THEORETICAL
POPULATION
BIOLOGY
16, 144-158
The Structure
(1979)
of Food Webs
STUART L. PIMM Department
of Biological
Sciences,
Texas
Received
Tech University,
January
Lubbock,
Texas
79409
1979
For nonrandom models of species interaction there is a precipitous decrease in stability as comrectance increases. However, the range of stability for different models of the same connectance is large; stability also depends on how the species interactions are organized. Systems with species feeding on more than one trophic level (omnivores) are likely to be unstable, the extent depending on the number and position of the omnivores. For systems of equal connectance, those that are completely compartmentalized are less likely to be stablg than those that are not.
INTRODUCTION
Food webssummarizethe interactions amongspeciesin natural communities. Many textbook examples look complicated (e.g., Wilson and Bossert, 1971, Figs. 12 and 13), although someof this complexity may be more apparent than real. However, the existence of real, complex webs does not preclude the existenceof pattern in the designof food webs. Elsewhere(Pimm, 1979),I shall examinepublished food websfor the existenceof patterns and test the theoretical results presented here, This is a purely theoretical paper whose objectives can be formulated asa seriesof questions: (a) Generally as the percentage of the number of possible interactions (connectance) within a random web increases,so doesthe probability that the system’sequilibrium (or equilibria) will be unstable (May, 1971; May, 1972; Roberts, 1974; Gilpin, 1975). However, this result is sensitive to details of the model. Webs in the real world are not random structures (DeAngelis, 1975; Lawlor, 1978), and web structure has a marked effect on stability (McMurtrie, 1975; Siljak, 1974, 1975). The various assumptionsand results of the growing number of studiesin this area are reviewed by May (1979). Here I ask: how are cormectance and stability related in a set of models whose structures more closely approximate reality than random webs? (b) I have argued that as the number of trophic levels increasesso does the relative time for the system to return to equilibrium after disturbance (Pimm and Lawton, 1977; DeAngelis et al., 1978; Vincent and Anderson, 1979). 144 004&5809/79/050144-15$02.00/O Copyright All rights
0 1979 by Academic Press, Inc. of reproduction in any form reserved.
THE
STRUCTURE
These results were for four-species greater complexity ?
OF FOOD
models:
145
WEBS
do they also hold
for models
of
(c) Many food webs nurture omnivorous species (by omnivory I mean feeding on more than one trophic level). Two patterns emerge from a consideration of four-species models; (i) more than one omnivore in any one food chain is improbable and (ii) if there is an omnivore in a food chain it is probable that it will feed on adjacent trophic levels and improbable that its prey will be separated by more than one trophic level (Pimm and Lawton, 1978). Do these results hold for more complex systems ? (d) For random systems, the product of average interaction strength and connectance must be “small” to ensure stability (May, 1972, 1979), but for a given level of connectance and interaction strength, the probability of stability may be enhanced if the species are arranged in compartments or blocks. To quote May (1972): “of the infinite ensemble of these particular 1Zspecies models, essentially none of the general ones are stable, whereas 35% of those arranged into three blocks are stable. Such examples suggest that our model multispecies communities, for a given average interaction strength and web connectance, will do better if the interactions tend to be arranged in blocksagain a feature observed in many natural systems.” A major objective of this paper is to ask: should model systems be organized into compartments of species characterized by strong interactions within compartments, but weak interactions among the compartments ?
METHODS
Briefly, the method involves generating a wide variety of food webs, for which I then calculate the percentage of models whose equilibria are unstable and, for the remainder, the distribution of their return times. I assume that if a system is ody stabIe for a small percentage of the possible parameter values, then real food webs with a similar structure will be rare. Conversely, I assume that the food webs in the real world will be those for which a high proportion of the model analogs are stable. I further assume that rapid returntimes enhance the probability of persistance. The method raises some technical issues which are considered next: they can be omitted without loss of continuity.
The Useof Structured Models Model structure is defined by a matrix which summarizes interactions between species in the webs. Each painvise element in this matrix, a,, , can be negative, zero, or positive. For n species, the matrix has na entries and so
146
STUART
L.
PIMM
there are 3”’ possible structures. Even for n = 2 the number of possible models is large (81), yet perhaps only four of these are ecologically interesting (predatorprey, prey-predator, competitor-competitor, and symbiont-symbiont). The use of random models leads to choices which are easy to create mathematically, but ecologically absurd (Lawlor, 1978). The advantages of structured models are that I can avoid structures of little interest and minimize the number of simulations required. They also permit various problems to be tackled independently. Furthermore, in comparing different models I shall frequently conclude that model A is more probable than model B, yet model B might be more probable (by having a statistically greater percentage of stable models, for example) than a randomly chosen model. Using structured models eliminates the need for complex multiple comparisons, necessitated by the generation of purely random interaction matrices.
Choiceof Parameters Consider
the familiar
Lotka-Volterra
xi = Xi
model:
bi + f
aijX,
= Fi ,
i=l
where & and X, are the rate of change and population level of the ith species, b, the instantaneous per capita growth rate in the absence of any other species (negative for predators, positive for species at the base of the food web), and aii the effect of the jth species on the growth rate of the ith species. The dynamics of this equation near equilibrium are determined by the eigenvalues of the Jacobian (J)
3 3Xj
for
i=l,nandj-1,n
evaluated at equilibrium. If the real part of maximum eigenvalue (RAmax) of J is greater than zero, the system will be unstable. Otherwise the system will return to equilibrium at a rate approximately dependent on l/Rhmax . I define return time (RT) as
RTy&, max
xmax-=c0 .
