Community assembly and food web stability

Community Assembly and Food Web Stability* W. M. POST

AND

S. I_..

PIMM+

Environmental Sciences Dioision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 Received 29 July 1982; revised 16 December I982

ABSTRACT The ecological

assembly

of food webs is considered

as a process

of predator

coloniza-

tions and extinctions. The results of computer simulations using predator-prey equations allow us to identify three types of food web stability and examine how they may change through development of food webs. Species turnover stability increases, stability to extensive species extinction remains constant, and local stability to population fluctuations decreases

as food web assembly

proceeds.

INTRODUCTION In this paper we model developing food web structures and how this development effects changes in some community functions. Empirical studies have demonstrated a wide variety of food web structures [ 1,8,20,24,28]. Research toward a theoretical understanding of food web structure is already well underway. Thus, in a series of papers, Lawton and Pimm [6,11,12- 14,16,18,19] have considered local stability as a necessary property of food webs. Using fixed population parameters and patterns of interactions, they have been successful in predicting novel community properties, many of which have later been confirmed by analysis of real food webs [ 151. In contrast, several studies have indicated that there are features of ecological communities that may be best appreciated by understanding the ways in which they develop. Yodzis [28] created a set of rules, based on energetic constraints, which express the random assembly of food webs by

*Research supported by the National Science Foundation’s Ecosystem Studies Program under Interagency Agreement No. DEB 77-25781 with the U.S. Department of Energy under contract W-7405-eng-26 with Union Carbide Corporation. Publication No. 2126, Environment Sciences Division, ORNL. +Department of Zoology and Graduate Program in Ecology, University of Tennessee, Knoxville, Tennessee. MA THEMA TICA L BIOSCIENCES

64: 169- 192

OElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017

(I 983)

169 00255564/83/$03.00

170

W. M. POST AND S. L. PIMM

sequentially arriving species. He shows that the real food webs have statistical properties that are consistent with random food webs assembled according to these rules. Robinson and Valentine [22], using linear equations of population dynamics, examined the likelihood of a single species colonizing a randomly constructed system. They found that invadability of communities decreased as they became more complex. Roberts and Tregorming [21,25] studied the properties of systems constructed by natural processes, operating on initially large collections of randomly interacting species. They successively eliminated those species that contributed to the large systems’ instability. The resulting smaller systems showed a preponderance of predator-prey interactions and were much more likely to be stable than randomly produced systems of comparable size and complexity. We will add to this body of theory by considering food webs as the result of an assembly process that involves both colonization and extinction. In a sense, we combine the considerations of Yodzis [28], Robinson and Valentine [22], and Roberts and Tregorming [21]; however, our procedure differs substantially. We have chosen population models as the basis to explore community assembly succession. Furthermore, we do not begin with totally random systems. Our models are of food webs assembled according to a small number of ecologically plausible rules. Our model food websare special in a number of ways; perhaps most important is that it is among the animals, and not the plants, that the major changes in species composition take place. Plant species are lost, but never gained in our models. This is done to eliminate complexity due to a shifting resource base. We also make an assumption, largely for ease of performing the simulations, about the time scales of density adjustment due to population dynamics relative to colonization frequency. We assume that the population density adjustments occur fast enough that equilibrium is reached between colonization events. We are aware that many biological details are lacking. For example, direct (interference) competition and mutualistic interspecies interactions are not included. Nor do we include feedbacks between the environment and organisms such as those that result from the accumulation of soil organic matter, nutrients, etc. Nonetheless, our models do allow us to explore several features associated with community assembly. We concentrate on various meanings of food web stability and how they may be expected to change during this assembly process. We examine the questions: (1) How should the rate of colonizations change as the total number of accumulated colonizations increases? That is, should it be easier or harder for a species to colonize a young or an old food web? (2) How will the properties (growth rates, trophic efficiencies, etc.) of successful colonists change with time?

FOOD

WEB STABILITY

171

(3) What will be the effects of the successful colonists? In particular, will the colonist cause greater or lesser species losses from the food web as time progresses? (4) How quickly will the densities of the species return to equilibrium following perturbation? How will this return time change with time? METHODS We constructed a computer program to simulate general features of how a food web may develop through time (Figure 1). In general terms, we view the development of food webs as a heterotrophic successional process beginning with a community composed solely of autotrophs. To generate somewhat large and interesting food webs we use equations representing six autotrophic species populations. To aid interpretation of the results we assume that these

FIG. 1.

