that pro- tected them from damage during passage through the gut of the zooplankton. She attributed their increase under grazing to a direct effect from the zooplankton, namely a fertilizer effect from zooplankton excrement. Since these experiments lasted over many generations of algal growth, they may be interpreted as PRESS experiments. Thus, the possibility exists that certain alga species increased with grazing because of indirect effects. If the different algae compete for resources, then those algae which are less preferred prey will benefit indirectly from grazers because they gain a rel- ative competitive advantage over more heavily-grazed algae.

As we have seen, except for two-species commu- nities, the presence of a net density gain in one species (following the removal of a presumed competitor) is neither a necessary nor sufficient demonstration of competition between those two species. Similarly, the absence of a density gain is neither a necessary nor

sufficient demonstration of the absence of competi- tion.

Procedural differences.-PRESS experiments con- trast with PULSE perturbations in a variety of ways. First, one can remove a population completely in PRESS designs (i.e., set Nff = 0). Second, the growth rate parameters ci cannot be determined in PRESS experiments (and need not be to find A) since they affect only the rate at which equilibrium is reached and not its actual value.

On the other hand, ci must be determined in PULSE experiments to obtain A. In fact, radically different population growth rates for the species can cause se- rious problems in estimating dNif/dt since we need two separate measurements, such that Ni will have changed enough to estimate dNif/dt but the percentage change in K, - laijNj will have been small.

We foresee other potential difficulties with the ap- plication of PULSE and PRESS experimental designs. PULSE experiments are more prone to experimental error since it is generally more difficult to measure the rate at which a function changes than the magnitude of that function at equilibrium. The initial rate at which species i responds to a perturbation in species j is proportional to ci. In seasonal environments ci may fluctuate severely, even reaching zero. Thus, it is im- portant that each experiment have simultaneous con- trols. By adopting a PRESS design, we eliminate this ambiguity, but each experiment may become prohib- itively long and expensive.

In PULSE experiments, the value of aij can be com- puted after two single species perturbation experi- ments; in PRESS we must have the entire AN matrix so that we can invert it in order to obtain a single aij, thus n experiments are required. There is, however, an important exception to the need to determine the entire AN matrix. Suppose one set R of species does not numerically respond following perturbations in another set P. Specifically, suppose the first p exper- iments have been performed, and we have Njf = Nj for f = I to p and j p + I to n. Then AN has the block form

AN |ANpp ANPR N 0 ANRR'

which is inverted to obtain

AN-' = ANpp1 U 0 1ANRR<1

where U = -ANppJ'ANPRANRR'1. Note that by doing only p experiments we have found p entire columns of A, and that many of the entries are zero. (See below for more discussion.)

This argument is fundamental to Hairston's (1981) interpretation of his long-term study on a guild of sev- en salamander species in the southeastern United States. Hairston followed a PRESS experimental de- sign. The experiment consisted of continual removals

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of the most numerically dominant species (Plethodon jordani) from one set of replicate plots and the second most dominant species (P. glutinosus) from another set. Plots were unfenced, and the experimental re- movals maintained the focal species at densities close to zero. Compared to control plots, Hairston observed that the removal of either of these two abundant species increased the numerical abundance of the other. How- ever, the remaining five salamander species showed no significant increase. Hairston concluded that the two abundant species compete strongly with one another but not with the other salamander species. This conclusion is correct as long as ANRP is zero. ANRPI

is not exactly zero in Hairston's results; rather, the remaining five species' numerical responses were sim- ply not statistically significant. The conclusions Hair- ston reached will still be valid unless ARR (i.e., the sub- block of A containing the interactions among the five unperturbed salamander species) contain entries much larger than one in absolute value. If this condition is violated then, theoretically, diffuse interactions might yield a relatively small change in unperturbed species following perturbation of others despite strong direct interactions between the species involved.

OVERLOOKED SPECIES

A perennial problem in field experiments is identi- fying the complete set or guild of species so as to delimit the experimental design. In PULSE experi- ments this problem is not serious since, as noted ear- lier, if there are n total species, one may focus on a subset of s species and, by doing s experiments, find all the ai1s (for i, j 1 to s). In a PRESS design this neglect can be catastrophic because the elements of AN are shaped by the direct and indirect "links" be- tween all the various species. There is no reason why these links should be limited to species within the same taxon or of similar body size. The magnitude and even the sign of many calculated aij's can be affected by neglecting a single species.