Parameter values were set by the biological details implicit in the model. The nonzero diagonal elements of the Jacobian (corresponding to species at the base of the food web which are limited by resources) are Xfaii = -blx/Ki where @ is the equilibrium population of the ith species and fi=$ its equilibrium in the absence of all other species. Since the Xf cannot exceed & , these diagonal
THE
STRUCTURE
OF FOOD
WEBS
147
values will be in the interval (0, --bi). The bi were arbitrarily set to + 1 for selflimited species. This scales the return times. The perturbation to the system will return to l/e of its initial value in a time that is the return time multiplied by the time it takes the basal species to produce one new individual. None of the other species were self-limiting. This is a critical assumption, because if I allow species at the second and higher trophic levels to be selflimited, webs of highly complex structures are possible (Saunders, 1978). However for most animal species such models are implausible if the self-limiting terms are large because, other than in special cases, they imply the widespread existence of sophisticated conventional behavior (e.g., territoriality) which replaces food supply in the proceeding trophic level as the main controlling influence on the population (Lawton and Pimm, 1978). The off-diagonal elements are also of the form Xfaij where the sign of aij will depend on whether i is feeding on j or vice versa, or whether i and j are competitors. For predator-prey interactions uij is the number of new predators produced per prey eaten or the number of prey killed per predator; the latter will usually be much larger than the former. The X* will also differ, since predators are likely to be fewer than their prey at equilibrium. To model this asymmetry in magnitude the effects of the predator on the prey population were uniformly distributed on the interval (0, -10) and the effects of the prey on the predator on the interval (0, +O.l). A detailed consideration of the problem of parameter values is provided elsewhere (Pimm and Lawton, 1978). For each model I ran 1000 simulations. Though this represents a considerable investment of computer time, 1000 is a small number compared with the half million or more possible combinations of low, medium, and high values for each of 12 parameters. Many of the models I shall analyze are more complex than this; my most complex models would require on the order of lOLo simulations to exhaust the combinations. Though there is no guarantee that the small selection of parameters is not hopelessly biased, one can gain confidence in the sample by running models with a different “seed” for the random-number generator. I ran two models that, for their similarity in structure, appeared to have the most discrepant percentages of unstable models (33x, 88%) through six different sets of simulations. All percentages were within 2% of the mean for unstable models and 11 out of 12 of the percentages within 3 y/o of the mean for return times; the exception was only 5% more than the mean return time. For every simulation each matrix was converted to upper Hessenberg form and the eigenvalues extracted using routines in the IBM Scientific Subroutine Package. These techniques are described in Francis and Strachey (1961) Francis (1962), and Wilkinson (1965). Though double precision FORTRAN was used, I checked the accuracy of the procedures. Certain models were known to be qualitatively stable (May, 1973; Je f f ries, 1974) from their structure. For these models any positive eigenvalues must indicate numerical error in the routines. The numbers of positive eigenvalues never exceeded 3 per 1000 models
148
STUART
L.
PIMM
analyzed, were generally zero, and when positive results were found, never larger than the minimum value printed by the program (0.0001).
THE “TAXONOMY”
were
OF FOOD WEBS
To facilitate the description of food webs I must introduce some terminology. I try to avoid such terms as “primary consumer,” or “herbivore,” because although these terms are usually unequivocal, there are cases where they are not. Such cases have led to criticism of the trophic level concept, and though in model webs there are no difficulties, I feel it is appropriate to use terms here that one can also use when testing my results with real food webs. (a) Types of species. Basal species are those which are resource limited and which feed on no other species in the web. Though they may be thought of as autotrophs, many food webs have as basal species animals whose limiting resource is detritus (and its associated microorganisms). Top predators are species on which nothing else feeds. Omnivores are species which feed on more than one trophic level; by this I mean that they consume energy which has flowed through a varying number of pathways from a basal species. (b) Position. Omnivores can feed in a variety of positions within the web. I number levels by the maximum number of connections plus one, from the basal species (level one). The position of the omnivor is defined by its level and that of the lowest level of its food. For example, a top-carnivore which feeds on a carnivore and on a herbivore on which the latter carnivore also feeds, is “2-4 omnivory.” Were the carnivore to feed on plants as well as the herbivore the web would include an example of “l-3 omnivory.” The final possibility is for the top predator to also feed on plants, “l-4 omnivory.” The number of predator-prey interactions in excess of those in the simple, straight chain models, I call Zinks. In a four-species, four-trophic-level model there can be up to three links (l-3,2-4, 14). With two or more food chains there can be many more links; either within or between chains. Links within chains are inevitably omnivore links, but those between chains can also include a predator feeding on prey species in different chains but at the same trophic level; these are nonomnivore links.
RESULTS AND DISCUSSION I have studied models of six and eight species. The interactions common to all models give an arrangement of two chains as shown in Fig. la. The models differed in the number of interactions in excess of the basic models (links), their kind (omnivore of nonomnivore), position and whether they were within
THE
STRUCTURE
OF FOOD
B ii
H 2
1
FIG.
1.
Diagrams
of models
149
WEBS
analyzed.
For further
details
see text.
100
P
10
1
FIG. 2. Mean percentages of stable models as a function of the number of links (zero through six), and connectance (C), for three trophic level models (circles; upper values on abscissa) and four level models (squares; lower values on abscissa). Ranges, (vertical lines) are truncated at 1 yO for all models where 1 %, or less, of the models were stable.
150
STUART
L.