Flowchart

of food web assembly

program.

172

W. M. POST AND

S. L. PIMM

autotrophic species do not interact directly with each other. This simple community is subject to invasion by other species, drawn at random from a finite species pool, which prey upon one or more of the autotrophs. The invasion is successful if the food web, augmented by the additional equation and interaction terms representing the colonist population, passes a number of tests: (1) The new species’ density must be able to increase when it is rare and the other species are at equilibrium. (2) If a colonizing species completes a loop, it is rejected. Loops are of the form “A eats species B, which eats species A,” or “A eats B eats C eats A,” etc. Such loops are rare in nature [ 161. (3) Next, we examine whether the new food web has a finite number of equilibria. This is not the case if, for example, two top-predators (species on which no other feeds) share exactly the same prey species, or n top-predators share m prey species and n > m. In such cases we reject the potential colonist and try another. (4) When a colonist invades, it may cause some of the other species to assume negative (i.e., impossible) equilibrium population densities. All populations with negative density are deleted from the food web, and the resulting equilibrium densities are checked again. This process is repeated until a food web is obtained with positive population densities for all the remaining species-including the colonist. In food webs with three or more interacting species, this process may eliminate the potential colonist. When this happens, we restore the food web to its structure prior to the attempted colonization and pick another colonist. (5) The final test is whether the new food web is locally stable at its new equilibrium. If not, the potential colonist is rejected and the food web returned to its previous state. The first predator species are followed in turn by other colonists that prey upon members of the food web and may themselves be eaten by previous colonists. The ranges of the strengths of these predator-prey interactions and their probability of occurrence are specified at the beginning. The program continues to introduce potential colonists until 200 colonization attempts have been completed. At this point we end the simulation. The entire process is repeated 200 times. We then modify the parameters which determine strength and frequency of interaction and perform another 200 simulations. The design of the simulations is shown in Table 1. There is a total of 24 runs divided into four sets of six runs each. In the six runs of each set the colonists have different probabilities of being prey or predator of existing food web species. In sets 1 and 2, these probabilities are fixed. In sets 3 and 4, these probabilities depend on the number of species in the food web, decreasing as the number of species in the food web increases. The latter case

FOOD

173

WEB STABILITY TABLE Organization

1

of the Simulations” Set

I PP-Y Ppd L Prey L pred

2

Constantb

Constant

Constant 0.1 1.0

Constant 0.05 2.0

4

3

Adjusted

AdjustedC Adjusted 0.1

Adjusted 0.05

1.0

2.0

determine connectance while L,,,, L,,, determine the aPprey. Ppred strength of those connections. bFor each set of simulations, 6 runs of 200 replications each were performed. The values of Pprey and Pprcd used in these runs were: PPrey Run P pred 1 2 3 4 5 6 ‘The connectance

was adjusted

0.1

0.1

0.5 0.9 0.1

0.1 0.1

0.5 0.5 0.5 0.1 0.9 so that when the community

contained

10

species, Pprey and Ppred assumed the same values as in the constant case. When there were fewer (or more) species, the connectance was higher (or lower) according to the formula connectance is the number of species. This was truncated number contrast number

= lO( Pprer + P,,,,)/n, at I .O. This procedure

where n keeps the

of species with which a colonist interacts approximately constant, in to the other runs, where, because Pprer and Ppred are fixed, this increases with n

can occur in nature [20] and leads to a hyperbolic relationship between the number of species in the web and the percentage of nonzero interactions [13]. Sets 1 and 3 differ from 2 and 4 by the magnitude of the effect predators and prey have on each other. By choosing a variety of parameter limits, we can tell whether these parameters critically affect our results. Generality of the results across the various sets and runs is required for confidence in our conclusions. TECHNICAL

ASPECTS

OF THE COMPUTER

PROGRAM

The ability of any model to yield insight into food web structure and function depends on how well the details of the model compare with real food webs. Examination of these details will indicate the limits of the model’s abilities. First, we examine how well the mathematical equations imitate our notions of community dynamics. Second, we present assumptions involved in our parameter selection. Finally, we assess the appropriateness of each of the five tests for successful colonization for the simulations.