Although it may be extremely difficult to identify a complete guild of interacting species, we may be able to at least identify a subset of species that strongly interact. Suppose F is this set of s species on which we focus, and R is the remaining n - s species. The interaction matrix A may then be partitioned into four blocks, such that the interactions among the focal species are in the s x s block A,,, the interactions among the neglected species are in block A,,, etc. In Appendix 2 we derive the following result: the effects of the ignored n - s species are negligible if either the interaction coefficients of ignored species on focal species (AFR) are all very small or the interaction coef- ficients of the focal species on ignored species (As,)

are all very small. For example, suppose we are interested in under-

standing the interactions between two species of ants. Suppose we know elephants also have a big effect on

these ants because they trample on their colonies, yet we are fairly confident that the direct effects of ants on elephants are low. If we conduct PRESS experiments on the ants only (ignoring elephants), how much error will result in our estimates of the ai~j's between ant species? A hypothetical interaction matrix with these features is the following:

Ant I Ant 2 Elephants 1 0.7 1.2

A 0.5 1 1.3 -0.01 0.02 1

Here both ant species have very small effects on elephants relative to their effects on each other. The effect of the elephants on the ants is very large (1.2 and 1.3); the exact values used are not critical as long as either AFR or Arek is very small. The inverse of A iS

1.524 -1.058 -0.454 A-1 = -0.803 1.584 -1.096

0.031 -0.04 1.017

We take this for our AN matrix. If we ignored ele- phants and adopted a PRESS perturbation design to estimate the ant-ant interactions, our interaction esti- mates will be given by Eq. 9, where the ANij'-l' s are entries in the inverse of the 2 x 2 matrix formed by the top left corner of AN (i.e., the boldface numbers above). This is

1.524 - 1.058 -0.803 1.584

Thus, taking the inverse and normalizing the diagonal elements to I yields our estimate for AF

1 0.668 0.527 1

which agrees very well with the corresponding ant-ant corner of A itself.

Overlooked species and ecological abstraction

In many cases, we purposely wish to consider the aij terms as potentially containing terms arising from indirect interactions through loops involving neglected species. The clearest example of this occurs in studies aimed at uncovering competitive interactions in a guild of consumer species. Rarely will we wish explicitly to consider all resource species in our experimental de- sign. Imagine a community of n consumer species and in resources. By performing n + m perturbations we could deduce the n + m dimensional A matrix (via Eqs. 5a or 9) explicitly containing terms for all con- sumer-resource, resource-consumer, consumer-con- sumer, and resource-resource interactions. The off- diagonal terms in the n x n subblock of A dealing only with the direct consumer-consumer interactions will

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be zero or small unless the consumers exhibit strong interspecific interference interactions. Interspecific consumer competition will show up as the similarity between different consumer's effects on different re- sources, not in the consumer-consumer interactions themselves. If we ignore the resource species in our experimental design, can we somehow deduce the competitive indirect effects between consumer species that are mediated through shared resources? That is, by purposely neglecting some species, can we experi- mentally parameterize their effects into the aij terms of the focal species?

We consider three cases, depending on whether the relaxation times (dynamics) of the neglected species are (1) much smaller than, (2) much larger than, or (3) the same order of magnitude as the relaxation times of the species on which we focus.

Neglected species have small relaxation times. This case was discussed by Schaffer (1981), and we have little new to add. Schaffer (1981) refers to the problem of overlooked species as "ecological abstrac- tion." He asks what set of parameters will reasonably predict the dynamics of an s species Lotka-Volterra approximation if the interactions are actually de- scribed by an n species Lotka-Volterra model (s < n). He concludes that when the neglected species have much smaller relaxation times than the focal species, a Lotka-Volterra model can be used, and PRESS ex- periments will give correct estimates (namely the NFF-1

of Eq. A2. 1). Notice that Schaffer's condition is not at all related to our condition A2.2, because we have different goals in mind. Our condition derives from requiring accuracy in the estimate of interaction coef- ficients. Schaffer is concerned with finding a parame- terization for the reduced system that yields dynamic accuracy of that subsystem. We emphasize, therefore, that estimates for AFF from PRESS experiments may be inaccurate even if the s species model accurately reflects community dynamics. In other words, if we estimate Acs, disturb the entire community and find that its dynamic behavior is accurately predicted by our estimate, then we still cannot conclude that our estimate of AFF is accurate!