PIMM
or between chains. Tables I-IV present the percentages of stable models, and of these, the percentage that have return times shorter than 200. I shall discuss the results in the order of the questions posed in the Introduction. I plot, in Fig. 2, the percentage of stable models against the connectance for the six and eight species models. The models are highly nonrandom models yet they still show a sharp transition from largely stable to largely unstable as connectance increases. This is characteristic of random webs. Furthermore, TABLE Six-Species
Links Only Links (a)
Models:
Percentage
I of Unstable
Models
= 0 one model
0 y0 unstable
= 1 No omnivore Between 0
(b)
Links (a)
One omnivore
Within’ 89
Between 0
= 2 No omnivore Between 17-Sl(30)b
(b)
One omnivore
Within 82-89(85)
(c)
Two
2 within 22
Links (a)
omnivores
Between 64-80(72) 1 within/l 87
between
2 between 92
= 4 No omnivore Between 75
LI Links the links * Range link is not
(b)
One
(c)
Two
omnivore omnivores
Within 94 2 within 99
Between 91 1 within/l 99
between
can be between or within chains; web compartmentalization are all within chains. (mean); several models are possible because the position fixed.
2 between 97
is greatest
when
of the nonomnivore
THE
STRUCTURE
OF FOOD
TABLE
151
WEBS
II
Six-Species Models: Percentage of Stable Models with Return Times > 200
Links = 0 Only on model:
24
Links = 1 (a)
No omnivory Between 18-21(20)
(b)
One omnivore
Within 14
Between 15
Links = 2 (a) No omnivore Between 14-23(19) (b) One omnivore (c) Two omnivores
Within 8-13(11) 2 within
1 within/l 8
between
Between 7-23(12) 2 between 16
Links = 4 (a) No omnivory
(b) One omnivore (c) Two omnivores
Within 17 2 within -
1 within/l -
between
Between 14 Between 11 2 between 8
value for the transition from largely stable to largely unstableoccurs at lower valueswith the higher number of species.Both theseresultswere found by Gardner and Ashby (1970) and May (1972). However, the results are quantitatively different in one important respect: applying May’s (1972) formula for
the critical
transition
from stable to unstable
one would
predict that all my models would
be
unstable. It seemsunlikely that precise quantitative tests of May’s prediction are particularly meaningful for real webs(cf. McNaughton, 1978). While the mean percentagesof stable models decreasewith connectance, the ranges show considerable overlap suggesting that connectance, though important, is only one factor which affects the stability of these models. Table V analyzes the data grouped by the number of trophic levels, and
152
STUART
L. PIMM
whether or not the model includes a l-4 omnivore. Food webs in this latter group have long return times (Pimm and Lawton, 1978); excluding these shows that there is a statistically significant negative relationship between short return times and the percentage of stable models. This is interesting. Though there is considerable variation, and the slopes of the relationships are small, the interpretation is that models with small ranges of parameters consistent with stability will show faster return to equilibrium (and hence, in variable environments less TABLE Eight-Species
Links
(a)
Percentage
of Unstable
Models
= 0
Only Links
Models:
III
one model =
0 y0 unstable
1
No omnivore Between 0
(b)
Links (a)
1 omnivore Position
Within
Between
l-3 2-4 l-4
81 91 96
0 0 0
= 2 No omnivore Between 21-51(32)
(b)
(c)
1 omnivore Position
Within
Between
1-3 2-4 1-4
77-81(78) 87-93(89) 94-96(95)
62-81(72) 74-87(80) 87-95(93)
2 omnivores Position
2 within
l-3 2-4 l-4 l-3,2-4 1-3, l-4 2-4,1-4
96 99 100 98 99 loo
1 within/l 78-81 86-90 94-86 81-83 80-82 89-93
between
2 between 82 92 100 33-88 80-96 82-98 Table
continued
THE
STRUCTURE
TABLE
OF FOOD
153
WEBS
III-Continued
Links=4 (a)
No omnivore Between 70-Sl(75)
(b)
1 omnivore Position
Within
Between
1-3 2-4 1-4
92 94 97
92 96 99
(c)
(d) Links (a)
2 omnivores Position
2 within
1-3 2-4 l-4 1-3, l-4 l-4,2-4 l-3,2-4
97 99-100 loo 99-100 99-100 99
93-95 97 99-100 100 100 95
than
99.8 ‘A unstable
4 omnivores-all
greater
1 within/l
between
2 between 97 99 loo 98 99 97-99
= 6 No omnivory Between 93
(b)
1 omnivore Position
Within
t
1-3 2-4 1-4 (c)
2 or more
same
+
Between
98 98 99 omnivores:
greater
than
98 o/O unstable
variation through time) than those having large ranges of parameters consistent with stability. Table V also illustrates the effect of adding an additional trophic level on the return times. Though the percentages of stable models span the same range, the return times for the three trophic level models are nearly always shorter than those for four-trophic-level models, irrespective of the details of the model. The earlier results on four-species models (Pimm and Lawton, 1977) seem robust. The effects of the numbers of omnivores and their position within a web can best be seen from an examination of Table I-IV, comparing models with the 653/r612-4
154
STUART
L.
TABLE Eight-Species
Links
Percentage
IV
of Stable
Models
with
Return
Times
> 200
= 0
One
model:
Links (a)
Models:
PIMM
=
51
1
No omnivore Between 46
(b)
One
Within
Between
l-3 2-4 l-4
40
42
36
45
Links (a)
omnivore
Position
= 2 No omnivore Between 39-54(44)
(b)
(c)
One omnivore Position
Within
Between
l-3 2-4 l-4
29-49(35) 16-49(26) 30-49(37)
39-54(44) 13-37(23) 36-61(48)
Two
omnivores
Position
2 within
1-3 2-4 l-4 I-3,2-4 l-3, l-4 2-4, l-4 Links (a)
-
1 within/l 30-36 21-29 38-41 30-31 35-40 22-43
between
2 between 64 26 22-24 52-64 26
= 4 No omnivore Between 35-60(43)
(b)
One
omnivore
Position
Within
Between
l-3 2-4 l-4
20 20 27
25 22 Table
continued
THE
STRUCTURE?