174 EQUATION

W. M. POST AND

S. L. PIMM

SELECTION

A view commonly held argues that the Lotka-Volterra model adequately describes population interactions only in the neighborhood of an equilibrium as a second order approximation of the actual and more complicated interactions. It has been used in this manner to make predictions of real food web properties [6,14-16,18,19]. Our use of these equations here is not in accord with this view. Our food web assembly algorithm models changes in species populations from one equilibrium to another including transient dynamics far from either equilibrium. However, we are not attempting to model any particular system where the richness of detailed interactions is of interest. Instead we are exploring general consequences of ecological processes thought to be important in shaping the structure and dynamics of food webs. For this level of abstraction we feel the Lotka-Volterra model is the simplest model of trophic interaction that adequately includes the ecological processes of establishment of initially rare colonists, local extinction of species due to infeasibility, and stability of the resultant food web. While the LotkaVolterra model offers an appraisal of nature that is idealized and incomplete, we believe that this model is suitable for our purposes. An advantage of the Lotka-Volterra equations lies in the ease of determining their equilibrium and stability properties. This is difficult for more sophisticated equations without resorting to numerical calculations of population size changes. We avoid these numerical calculations by analyzing the properties of successive equilibria. Consequently, we must assume that the population sizes will move from the old equilibrium to the new one following a successful colonization. This is not assured. What actually happens is that the old equilibrium becomes unstable due to the introduction of a colonist, causing population sizes to move away from this equilibrium to some other equilibrium. Without numerical simulation we cannot determine which equilibrium [5]. If it is possible for the new food web to have all the population sizes of the original food web positive as well as the population size of the colonist at the equilibrium with ah species densities positive, then our method always has the same result as numerical simulation. If this is not possible, one or more species must be deleted. This may result in several alternative equilibria, depending on which species are deleted to arrive at a feasible equilibrium. Tregonning and Roberts [26] suggest the somewhat arbitrary deletion of the species with the most negative equilibrium value. To decrease the computer time involved we chose a different though still arbitrary deletion process. We delete all species with negative population sizes at once. In most cases this is only one species and therefore does not differ much from the Tregonning-Roberts [26] process. With a limited computer run using Tregonmng and Roberts’s deletion rule, we did not notice large differences from our results.

175

FOOD WEB STABILITY

Clearly, given unlimited budgets to use large, fast computers, the best way to perform the food web assembly simulations is to numerically compute changes in species population sizes using a diversity of equations in addition to the Lotka-Volterra model. However, without a clear idea of what to expect, much time and money can be wasted. Our approach is to construct the simplest model possible, to see if this provides adequate insight into the nature of food webs before introducing other complications. PARAMETER

SELECTION

The dynamics of the autotrophic assumed to be logistic; that is,

species, in the absence of predation,

fi=xi(b;+aiixi),

are

(1)

where xi is the density of the ith species, jc, is its rate of change, and bi and aii are species-specific parameters. These species have a positive intrinsic rate of increase (bi > 0) and a negative self-limiting term (ai, < 0), implying intraspecific competition for resources. The magnitudes of the b, and aii are chosen randomly on the interval (0,l) and (- 1,0) respectively, thereby scaling the autotroph densities so that their median carrying capacity ( Kj = - bi/aii) is unity. Subsequent colonists have equations of the form

Thus, a colonist may be represented by adding a row and column onto the matrix A whose elements uij represent the effects of an individual of speciesj on the growth rate of species i. The signs of the nonzero elements of the new row and column indicate which of the existing species feed upon, or are fed upon by, the potential colonist. The occurrence of nonzero elements is decided randomly with probability Ppred that a species in the food web will be a predator of the colonizing species and probability Pprer that a species in the food web will be a prey of the colonizing species. The magnitudes of the uij’s are chosen at random from the interval (0, Lprey) and ( Lprcd,O) for different runs of the simulations as given in Table 1. Note that the effects that the predators have on prey growth rates (L,,,) and vice versa (Lprey) are asymmetrical, because predators usually must consume many units of prey before they produce an offspring [2,18]. We define the trophic ratio as &red /Lprey (note that ratios greater than 1 can occur). In two sets of simulations we chose values of L,, and Lpred that resulted in moderate trophic ratio (a median of 0.1). In two other sets the trophic ratios were low (a median of 0.025).