What happens with PULSE experiments? As noted earlier, these are hard to perform correctly when species relaxation rates differ radically. However, if we allow a time between the two measurements for estimating dNif/dt at least comparable to the relaxation times of the neglected species, reasonably accurate estimates will be obtained for the reduced Lotka-Vol- terra model; that is, the estimated aij's will "absorb" reasonably well the indirect effects between focal species that are mediated through neglected species.

Neglected species have longer relaxation times. If the time differential is great enough, we can treat these species as constant and use a Lotka-Volterra model that does not contain any effects of the neglect- ed species. Either PULSE or PRESS experiments can

be used. We must then ask how much error is made in applying the ci and aij estimates at a much later time when the levels of neglected populations have changed? Surprisingly, the error is small since the ne- glected species only affect the "carrying capacities" in the reduced model. This can be seen by regrouping the bracketed term in Eq. 1:

Ki Y aNj= - = E' aijNj -Y aijN. (11) j=1 j=s+l

The parenthetic term is the "carrying capacity' for the model in which the last n - s species are neglect- ed. We can view this as an example of temporal vari- ation in Ki (see Temporal and Random Variation).

As an example of slow relaxation, consider two species of rabbits feeding on lichens. Suppose we are interested in the amount of competition among these rabbits. We perform PRESS experiments on rabbits only; in separate experiments, we depress and main- tain the numbers of each rabbit at some new level. After a while a quasi-equilibrium will be reached whereby rabbit numbers are relatively stable. Lichens, however, are still slowly adjusting. If we make our PRESS measurements of N1j at this quasi-equilibrium, our conclusions regarding rabbit-rabbit interactions will not be seriously in error.

Suppose we adopt a PULSE design for this same system by perturbing the numbers of one rabbit species and immediately afterward measuring the change in the numbers of the two rabbits. This growth rate should be measured over a brief time interval, approximately one rabbit generation. If the rabbit species do not ex- hibit direct interference between each other, our PULSE experiments will correctly indicate no direct rabbit-rabbit interactions, but we cannot conclude from this that rabbits do not compete. The competition through lichens could be uncovered by measurements of both lichen density and single-species rabbit den- sity. To do this we could parameterize the effects of lichens into variable "carrying capacities" for the rab- bits. If the rabbits are species I and 2, and we lump all lichens as species 3, then

K i(t') - i(t) = aO3 N3(t)JO -NJO),

for i = 1,'. (12)

Thus, we can estimate the effects of lichens on rabbits and thus rabbit exploitation competition if we can de- termine Ki's and N3 at two separate times. The time interval required will be longer than that normally used for a typical PULSE experiment.

Neglected and focal species have comparable re- laxation rates.-In this case it is not possible to find a Lotka-Volterra model for the focal species alone. PULSE and PRESS experiments can be expected to give radically different estimates for A, with PULSE giving correct estimates for the direct interactions only in the Ae block of the full n species matrix and PRESS giving estimates for the combined direct and indirect

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effects through focal and neglected species (i.e., the of Eq. A'2.1).

LUMPED SPECIES

Suppose we are only interested in the interactions between broad categories of species on the same trophic level. If we lump species together into cate- gories and perform PRESS experiments involving cat- egories instead of species, would the qualitative inter- pretation of the interactions between species necessarily be correct'?

This was the tack taken experimentally by Brown and Davidson (1978) and Brown et al. (1979) in their studies on the competitive effects of granivorous des- ert ants and rodents. For purposes of their removal experiments, all ant species were considered as one category and all such rodents as another. By lumping species into only two broad categories and by con- ducting experiments involving the total removal of one category or the other, it is possible that the net qual- itative interaction between species categories can be confused. We illustrate how this might happen with the following hypothetical example. Suppose we have a single rodent species and two ant species whose A

matrix and K vector are

Rodlent A/it / Ant 2 1 '/2 0 36

A= 0 1 2 K 36. I/8 0 1 9

We have contrived ant 2 to have a large depressive effect on ant 1, but ant 2 has no effect on the rodents. Ant I, on the other hand, has a competition coefficient of 1/2 on the rodents. This system has a stable equilib- rilum at