TABLE
OF FOOD
155
WEBS
IV-Continued
(c) Two omnivores Position l-3 2-4 1-4
2 within 1 within/l between 2 between 39 32 39 30 - (remainderof four-
link modelsare all unstable) Links = 6 (a) No omnivory
Between 36
samenumber of links. The total number of omnivores is important; the more omnivores there are within a web, the less likely it is to be stable. An eightspeciesmodel with one omnivore (l-3 or 2-4) is about as likely to be stable as a similar model with six between-chain, nonomnivore links. The connectance of the latter model is 25% greater than the former. The position of the omnivory is also important. Considering webswith four trophic levels and one omnivore, for example, one has a sequenceshowing increasing stability in the order l-4, 2-4, and l-3 omnivory (Table III). Models with 1-4 omnivores also have very long return times (Table V) which would also make this pattern of omnivory unlikely to be found in the real world. Again, the earlier results seemrobust. The effects of compartmentalization are also important in determining the percentageof stablemodels. Generally, as connectanceincreases,the degreeto which the speciesare compartmentalizedinto the two chainsdecreases.However, it is possible to vary compartmentalization independently of connectance. An example is shownin Fig. 1b; both modelshave a l-3 omnivore; in (1) the omnivore link is within one chain, and the system is completely compartmentalized, but in (2) the omnivore link is between chains. Though model (1) has only a 19% chance of being stable, model (2) is always stable. Indeed, model (2) is qualitatively stable, in the senseof May (1973) and Jeffries (1974). This suggests that completely compartmentalizedmodelsare not the most stablearrangement. Inspection of Tables I and III showsthis to be the case.In most casesfor a given degree of connectancethe most compartmentalizedmodels are the least stable. The least compartmentalized models may be less stable than those of intermediate compartmentalization. The few exceptions are from models with a low probability of stability. The complexity of even thesesimplemodelsmakes if difficult to display theseresults. Tables I and III are for six- and eight-species models. Within the Tables modelsare grouped that have the samenumber of links-interactions in excessof the basic models. Models are further grouped
156
STUART
L. PIMM
TABLE Analysis
V
of Variance for Return Times Trophic Levels and Percentage
Source
sums of squares
df
A.
Excluding
as a Function of Unstable
data with
of Number Models
of
F
P
1-4 omnivores”
Difference between trophic levels (6 and 8 species models)
1
6348
93
200
B. Data “/b of unstable
Sums of squares
df
Source
for l-4
models
1
33
Residual
omnivores
F
P
only”
294 4266
2.28
NS
Equations Mean
value
for
y0 return
times
>200
= 37.5 (Compare
with
Eq. (ii))
(i The I-4 omnivore data are significantly different from the remainder of the four omnivore models at P < 0.001. b Examination of residual plots showed no curvilinear trends. Partitioning of the residual variability into lack of fit and pure error confirmed that the linear model had no prominent bias. c Terms in parentheses are standard errors of the estimate. d Analysis A fits only a three parameter model to data (two intercepts, one slope) since slopes are not significantly different.
THE
STRUCTURE
OF FOOD
WEBS
157
by the kind and position of these links. Models on any row of the table have the same number of trophic levels and have identical numbers of omnivore and nonomnivore links. Furthermore, the omnivore links differ only in whether they are within or between chains; the trophic levels connected are the same. Different degrees of compartmentalization can be simply compared by reading across the table; compartmentalization decreases from left to right. This result, of the most compartmentalized models being least stable, runs counter to a number of suggestions about the organization of ecological systems (May, 1972, 1979). I conclude that species interactions should not be arranged in tight compartments. ACKNOWLEDGMENTS Dr. J. H. Lawton and Dr. R. M. May and three anonymous helpful suggestions. Computer facilities were provided by the Computer Center; I am greatful to all of these.
reviewers made many Texas Tech University
DEANGELS, D. L. 1975. Stability and connectance in food web models, Ecology 56, 238-243. DEANGELIS, D. L., GARDNER, R. H., MANKIN, J. B., POST, W. M., AND CARNNEY, J. H. 1978. Energy flow and the number of trophic levels in ecological communities, Nature 237, 123. FRANCIS, J. F. F., AND STARCHY, C. 1961. Reduction of a matrix to co-diagonal form by eliiination, Comput. J. 4, 168-174. FRANCIS, J. F. F. 1962. The Q-R transformation: a unitary analogue to the L-R transformation: -I. Comput. J. 4, 462-471. GARDNER, M. R., AND ASHSY, W. R. 1970. Connectance of large dynamical (cybernetic) systems: critical values for stability, Nature 228, 784. GILPIN, M. E. 1975. Stability of feasible predator-prey systems, Nature 254, 137-138. JEFFRIES, C. 1974. Qualitative stability and digraphs in model ecosystems, Ecology 55, 1415-1419. LAWLOR, L. R. 1978. A comment on randomly constructed ecosystem models, Amer. Natur. 112, 445-447. LAWTON. J. H., AND PIMM, S. L. 1978. Population dynamics and the length of food chains, Nature 212, 189-190. MAY, R. M. 1971. Stability in multispecies communities, &&r/z. Biosci. 12, 59-79. MAY, R. M. 1972. Will a large complex system be stable? Nature 238, 413-414. MAY, R. M. 1973. Qualitative stability in model ecosystems, Ecology 54, 638-641. MAY, R. M. 1979. The structure and dynamics of ecological communities, in “Population Dynamics,” British Ecological Society Symposium (R. M. Anderson, and B. R. Turner, Eds.), Blackwell Oxford. MCMURTRIE, R. E. 1975. Determinants of stability of large randomly connected systems, 1. Theoret. Biol. 50, l-11. MCNAUGHTON, S. J. 1978. Stability and diversity in ecological communities, Nature 274, 251-252.