176

W. M. POST AND

S. L. PIMM

Each colonist is assigned a self-limiting term of zero (aii = 0, heterotrophic species). Arguments for the lack of self-limiting terms for predators are given by Lawton and Pimm [6]. Biologically reasonable self-limiting terms of small magnitude generated by predator interference, satiation, territoriality, etc., will not seriously change the results of our study. All heterotrophic species are assigned an intrinsic rate of increase chosen at random born the interval ( - 0.l,O),indicating that the predators depend entirely on specnes in the food web for their food. Throughout the simulations, the ranges from which parameters were selected remained constant (although the parameters required for successful colonizations may change through time). This assumption of a fixed range of parameter values can be cnticized on several grounds: (a) In many natural communities, early colonists may physically modify the habitat and, in so doing, facilitate the entrance of species whose parameters are distributed over ranges different from those of early colonists. (b) Different colonists may have parameters that vary over different limits. Thus, on average, omnivores or scavengers might interact with different numbers of prey and predators than would herbivores or carnivores. (c) Early colonists persist for shorter periods than those that arrive later. To survive, early colonists will require more rapid and effective dispersal than later coiomsts. Adaptations to dispersal will likely be at the expense of other adaptations-there will be “tradeoffs.” We could specify several classes of colonists by assigning them different parameter limits, different chances of interacting with existing species, and different chances of colonizing. We could allow parameters to change through time, and we could incorporate a variety of tradeoffs. But this would require a host of ad hoc assumptions with many new variables. Our strategy is to keep things as simple as possible and see if our simple models are sufficient to predict some general patterns that are known from nature. APPROPRIATENESS

OF COLONIZATION

TESTS

There are a variety of factors involved when our model determines a species to have failed in its attempt to colonize a community. Some of these factors have a biological explanation, such as the need to be able to increase when rare. Others are for mathematical convenience, such as the existence of a finite number of equilibrium points. The biologically reasonable decisions are more common. The most important reason for a species failing to colonize a community is its inability to increase when rare (usually more than 80% of the failures). The remaining reasons for failure are too few to alter our conclusions.

FOOD

177

WEB STABILITY

RESULTS For convenience we will use the number of attempted colomzations (whether or not they are successful) as a measure of time during an assembly. We present our results as a function of this measure, which we call successional time. In Figures 2-5 we have averaged the food web statistics of interest over each of 200 runs and by 5 time units.

COLONIZATION

DIFFICULTY

AND

TIME

Figure 2 shows the mean number of unsuccessful colonization attempts that occur between two successive successful colonization events. We call this difficulty of colonization, and it is plotted against successional time as defined above. The major features of this and other figures will be numbered so that we can refer to them later. From Figure 2 we observe: 1A. The difficulty of colonization increases approximately linearly with time; older food webs are harder to colonize than younger ones. 1B. Result 1A is independent of the particular combination of Ppred and Pprey used. However, the rate at which the difficulty of colonization increases with time does differ between runs. As one might expect, colonization is most difficult when the probability is highest that an existing species is the predator of the colonist. Colonization is easiest when the probability of an existing species being a prey to the colonist ( Ppr,) is highest. 1C. Differences between the four sets of runs are small. Thus changes in trophic efficiency and the precise patterns of how connectance varies with species number do not affect the results.

SPECIES

NUMBER

AND

TIME

Figure 3 shows a plot of the mean number of species as a function number of attempted colonizations. Its major features are:

of the

2A. The number of species approaches an equilibrium in all runs. Where the probability of an existing species being a prey to the colonist is 0.1 (which means the invader nearly always feeds on only one species), the equilibrium number of species is greater than the original six autotrophs. Where the probability is higher (0.4,0.9), the equilibrium number of species is less than the original six autotrophs. At constant levels of this probability, the smaller the probability of an existing species being the predator of the colonist, the greater the equilibrium number of species in the web. 2B. Again, the differences among the simulation sets are small.