24 N 2 4

6

If we lump both ant species into a single category "ants" and then remove them from experimental plots, rodents will increase to their carrying capacity of 36. On the other hand, if we remove rodents, the two ant species will reach a new stable equilibrium with den- sities of 18 and 9, giving a summed density of 27 for ants. That is, the removal of rodents causes ant num- bers to decrease also. Using Eq. 9 to calculate the xij's for the lumped system yields:

Rodent Ants A= 1 2/5

Thus, our interpretation of these PRESS experi- ments would be one of a predator (rodents) and their prey (ants) rather than the strictly competitive system that we have modeled it to be. As a rule of thumb, lumping appears to be inappropriate when the species

in any one category have grossly different effects on one another and on species in the other category. For the case of Brown et al.'s rodent-ant system, we doubt such inequalities exist and believe that their original conclusion of ant-rodent competition is probably cor- rect.

NONLINEAR MODELS

Many have argued that the dynamics of real com- munities is probably not adequately described by equations as simple as the Lotka-Volterra equations (Eq. 1), since the aij terms are probably not constant but themselves functions of population density and species frequencies (see Abrams [1980b] and Thomson [1980] for recent reviews). Even so, we still expect Eq. 1 to be valid for perturbations in a small neigh- borhood of an equilibrium. In this case, one would design PULSE and PRESS experiments with small per- turbations. Unfortunately, as already noted, this makes it more difficult to obtain accurate results. How might one check the validity of Eq. 1 far from equilibrium'? An investigator performing a PRESS experiment in- volving large perturbations could collect data on both the initial rate of change of the n - I unperturbed species, as well as their net density change once the community comes to rest. The values of aij could be estimated from Eq. 9 and then used in Eq. 4a to es- timate ci:

dNif _ dt

c, , f i. (13) N1 ajNJ - N.)

This provides n - I estimates of each of the ci's. The degree of similarity between the various estimates for each ci is indicative of the adequacy of the gen- eralized Lotka-Volterra equations as a description of the observed community dynamics. However, if the central purpose of the experiments is to test for the presence of "higher-order interactions," more defini- tive experimental designs are available (Case and Bender 1981).

PRESS experiments do not allow rate constants to be determined because only equilibrium values are studied. The underpinnings of PULSE experiments are closely tied to the adequacy of differential equations as descriptions of the population dynamics. PRESS experiments, however, do not make this assumption. Since PRESS experiments rely only on the equilibrium condition K = AN, our analysis is correct as stated for a variety of models such as Lotka-Volterra difference equations, discrete Ricker equations, and time-delay logistic differential equations.

TEMPORAL AND RANDOM VARIATION

One might argue that since all the foregoing assumes that communities are at or approaching some ideal "equilibrium," the results have little bearing on the

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10 EDWARD A. BENDER ET AL. Ecology, Vol. 65, No. 1

real world where populations are in a greater state of flux.

Suppose the Lotka-Volterra equations (Eq. 1) still hold, but ci and/or Ki are functions of time. PULSE experiments will be virtually unaffected. We simply compare experimental populations with controls rath- er than with the preperturbed state. Imagine a control population with population levels Ni' and per capita growth rates Pi" at some time t. At this time we ex- perimentally perturb species f in experiment f to a population level Nf. We then measure the resulting new per capita growth rate Pif. Subtract the control population value of Eq. 1 from that for the perturba- tion of speciesf to obtain:

Pif - Pi" ci(t)aiANf' - Nf) (14)

in place of Eq. 4. Proceeding as in the derivation of Eq. 5 we obtain:

aij = (Pij - P j')(N( - N) (1)5) (P.i

- PC")(NJV

- ANi)(I)

C(t) = i, -N, (15b)

In the event that the Ki(t)'s are constants, we may take the control population to be in equilibrium, and so Ni' = Ni and Pi" = 0. Since Pij = dNij/Nidt, if i #1 j and Pji = dNiilNidt, Eq. 15 reduces to Eq. 5.