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PIMM
PIMM, S. L., AND LAWTON, J. H. 1977. The number of trophic munities, Nature 268, 329-331. PIMM, S. L., AND LAWTON, J. H. 1978. On feeding on more
levels
in ecological
com-
than
one trophic
level,
Nature 269. PIMM, S. L. 1979. Properties of food webs. Ecology (in press). ROBERTS, A. P. 1974. The stability of a feasible random ecosystem, Nature 251, 607-608. SAUNDERS, P. T. 1978. Population dynamics and the length of food chains, Nature 272, 189. SILJAK, D. D. 1974. Connective stability of complex systems, Nuture 249, 280. SILJAK, D. D. 1975. When is a complex ecosystem stable ? Math. Biosci. 25, 25-50. VINCENT, T. L., AND ANDERSON, L. R. 1979. The vulnerability of long food chains, Theoret. Pap. Biol. (in press). WILKINSON, J. H. 1965. The algebraic eigenvalue problem. Clarendon, Oxford. WILSON, E. O., AND BOSSERT, W. H. 1971. A primer of population biology, Sinauer Associates, Stamford, Corm.
POPULATION
BIOLOGY
16, 144-158
The Structure
(1979)
of Food Webs
STUART L. PIMM Department
of Biological
Sciences,
Texas
Received
Tech University,
January
Lubbock,
Texas
79409
1979
For nonrandom models of species interaction there is a precipitous decrease in stability as comrectance increases. However, the range of stability for different models of the same connectance is large; stability also depends on how the species interactions are organized. Systems with species feeding on more than one trophic level (omnivores) are likely to be unstable, the extent depending on the number and position of the omnivores. For systems of equal connectance, those that are completely compartmentalized are less likely to be stablg than those that are not.
INTRODUCTION
Food webssummarizethe interactions amongspeciesin natural communities. Many textbook examples look complicated (e.g., Wilson and Bossert, 1971, Figs. 12 and 13), although someof this complexity may be more apparent than real. However, the existence of real, complex webs does not preclude the existenceof pattern in the designof food webs. Elsewhere(Pimm, 1979),I shall examinepublished food websfor the existenceof patterns and test the theoretical results presented here, This is a purely theoretical paper whose objectives can be formulated asa seriesof questions: (a) Generally as the percentage of the number of possible interactions (connectance) within a random web increases,so doesthe probability that the system’sequilibrium (or equilibria) will be unstable (May, 1971; May, 1972; Roberts, 1974; Gilpin, 1975). However, this result is sensitive to details of the model. Webs in the real world are not random structures (DeAngelis, 1975; Lawlor, 1978), and web structure has a marked effect on stability (McMurtrie, 1975; Siljak, 1974, 1975). The various assumptionsand results of the growing number of studiesin this area are reviewed by May (1979). Here I ask: how are cormectance and stability related in a set of models whose structures more closely approximate reality than random webs? (b) I have argued that as the number of trophic levels increasesso does the relative time for the system to return to equilibrium after disturbance (Pimm and Lawton, 1977; DeAngelis et al., 1978; Vincent and Anderson, 1979). 144 004&5809/79/050144-15$02.00/O Copyright All rights
0 1979 by Academic Press, Inc. of reproduction in any form reserved.
THE
STRUCTURE
These results were for four-species greater complexity ?
OF FOOD
models:
145
WEBS
do they also hold
for models
of
(c) Many food webs nurture omnivorous species (by omnivory I mean feeding on more than one trophic level). Two patterns emerge from a consideration of four-species models; (i) more than one omnivore in any one food chain is improbable and (ii) if there is an omnivore in a food chain it is probable that it will feed on adjacent trophic levels and improbable that its prey will be separated by more than one trophic level (Pimm and Lawton, 1978). Do these results hold for more complex systems ? (d) For random systems, the product of average interaction strength and connectance must be “small” to ensure stability (May, 1972, 1979), but for a given level of connectance and interaction strength, the probability of stability may be enhanced if the species are arranged in compartments or blocks. To quote May (1972): “of the infinite ensemble of these particular 1Zspecies models, essentially none of the general ones are stable, whereas 35% of those arranged into three blocks are stable. Such examples suggest that our model multispecies communities, for a given average interaction strength and web connectance, will do better if the interactions tend to be arranged in blocksagain a feature observed in many natural systems.” A major objective of this paper is to ask: should model systems be organized into compartments of species characterized by strong interactions within compartments, but weak interactions among the compartments ?
METHODS
Briefly, the method involves generating a wide variety of food webs, for which I then calculate the percentage of models whose equilibria are unstable and, for the remainder, the distribution of their return times. I assume that if a system is ody stabIe for a small percentage of the possible parameter values, then real food webs with a similar structure will be rare. Conversely, I assume that the food webs in the real world will be those for which a high proportion of the model analogs are stable. I further assume that rapid returntimes enhance the probability of persistance. The method raises some technical issues which are considered next: they can be omitted without loss of continuity.