178

W. M. POST AND

S. L. PIMM

INVASION DIFFICULTY WITH SUCCESSIONAL TIME 200

I

I

I

I

-(2) 160

80

4‘- _______ 2-S__-__

0

40

80

160 2000 40 SUCCESSIONAL TIME

80

420

3--S-

-

(60

200

FIG. 2. Plot of average invasion difficulty against successional time. Each subplot (1,2, 3, and 4) represents a different set of colonist parameters Lprey, Lpred. The different lines on each subplot represent the time course of runs with different values of Ppred and Pprey (see Table 1). Successional time on the ordinate is expressed in units of attempted colonizations, whether or not they are successful.

x

S3133dS JO kl38#nN

iP3b’U

180

W. M. POST AND

S. L. PIMM

with its prey and predators. Each colonist (xk) has an intrinsic rate of increase b, , which is negative. Each colonist will also prey on at least one species. We call the average per capita effect of a colonist on the growth rate of its prey u,~. The average effect a prey has on the growth of its predator colonists is called ukr. The average trophic ratio of the colonist is Tk and is calculated from the ratios uki/urk. Now a colonisr may also be the prey to one or more species in the community. If so, there are three additional parameters that describe the colonist. These are: the average effect the predators have on the colonist, akj; the average effect of the colonist on the predators, uJk; and the average trophic ratio of the colonist’s predators, Pk. We examined how these parameters changed through time. Statistical analyses nor presented here confirmed our impressions of the computer runs. 3A. There were no discernable ters through time.

changes in colonist

population

parame-

Parameters typical of successful colonists were esrabkhed at the start of the simulations and remained throughout the assembly sequence. Table 2 contains the means for these parameters. While we were surprised that we could not uncover any simple trends through time, the parameters do differ from those expected from a random sampling of those parameters available: 3B. The effects that a prey has on the predator (uki, ajk) are higher than the mean, while the effects that a predator has on its prey ( uik, uk,) are less negative than the mean. Both of these contribute to increased trophic ratio-particularly of the colonist’s predators. The intrinsic rate of increase of the colonist (bk) is smaller (less negative) than the mean. These features are to be expected-they all enhance the colonists’ chances of invading when rare. RESISTANCE

TO SPECS

EXTINCTION

Next we consider the effects of a colonist on species already in the food web. In Table 3 we display the proportions of the successful colonizations that result in ‘extensive’ extinctions in which three or more species were eliminated from the food web. This proportion was averaged over four periods (the first 50 attempted invasions, the second 50, etc.) to see if serious extinctions were more frequent early or late in the assembly sequence. The statistical analyses of these data confirm what can be seen by inspection: 4A. Successful colonists do not cause serious extinctions any more commonly early or late in successional time. Of course, successful colonists become scarcer later in the community assembly, so the number of serious extinctions will decrease through time.

FOOD

181

WEB STABILITY TABLE 2 Some Characteristics Seth 1

Runb

n

c,k

CT/k

iik,

‘k

i ii “1

10.3

.060

-.486

,053

-.311

0.594

.339

- ,037

4.0 2.8

,056 ,051

-.483 - ,489

,053 .062

- ,314 -.217

,051 ,058

- ,363

.316 ,312 ,354

- ,040 - ,043

- ,484 -.462

0.545 1.229 0.622

-.496

,049

- ,394 - ,274

1.209 2.362

,353 .361

-.969

,023

- .648

0.328

.I02

-.988 -.906 -.971

,023 ,032 ,029

- ,636 - ,206 - ,488

0.100 0.216 0.561

,100 ,146 ,141

-

- ,900 -.983

,028 ,025

- .553 - .421

1.477 0.945

.1971 ,128

- ,036 - ,028

-.484 - ,491 -.483

,044 .064 .058

- ,038 - .042 - .041

-.475 -.471 - ,471

.051 ,053 ,050

-.982 -.901

,026 .033

-.894 -.960

,031 ,025

-.904 -.954

,028 ,026

iv ” vi 2

4

akl

8.8

,065

4.1 1.5

,057 ,065

9.2 4.4 3.4

,031 ,029 .025

8.4 4.6 7.4

,031 ,028 ,032

i ii ill iv ”