PRESS experiments appear to be useless if Ki is highly variable since we always assume that the pop- ulation is in equilibrium. On the other hand, if the ci's are variable, then our analysis of PRESS experiments is unaltered since these rate parameters will not affect equilibria.

NATURAL VARIATION TECHNIQUES

Techniques based on natural variation are a third category of methods for estimating interaction coeffi- cients. They are not experimental in the sense that the investigator manipulates populations; rather, they rely on nature doing the perturbations. Yet, how should this stochasticity be interpreted'? Should we visualize the natural variation as affecting only the N's while the basic parameters describing species interactions remain constant? Or should we visualize the pertur- bations as affecting the parameters themselves? In the latter case, which parameters are affected? The math- ematical analysis needed to proceed depends on our answers to these questions.

Can anything be done if natural variation affects only the N's? Hallett and Pimm (1979) considered the fol- lowing scenario. Imagine two competitor species ob- eying discrete forms of Eq. 1, at least in the neigh- borhood of the equilibrium N1, N2. Using primes to denote the value of Ni at the next generation,

Ni = Ni + riNi(Ki - aiNV -ai2N2)lKi,

i= 1,2. (16)

The system begins in equilibrium. Hallett and Pimm then randomly altered N1 and N2 by amounts x and v, respectively, where x and v are random variables with zero means, equal variances, and zero covariance. Following these random perturbations N, and N, grow by amounts Ax and Ay, respectively, in a single gen- eration where these are given by Eq. 16. It is assumed that only N, = (N1 + x + Ar) and N2 (N9 + w + Ayx) are observed by the investigator. After observing many such pairs of N1 and N2, the regression coeffi- cients KI* and a12* in

N1= K* - al2*N2

may be estimated. A similar regression provides K9,* and a21*

Hallett and Pimm (1979) explored this stochastic process, using Monte Carlo simulations for different values of a12, a,1, K,1 and K, with r, = r2 = 1.0. They suggest that census data yielding the number of indi- viduals of two species across different sites or at the same site but over different times could be used to calculate these regression coefficients, claiming that (li,* and Kj* are estimates for aij and Ki and that the assumption of only one time step is not critical.

Actually, there is no need to perform Monte Carlo simulations since the expected regression coefficients for this single-step process can be calculated analyti- cally; they do not equal a12 or a2l- Using a linearized form of Eq. 16 near equilibrium:

r1N

N, = N2+ no- _ 2 (y + a(2X). K.,

(The use of the exact form of Eq. 16 complicates the mathematics but does not alter the conclusions.) The regression coefficient a12* is cov(N1, N.,)/var N., which is

a 12* =

K., (KrN)Na, + (K., rN2)r,&,a, Is.., 151-

1N1r9Nci1~. ?

(K9- 2N)rN'1 . (17)

K, (K.- rN9)2 + (r2N2a21)2

Of course, a,21* is obtained from Eq. 17 by inter- changing the indices I and 2 wherever they occur. It is apparent from Eq. 17 that a12 X( a12*. (This holds even if r, = r2 = 1.0, as in Hallet and Pimm's simu- lations.) In cases where the two species being com- pared have very different r's and K's, the discrepancy between a 12 and a12* will be large. The values of a12

and a(12* can even be of opposite sign. We might try to obtain al2, a21, K1, and K2 by estimating a12*, a.1*,

K,*, and K.,* and then solving equations such as Eq. 17. Unfortunately, this requires a knowledge of r1 and rev and a guarantee that only a single time step was taken. A further complication arises when more than two species are involved, since, as in PRESS experi-

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February 1984 PERTURBATION EXPERIMENTS 11

ments, the calculated interaction coefficients between any two species are functions of the interactions of these species with all other species in the guild. We can think of no safe method for utilizing natural varia- tion in the N's to estimate the parameters in Eqs. I or 16.

Another way of visualizing observed spatial (or tem- poral) variation in species numbers is to assume that density differences between sites arise because differ- ent sites have somewhat different sets of K's, and each community is in equilibrium. Schoener (1974) men- tioned the situation in which there are two species in equilibrium at two sites where K, and a,2 are constant and K.) differs. If N1j denotes the equilibrium popula- tion density of species i at site, then

NDj = A - 112N21 (18)

for] = 1, 2. These equations can be solved for K, and .>12 If more than two sites are available, regression

can be used to find the best estimates of K, and a,2 . This readily extends to n species and n sites with K, and aj, constant. A major defect of this method is locating sites where certain parameters differ and oth- ers do not. Such studies can probably only be con- ducted under carefully controlled laboratory condi- tions. Application to field situations is generally impossible.