The Useof Structured Models Model structure is defined by a matrix which summarizes interactions between species in the webs. Each painvise element in this matrix, a,, , can be negative, zero, or positive. For n species, the matrix has na entries and so
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there are 3”’ possible structures. Even for n = 2 the number of possible models is large (81), yet perhaps only four of these are ecologically interesting (predatorprey, prey-predator, competitor-competitor, and symbiont-symbiont). The use of random models leads to choices which are easy to create mathematically, but ecologically absurd (Lawlor, 1978). The advantages of structured models are that I can avoid structures of little interest and minimize the number of simulations required. They also permit various problems to be tackled independently. Furthermore, in comparing different models I shall frequently conclude that model A is more probable than model B, yet model B might be more probable (by having a statistically greater percentage of stable models, for example) than a randomly chosen model. Using structured models eliminates the need for complex multiple comparisons, necessitated by the generation of purely random interaction matrices.
Choiceof Parameters Consider
the familiar
Lotka-Volterra
xi = Xi
model:
bi + f
aijX,
= Fi ,
i=l
where & and X, are the rate of change and population level of the ith species, b, the instantaneous per capita growth rate in the absence of any other species (negative for predators, positive for species at the base of the food web), and aii the effect of the jth species on the growth rate of the ith species. The dynamics of this equation near equilibrium are determined by the eigenvalues of the Jacobian (J)
3 3Xj
for
i=l,nandj-1,n
evaluated at equilibrium. If the real part of maximum eigenvalue (RAmax) of J is greater than zero, the system will be unstable. Otherwise the system will return to equilibrium at a rate approximately dependent on l/Rhmax . I define return time (RT) as
RTy&, max
xmax-=c0 .
Parameter values were set by the biological details implicit in the model. The nonzero diagonal elements of the Jacobian (corresponding to species at the base of the food web which are limited by resources) are Xfaii = -blx/Ki where @ is the equilibrium population of the ith species and fi=$ its equilibrium in the absence of all other species. Since the Xf cannot exceed & , these diagonal
THE
STRUCTURE
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WEBS
147
values will be in the interval (0, --bi). The bi were arbitrarily set to + 1 for selflimited species. This scales the return times. The perturbation to the system will return to l/e of its initial value in a time that is the return time multiplied by the time it takes the basal species to produce one new individual. None of the other species were self-limiting. This is a critical assumption, because if I allow species at the second and higher trophic levels to be selflimited, webs of highly complex structures are possible (Saunders, 1978). However for most animal species such models are implausible if the self-limiting terms are large because, other than in special cases, they imply the widespread existence of sophisticated conventional behavior (e.g., territoriality) which replaces food supply in the proceeding trophic level as the main controlling influence on the population (Lawton and Pimm, 1978). The off-diagonal elements are also of the form Xfaij where the sign of aij will depend on whether i is feeding on j or vice versa, or whether i and j are competitors. For predator-prey interactions uij is the number of new predators produced per prey eaten or the number of prey killed per predator; the latter will usually be much larger than the former. The X* will also differ, since predators are likely to be fewer than their prey at equilibrium. To model this asymmetry in magnitude the effects of the predator on the prey population were uniformly distributed on the interval (0, -10) and the effects of the prey on the predator on the interval (0, +O.l). A detailed consideration of the problem of parameter values is provided elsewhere (Pimm and Lawton, 1978). For each model I ran 1000 simulations. Though this represents a considerable investment of computer time, 1000 is a small number compared with the half million or more possible combinations of low, medium, and high values for each of 12 parameters. Many of the models I shall analyze are more complex than this; my most complex models would require on the order of lOLo simulations to exhaust the combinations. Though there is no guarantee that the small selection of parameters is not hopelessly biased, one can gain confidence in the sample by running models with a different “seed” for the random-number generator. I ran two models that, for their similarity in structure, appeared to have the most discrepant percentages of unstable models (33x, 88%) through six different sets of simulations. All percentages were within 2% of the mean for unstable models and 11 out of 12 of the percentages within 3 y/o of the mean for return times; the exception was only 5% more than the mean return time. For every simulation each matrix was converted to upper Hessenberg form and the eigenvalues extracted using routines in the IBM Scientific Subroutine Package. These techniques are described in Francis and Strachey (1961) Francis (1962), and Wilkinson (1965). Though double precision FORTRAN was used, I checked the accuracy of the procedures. Certain models were known to be qualitatively stable (May, 1973; Je f f ries, 1974) from their structure. For these models any positive eigenvalues must indicate numerical error in the routines. The numbers of positive eigenvalues never exceeded 3 per 1000 models
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analyzed, were generally zero, and when positive results were found, never larger than the minimum value printed by the program (0.0001).
THE “TAXONOMY”
were
OF FOOD WEBS
To facilitate the description of food webs I must introduce some terminology. I try to avoid such terms as “primary consumer,” or “herbivore,” because although these terms are usually unequivocal, there are cases where they are not. Such cases have led to criticism of the trophic level concept, and though in model webs there are no difficulties, I feel it is appropriate to use terms here that one can also use when testing my results with real food webs. (a) Types of species. Basal species are those which are resource limited and which feed on no other species in the web. Though they may be thought of as autotrophs, many food webs have as basal species animals whose limiting resource is detritus (and its associated microorganisms). Top predators are species on which nothing else feeds. Omnivores are species which feed on more than one trophic level; by this I mean that they consume energy which has flowed through a varying number of pathways from a basal species. (b) Position. Omnivores can feed in a variety of positions within the web. I number levels by the maximum number of connections plus one, from the basal species (level one). The position of the omnivor is defined by its level and that of the lowest level of its food. For example, a top-carnivore which feeds on a carnivore and on a herbivore on which the latter carnivore also feeds, is “2-4 omnivory.” Were the carnivore to feed on plants as well as the herbivore the web would include an example of “l-3 omnivory.” The final possibility is for the top predator to also feed on plants, “l-4 omnivory.” The number of predator-prey interactions in excess of those in the simple, straight chain models, I call Zinks. In a four-species, four-trophic-level model there can be up to three links (l-3,2-4, 14). With two or more food chains there can be many more links; either within or between chains. Links within chains are inevitably omnivore links, but those between chains can also include a predator feeding on prey species in different chains but at the same trophic level; these are nonomnivore links.