9.4

,062

3.0 2.9 8.2 4.1

,049 ,051 ,061 ,056

vi

1.5

,065

i ii lu

8.5

,031

iv

3.5 3.5 7.9

.026 ,026 .032

” vi

5.0 7.4

,028 ,033

i ii Lu iv ” vi

3

of the Simulations” Tk

bk

- ,041 - .044 - ,041 ,027 ,029 .034 .027

- ,382

1.068

,338

-.391 -.I66 - ,322 -.310

0.495 1.660 0.854 1.040

,263 ,278 .354 .397

- ,309

1.935

.371

- ,042

- ,619

0.403 0.365

.109 ,131

- ,026 - ,035

- ,686 - ,506

0.231 0.192 1.625

,152 ,095 ,128

- ,033 - .033

- ,342

1.053

,078

- ,273 - .407

- ,039 - ,044

- ,035 - .034

a, = the mean number of species at the end of the simulations, ok, = the mean effect of a successful colonist’s prey’s density on the growth rate of the colonist, alk = the mean effect of a successful colonist on tne growth rate of its prey populations, a,, = the mean effect of a colonist on the growth rate of its predators, uk, = the mean effect of a colonist’s predators on the growth rate of the colonist, Pk = the mean trophic ratio of a colonist’s predators, Tk = the mean trophic ratio of the colonist, b, = the mean rate of increase of the colonist in the absence of its prey. From the choice of the parameters, it is possible to derive the expected parameter values. Midpoints of the ranges are: for sets 1 and 3, ak, = a,k = 0.05, alk = ak, = -0.5; for sets 2 and 4, ok, = a,k = 0.025, (I,~ = nkJ = - 1.0; for all sets, b, = - 0.05. bFor details see Table 1.

182

W. M. POST AND

S. L. PIMM

TABLE 3 Proportions of Colonizations that Lead to Three or More Species Becoming Extinct Periodb Seta I

I st

2nd

3rd

4th

i ii .

,097 ,135

,116 ,093

111

,044 ,154 ,211

iv ” vi

.038 ,140 ,014

,065 ,086 ,106

,065 ,088 ,067

,137 ,108 .I31

,063

.038

,119 ,150 ,145

,117 ,204

Runa

i

2

ii . ..

111

iv v vi 3

i ii 111

iv ” vi ‘See Table bPeriods

.05 1 ,194 ,215 ,015

.120 ,064 .059 ,094 ,126 .I73 .054

,031 ,113

.077 ,053 ,182

.OOO

,000

,100

,058

,121 ,067 ,108 ,072

,153 ,124

,116

,200 ,265 ,040 .072

,075 ,142

.194 .009

.008

,115 ,086 ,216 ,071

,125

,040 .I38 ,100 .068 .OOG

1 for details.

are the first 50 attempted

colonizations,

second

50, etc.

4B. Successful colonists cause a higher proportion of serious extinctions when the colonist has a greater chance of being a predator to species already in the web. RETURN

TIME

Establishment of new colonists also modifies the return time of the food web to equilibrium after disturbance. A measure of the return time is the time it takes the difference between species’ densities and their equilibrium densities to fall to a specified fraction (l/e) of the difference immediately following a perturbation. The approximate value can be calculated from:

T, = - l/Re(L,,),

(11)

where Re( A,,,,) is the real part of the maximum eigenvalue of the food web’s stability matrix [17]. Clearly, long return times imply low stability to per-

FOOD

WEB STABILITY

183 LOGARITHM OF AVERAGE RETURN TIMES

2.8

24

____-_ 5_____ 04080

(20

(60 2000 40 SUCCESSIONAL TIME

80

6-

G?o

-

(60

2Qo

FIG. 4. Logarithm of average return time, an inverse measure of food web stability to perturbation in population size, as a function of successional time (see Figure 2 for additional details).

turbations and vice versa. Figure 4 shows the logarithm of the average return times for the models as a function of time. The figure displays the result: 5A.

The return time of communities

increases through successional

time.

DISCUSSION From our results we draw these main conclusions: (1) As the community develops, the turnover rate of the species decreases: the food web resists the colonists better, and consequently species within the

184

W. M. POST AND

S. L. PIMM

food web persist longer (result 1A). This trend continues even after the number of species in the system has equilibrated. (2) The population parameters of successful colonists differ from mean values of the colonist pool. The differences are in the direction of higher trophic ratios and lower density-independent mortalities (result 3B). The parameters, however, do not obviously change during the course of community assembly (result 3A). (3) The sensitivity of the food web to extinctions of large numbers of species when a colonist is successful does not change through time, though from conclusion (1) the number of successful colonizations does decline (result 4A). (4) The return time of the population densities after perturbations increases with time (result 5A). DIFFICULTY

OF COLONIZATION

Conclusion (1) would seem to indicate a selection effect of successive colonizations on the allowable parameters of future colonists. When a new

2

-

0 1 3

+

+

1

1

INITIAL FOOD

WEB

FIG. 5. Simple example 3 are predators.