Davidson (1980) assumed that all K's could vary from site to site in her study of desert ant communi- ties. She hoped to estimate the elements of aij,-l' using Eq. 7 and the assumption that the observed spatial correlations between ant species i andj would be pro- portional to aij2'". Since equilibrium at all sites is im- plicitly assumed, the observed covariance between Ni and Nj is independent of the ri terms and the amplitude of the variation in the K's. (Without this assumption it is necessary to model the community dynamics with stochastic difference or differential equations (May 1975a, Turelli 1979). Yet, even with these simplifying assumptions, the relationship between the covariance in species K's and the resulting covariance in N.'s is not simple. For a linear system K = AN,

COV(K) = A COV(&)A" (19(1)

and

COV(&) = A--' COV(K)A- (I 9b)

where cov(&) and COV(K) are the covariance matrices such that the i, j h entry of cov(&) gives the covariance between Pi and rj when i #A j and the variance (i,)2 when i = j. A simple proof of Eq. 19 is in Feller (1966: 81-82). In our case, we would like to solve for A (or A-') given the observed cov(?), but clearly this is im- possible unless we also know cov(K). Even if COV(K)

is given, Eq. 19 has an infinite number of very different solutions, and even the sign pattern of A (or A- 1) is not determinable. This is true even in the simplest possible case where cov(K) = 21 (I being the identity matrix).

CONCLUSIONS

As community ecology emerges from a largely de- scriptive science, controlled experiments are more and more used to decide issues instead of simply in- terpreting existing patterns in nature. With respect to perturbation experiments, there is still much confusion regarding the proper interpretation of results and how they bear on the relative strength of present-day species interactions as well as the forces shaping community structure. Some believe that such experiments can provide "necessary and sufficient" conditions to dem- onstrate the presence or absence of particular biotic interactions such as competition (e.g., Connell 1980, Newton 1980, Tinkle 1982) and that perturbation ex- periments are the only road to ecological knowledge. But we find that too much of the thinking about per- turbation experiments is "two dimensional," that is, that it sees only one course of reaction following the experimenters' action. In the foregoing, we discuss the often hidden assumptions behind perturbation studies, the shortcuts that are usually taken by investigators, the pitfalls that may arise, and some of the rather counter-intuitive conclusions that emerge. For exam- ple, with PRESS designs, two species may each reach higher equilibrium densities in the absence of the oth- er, yet the two may not compete or, for that matter, directly interact at all. The adjustment in their equilib- rium densities is a result of the mediation of additional, unstudied species. Conversely, two species that do compete intensely may show no net increase in density when one or the other is removed or experimentally maintained at some low density, again due to the pos- sible buffering effects of species intermediaries. Using PULSE experimental designs, it is possible that nei- ther of two resource competitors will immediately in- crease in density following the removal or suppression of the other. This would be the case if resource dy- namics are very slow and if the competitors do not exhibit overt interference.

If, through PRESS perturbation experiments, we ar- rive at estimates for the interaction coefficients be- tween some set of species, and if these interaction coefficients accurately predict community dynamics in still further experiments, we still cannot conclude that our estimates even approximately represent the actual mechanistic coefficients of interaction.

In practice, no community ecologist can measure the density of every potentially interacting species in a community, yet once some species are neglected and others lumped into composite categories, there is a real danger that indirect effects can confuse and con- found the interpretation of results. It is not easy to prescribe how one should look for sets of species be- tween which interactions are strong, for the impor- tance of one species on another may be independent of its body size, taxonomic position, or biomass; mi- croscopic parasites may radically alter the nature of