RESULTS AND DISCUSSION I have studied models of six and eight species. The interactions common to all models give an arrangement of two chains as shown in Fig. la. The models differed in the number of interactions in excess of the basic models (links), their kind (omnivore of nonomnivore), position and whether they were within
THE
STRUCTURE
OF FOOD
B ii
H 2
1
FIG.
1.
Diagrams
of models
149
WEBS
analyzed.
For further
details
see text.
100
P
10
1
FIG. 2. Mean percentages of stable models as a function of the number of links (zero through six), and connectance (C), for three trophic level models (circles; upper values on abscissa) and four level models (squares; lower values on abscissa). Ranges, (vertical lines) are truncated at 1 yO for all models where 1 %, or less, of the models were stable.
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or between chains. Tables I-IV present the percentages of stable models, and of these, the percentage that have return times shorter than 200. I shall discuss the results in the order of the questions posed in the Introduction. I plot, in Fig. 2, the percentage of stable models against the connectance for the six and eight species models. The models are highly nonrandom models yet they still show a sharp transition from largely stable to largely unstable as connectance increases. This is characteristic of random webs. Furthermore, TABLE Six-Species
Links Only Links (a)
Models:
Percentage
I of Unstable
Models
= 0 one model
0 y0 unstable
= 1 No omnivore Between 0
(b)
Links (a)
One omnivore
Within’ 89
Between 0
= 2 No omnivore Between 17-Sl(30)b
(b)
One omnivore
Within 82-89(85)
(c)
Two
2 within 22
Links (a)
omnivores
Between 64-80(72) 1 within/l 87
between
2 between 92
= 4 No omnivore Between 75
LI Links the links * Range link is not
(b)
One
(c)
Two
omnivore omnivores
Within 94 2 within 99
Between 91 1 within/l 99
between
can be between or within chains; web compartmentalization are all within chains. (mean); several models are possible because the position fixed.
2 between 97
is greatest
when
of the nonomnivore
THE
STRUCTURE
OF FOOD
TABLE
151
WEBS
II
Six-Species Models: Percentage of Stable Models with Return Times > 200
Links = 0 Only on model:
24
Links = 1 (a)
No omnivory Between 18-21(20)
(b)
One omnivore
Within 14
Between 15
Links = 2 (a) No omnivore Between 14-23(19) (b) One omnivore (c) Two omnivores
Within 8-13(11) 2 within
1 within/l 8
between
Between 7-23(12) 2 between 16
Links = 4 (a) No omnivory
(b) One omnivore (c) Two omnivores
Within 17 2 within -
1 within/l -
between
Between 14 Between 11 2 between 8
value for the transition from largely stable to largely unstableoccurs at lower valueswith the higher number of species.Both theseresultswere found by Gardner and Ashby (1970) and May (1972). However, the results are quantitatively different in one important respect: applying May’s (1972) formula for
the critical
transition
from stable to unstable
one would
predict that all my models would
be
unstable. It seemsunlikely that precise quantitative tests of May’s prediction are particularly meaningful for real webs(cf. McNaughton, 1978). While the mean percentagesof stable models decreasewith connectance, the ranges show considerable overlap suggesting that connectance, though important, is only one factor which affects the stability of these models. Table V analyzes the data grouped by the number of trophic levels, and
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STUART
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whether or not the model includes a l-4 omnivore. Food webs in this latter group have long return times (Pimm and Lawton, 1978); excluding these shows that there is a statistically significant negative relationship between short return times and the percentage of stable models. This is interesting. Though there is considerable variation, and the slopes of the relationships are small, the interpretation is that models with small ranges of parameters consistent with stability will show faster return to equilibrium (and hence, in variable environments less TABLE Eight-Species
Links
(a)
Percentage
of Unstable
Models
= 0
Only Links
Models:
III
one model =
0 y0 unstable
1
No omnivore Between 0
(b)
Links (a)
1 omnivore Position
Within
Between
l-3 2-4 l-4
81 91 96
0 0 0
= 2 No omnivore Between 21-51(32)
(b)
(c)
1 omnivore Position
Within
Between
1-3 2-4 1-4
77-81(78) 87-93(89) 94-96(95)
62-81(72) 74-87(80) 87-95(93)
2 omnivores Position
2 within
l-3 2-4 l-4 l-3,2-4 1-3, l-4 2-4,1-4
96 99 100 98 99 loo
1 within/l 78-81 86-90 94-86 81-83 80-82 89-93
between
2 between 82 92 100 33-88 80-96 82-98 Table
continued
THE
STRUCTURE
TABLE
OF FOOD
153
WEBS
III-Continued
Links=4 (a)
No omnivore Between 70-Sl(75)
(b)
1 omnivore Position
Within
Between
1-3 2-4 1-4
92 94 97
92 96 99
(c)
(d) Links (a)
2 omnivores Position
2 within
1-3 2-4 l-4 1-3, l-4 l-4,2-4 l-3,2-4
97 99-100 loo 99-100 99-100 99
93-95 97 99-100 100 100 95
than
99.8 ‘A unstable
4 omnivores-all
greater
1 within/l
between
2 between 97 99 loo 98 99 97-99
= 6 No omnivory Between 93
(b)
1 omnivore Position
Within
t
1-3 2-4 1-4 (c)
2 or more
same
+
Between
98 98 99 omnivores:
greater
than
98 o/O unstable
variation through time) than those having large ranges of parameters consistent with stability. Table V also illustrates the effect of adding an additional trophic level on the return times. Though the percentages of stable models span the same range, the return times for the three trophic level models are nearly always shorter than those for four-trophic-level models, irrespective of the details of the model. The earlier results on four-species models (Pimm and Lawton, 1977) seem robust. The effects of the numbers of omnivores and their position within a web can best be seen from an examination of Table I-IV, comparing models with the 653/r612-4
154
STUART
L.