TRANSITION DURING COLONIZATION

of food web assembly.

Species

NEW COMMUNITY RESULTING FROM EXTINCTION OF POPULATION 2

1is an autotroph.

Species 2 and

FOOD

WEB STABILITY

185

species becomes established, it appears to alter the range of parameters available to another colonist so that it is more difficult for the new colonist to become established. This phenomenon can be demonstrated analytically for the simple food web process depicted in Figure 5, where a predator on an autotroph is replaced by another predator (see Appendix A for details). This conclusion is not initially obvious. We could argue that, as assembly proceeded, the resource base of the food web might become more finely divided among consumers so that a new species would have more combinations of predators and prey available to it and a better chance of colonizing. However. our invasion rule prevents this subdividing from occurring. A species is more likely to be successful if it utilizes as much resource as is available, resulting in continual concentration of resources through succession. This also is largely responsible for the difficulty of colonization after the number of species has equilibrated. Perhaps the inclusion of interference competition or mutualism or allowing for colonization of new autotrophs would reverse this result, or at least make colonization equally likely throughout succession. Empirical observations, however, are in keeping with our theoretical result. Shugart and Hett [23] calculated the rate of species turnover against, time for a wide range of successional communities. In all examples for which there were sufficient data, the species turnover rates decreased with time. PARAMETER

CHANGES

Theoretically, it should be possible to detect trends in the colonists’ parameters through assembly, reflecting the increase of the food web’s resistance to further colonizations. However, we did not find any such trends (result 3A). We attribute this to two factors. First, successive colonists may become established in many positions in a food web. Each position requires a different configuration of parameters for a successful colonist, and, as the food web changes, the kinds of positions available for colonization change. Second, the criteria for successful colonization are complicated functions of the population equation parameters (see the example in Appendix A). The condition for colonization can be met by a wide variety of relationships among the parameters. Changes in such a condition could be difficult to detect from the single parameters or the simple combinations of parameters we examined. It is possible that the cause of the difficulty of invasion increasing through assembly could be related to a trend in some macroscopic system parameter. In the example analyzed in Appendix A, successive colonizations require a continual decrease in the biomass of the autotroph. As indicated in Figure 6, there is a reduction in autotrophic biomass during the course of the simulations. It seems intuitively correct that increasing utilization and consolidation

W. M. POST AND S. L. PIMM

186

LOGARITHM OF MEAN AUTOTROPHIC WITH SUCCESSION I

1.6

I

I

BIOMASS

I

tt 1 1.2 0.8

5-b___ I

04080

120

160 2000

6-e

I

I

I

40

80

120

160

200

SUCCESSIONAL TIME

FIG. 6. Logarithm of average of the total autotrophic biomass in the food web with successional time.

of a fixed autotrophic population should lower the autotrophic biomass and thereby make it harder for later-arriving species to satisfy the criteria for colonization. Admittedly, the data in Figure 6 do not represent a general proof of this notion, but the downward trend does lend it some support. STABILITY

As shown in Figure 4, the return time of food webs increases during successional time. This is presumably due to changes in the parameter values of the food web as new colonists become established and other species are

187

FOOD WEB STABILITY

-0.15

0

I 0.1

I 0.2

I 0.3

I 0.4

I 0.5

I 0.6

I 0.7

I 0.8

I 0.9

1.0

XT FIG. 7. Plot of the maximum real part of the eigenvalues of a predator-prey system as a function of prey biomass, x,!. In this plot, a,, (see Appendix B) is allowed to vary while aI1 the other parameters were held constant with the parameter values a,, = 0.5, h, = 0.5, b, = 0.05.

eliminated. Again, a general analytical analysis of this phenomenon is impossible, due to the complexity of the food webs involved. However, analysis of the simple assembly process presented in Figure 5 is suggestive. Analysis which shows that the return time may either increase or decrease with assembly in this example appears in Appendix B. The essence of the results is summarized by Figure 7. From Appendix A we know that the equilibrium density of the autotroph, x, , decreases with succession. Figure 7 indicates that initial parameters determine whether the return time will increase or decrease with continued assembly. Consider the trivial but possible example of six independent predator-prey systems following the sequence depicted in Figure 5. It is likely that one of the pairs of species will have a prey density less than the value for which the return time is a minimum (that is, A,, is most negative). Decreasing prey density would result in increasing return time as the prey density of this pair moved further and further to the left in Figure 7. CONCLUSION Odum [lo] presented a tabular model of ecological succession which indicated expected trends in the development of ecological communities. We construct an abstract computer-based model that simulates several natural processes associated with food web development through time. Our model includes both colonizations and species losses according to various dynamical considerations. Our model is not capable of producing results applicable to