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12 EDWARD A. BENDER ET AL. Ecology, Vol. 65, No. 1

the measured interaction between alternative host species in PRESS designs. A classic example involves the semicontrolled introduction of white-tailed deer into Nova Scotia in 1894 and the consequent extinction of caribou (Embree 1979). Interpreting the result as a PRESS experiment, involving only two focal species, deer and caribou, it is easy to reach the conclusion that these two focal species severely compete to the extinction of one. In fact, the extinction of caribou probably had nothing to do with competition but rather with differential susceptibility of these two ungulates to a meningeal parasitic nematode (Anderson 1965). A proper three-species PRESS design (e.g., one includ- ing the removal of the parasite) would have revealed this result, but in practice, we can probably never de- fine the complete set of interacting species in an area. Unfortunately, without such knowledge, we are prone to misinterpret the interaction between any arbitrary species pair. This is a sobering thought, and one that underscores the importance of supplementing ecolog- ical experiments with the fullest amount of descriptive natural history and common sense. There is probably no one "royal road" to ecological knowledge; insights will best be gained by a pluralistic approach.

ACKNOWLEDGMENTS

We thank Will Thomas and Eric Pianka for helpful com- ments. We acknowledge the support of National Science Foundation grant DEB-79-05085 to M. E. Gilpin.

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February 1984 PERTURBATION EXPERIMENTS 13

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APPENDIX 1

Given the observable quantities ANjf = N, - Nif from a series of n PRESS perturbation experiments, one for each species, we may solve for the interaction terms aij as follows. Let AN be the matrix with all the ANij terms from our n ex- periments. Define a new diagonal matrix D where

dif = afjZNjf. (Al.l)

In words, dff represents the amount by which speciesf would change in density if all the other species were held at densities Njf and species f was free to reach a new equilibrium. This quantity dff is not observable since in our experiments it is species f that is held constant, yet by defining such an ab- stract quantity we will be able to reach a solution for aij in terms solely of observable quantitites, namely the ASNjf's. Using these definitions, we can write Eqs. 8 and Al.l in matrix form as

AAN = D, (A1.2a)

and so

A = DAN -. (A I.2b)

Hence aij = dii ANjj3-" where ANjj1- is the WJth element of the inverse matrix. Since aii and dii = l/ANi'-"1 we have

aij = ANjj -1-1/ANA'- " (A 1.3)

Another way of expressing the relationship between the mea- surable values Nj - Nif following the perturbation experi- ments and their quantitative relationship to the interaction coefficients, aij, is as follows: From Eq. A.2,a, AN = A-ID

and so

ANIf NNi = ai`-,l)dff.

Hence

AN f = a /aff 1), (A 1 .4)

which is the expression in Eq. 10.

APPENDIX 2 To explore the effect of neglected species on the analysis

of PRESS experiments, we first partition the matrices AN and AN-' (=M) using the subscript F for the first s species on which we focus and R for the remaining n - s:

AN = ANFF ANFR AN-' = M = MFF MFR

ANRF ANRR MRF MRR

Since MAN = 1, we have:

ME'EiANFF + MFRANRF MF'FANFR + MFFANRR - I 0

MRFAN'FE + MRRANRF MRFANFR + MRRANRR 0 1

From the lower left entry, MRFANFF + MRRANRE 0, and so ANRF = -MRR 1MREANEF. Substituting this into the upper left entry, multiplying on the right by ANFF1 and rearranging

MFF} - \NFF1 = MFRMRR1MRF- (A2. I) The matrix MEF should be the source of the numbers on

the right side of Eq. 9 in determining aij; however, if we neglected to include the last n - s species in our model, we would inadvertently use \NFF- instead. Thus we need to know by how much the corresponding entries in MF, and ANFF-J differ. This is given by Eq. Al.l. Thus to minimize error we need

MFF > MFR MRR MRP'

where the symbol > means "is much greater than." We can divide the i'h row of M by min to convert this to an A matrix equation:

AXF > AFRARR1ARF . (A2.2)

If we make the rather reasonable assumption that ARR is not ill conditioned, we can say that the effects of the ignored n - s species are negligible if either the interaction coefficients of ignored species on focal species (AFR) are all very small, or the interaction coefficients of the focal species on ignored species (ARF) are all very small. Ill conditioning in ARR may arise if the neglected species have very similar intra- and interspecific competitive effects on one another. For exam- ple, if

11 1.1 A -0.9 1

AR, I100 -110 ARR |-90 100

Thus, the right-hand side statement of A2.2 may become ex- cessively large even if neglected species do not interact very strongly with focal species.

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