TABLE Eight-Species
Links
Percentage
IV
of Stable
Models
with
Return
Times
> 200
= 0
One
model:
Links (a)
Models:
PIMM
=
51
1
No omnivore Between 46
(b)
One
Within
Between
l-3 2-4 l-4
40
42
36
45
Links (a)
omnivore
Position
= 2 No omnivore Between 39-54(44)
(b)
(c)
One omnivore Position
Within
Between
l-3 2-4 l-4
29-49(35) 16-49(26) 30-49(37)
39-54(44) 13-37(23) 36-61(48)
Two
omnivores
Position
2 within
1-3 2-4 l-4 I-3,2-4 l-3, l-4 2-4, l-4 Links (a)
-
1 within/l 30-36 21-29 38-41 30-31 35-40 22-43
between
2 between 64 26 22-24 52-64 26
= 4 No omnivore Between 35-60(43)
(b)
One
omnivore
Position
Within
Between
l-3 2-4 l-4
20 20 27
25 22 Table
continued
THE
STRUCTURE?
TABLE
OF FOOD
155
WEBS
IV-Continued
(c) Two omnivores Position l-3 2-4 1-4
2 within 1 within/l between 2 between 39 32 39 30 - (remainderof four-
link modelsare all unstable) Links = 6 (a) No omnivory
Between 36
samenumber of links. The total number of omnivores is important; the more omnivores there are within a web, the less likely it is to be stable. An eightspeciesmodel with one omnivore (l-3 or 2-4) is about as likely to be stable as a similar model with six between-chain, nonomnivore links. The connectance of the latter model is 25% greater than the former. The position of the omnivory is also important. Considering webswith four trophic levels and one omnivore, for example, one has a sequenceshowing increasing stability in the order l-4, 2-4, and l-3 omnivory (Table III). Models with 1-4 omnivores also have very long return times (Table V) which would also make this pattern of omnivory unlikely to be found in the real world. Again, the earlier results seemrobust. The effects of compartmentalization are also important in determining the percentageof stablemodels. Generally, as connectanceincreases,the degreeto which the speciesare compartmentalizedinto the two chainsdecreases.However, it is possible to vary compartmentalization independently of connectance. An example is shownin Fig. 1b; both modelshave a l-3 omnivore; in (1) the omnivore link is within one chain, and the system is completely compartmentalized, but in (2) the omnivore link is between chains. Though model (1) has only a 19% chance of being stable, model (2) is always stable. Indeed, model (2) is qualitatively stable, in the senseof May (1973) and Jeffries (1974). This suggests that completely compartmentalizedmodelsare not the most stablearrangement. Inspection of Tables I and III showsthis to be the case.In most casesfor a given degree of connectancethe most compartmentalizedmodels are the least stable. The least compartmentalized models may be less stable than those of intermediate compartmentalization. The few exceptions are from models with a low probability of stability. The complexity of even thesesimplemodelsmakes if difficult to display theseresults. Tables I and III are for six- and eight-species models. Within the Tables modelsare grouped that have the samenumber of links-interactions in excessof the basic models. Models are further grouped
156
STUART
L. PIMM
TABLE Analysis
V
of Variance for Return Times Trophic Levels and Percentage
Source
sums of squares
df
A.
Excluding
as a Function of Unstable
data with
of Number Models
of
F
P
1-4 omnivores”
Difference between trophic levels (6 and 8 species models)
1
6348
93
200
B. Data “/b of unstable
Sums of squares
df
Source
for l-4
models
1
33
Residual
omnivores
F
P
only”
294 4266
2.28
NS
Equations Mean
value
for
y0 return
times
>200
= 37.5 (Compare
with
Eq. (ii))
(i The I-4 omnivore data are significantly different from the remainder of the four omnivore models at P < 0.001. b Examination of residual plots showed no curvilinear trends. Partitioning of the residual variability into lack of fit and pure error confirmed that the linear model had no prominent bias. c Terms in parentheses are standard errors of the estimate. d Analysis A fits only a three parameter model to data (two intercepts, one slope) since slopes are not significantly different.
THE
STRUCTURE
OF FOOD
WEBS
157
by the kind and position of these links. Models on any row of the table have the same number of trophic levels and have identical numbers of omnivore and nonomnivore links. Furthermore, the omnivore links differ only in whether they are within or between chains; the trophic levels connected are the same. Different degrees of compartmentalization can be simply compared by reading across the table; compartmentalization decreases from left to right. This result, of the most compartmentalized models being least stable, runs counter to a number of suggestions about the organization of ecological systems (May, 1972, 1979). I conclude that species interactions should not be arranged in tight compartments. ACKNOWLEDGMENTS Dr. J. H. Lawton and Dr. R. M. May and three anonymous helpful suggestions. Computer facilities were provided by the Computer Center; I am greatful to all of these.
reviewers made many Texas Tech University
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