188

W. M. POST AND

S. L. PIMM

all the features listed in Odum’s [lo] table, because many features of succession are not considered by our model. For example, we do not incorporate decomposer cycles, mutualism, direct (interference) competition, or spatial structure. Our model, therefore, excludes various components of niche specialization and spatial diversity. Also, because we assume that the food web equilibrates after each successful colonization, the model does not follow time changes of such features as the ratio of production to respiration. Our approach, however, allows us to explore hypotheses concerning food web properties from a perspective of population dynamics. The conclusions that we wish to discuss here involve four notions of stability and how stability, variously defined, changes through community assembly. (a) The first conclusion is that simple processes of independent introductions of colonizing species and elimination of other species, according to population dynamics, result in a progression towards increased food web stability. “Stability” in this sense means a very slow rate of species turnover and the increasing difficulty of establishing new colonists even though the parameters of the colonists are being chosen from between the same bounds. Thus, the population dynamics of interacting species is sufficient to produce a process of food web development without invoking some abstract goalseeking behavior in the part of the system-such as increased homeostatic control [lo] or increased information content [9]. This turnover stability is due to the process of resource consolidation by successive colonists, needed in order for their density to increase in a food web when they are initially rare. (b) The second type of stability that appears in our model is resistance to extensive species extinctions in which three or more species are eliminated from the food web. Extensive extinctions caused by a recent colonist indicate a major shift in the organization of the food web. The exact cause of such a reshuffling is not known, but probably involves feedback processes between all of the species of the food web which is disrupted by the addition of a new colonist. Extensive extinctions occur with relatively low frequency throughout succession. A successful colonist is as likely to effect extensive extinctions early as late in the succession, though, as we have discussed, successful colonizations are much rarer late in the succession. (c) We also examine the model’s predictions about the return time to equilibrium following disturbance. In the simple system of Figure 5, one could observe an increasing or a decreasing trend in return time through time, or even a decrease followed by an increase. Odum [lo] cites several studies that indicate increasing resilience. Hurd et al. [4] report a field experiment in which older, more floristically rich communities showed a greater change in species abundances than younger, less floristically rich communities. Loucks [7] draws his evidence of a decrease followed by an

189

FOOD WEB STABILITY

increase in return time from analysis of forest data from Wisconsin. For more complex food webs than that of Figure 5, our simulations indicate that return time increases through community assembly. This may also be attributable to resource consolidation by successive colonists, because in simple cases it is linked to a decrease in autotrophic biomass. The role of resource consolidation in increasing the food web return time, however, is less clear than its role in increasing turnover stability. (d) Finally, using a theoretical result obtained by DeAngelis [3], we can also determine the effect of assembly on nutrient cycling. DeAngelis showed that the shorter the time a unit of nutrient resides in an ecosystem, the shorter the return time of the ecosystem to perturbations of that nutrient. During assembly, return time of the food web increases, indicating that the length of time a unit of nutrient remains within the trophic part of the ecosystem increases. Given that the residence time of a nutrient in the remainder of the system remains relatively constant, the proportion of nutrients retained by the system increases with time. Simply, during assembly nutrients would become more tightly bound-a result that matches Odum’s [lo] assertion about autotrophic succession. Robinson and Valentine [22] consider assembly by examining a single colonist invading random communities of differing complexity. Using a linear model, they also derived results which are essentially the same as (a) and (c) above. Our computer experiments with a different model and more extensive assembly process confirm their results. This concurrence argues for the generality of these results. APPENDIX

A

In Figure 5, species 3 enters the community as an omnivorous predator of the two existing species: species 2, a predator; and species 1, its prey. With Lotka-Volterra dynamics, the condition required for species 3 to enter the community when x, and x2 are at their equilibrium densities (XT, xz) is a,,x:

+ a32x; > b,.

(Al)

Similarly, when species 2 becomes extinct and species 1 and species 3 are at their equilibrium densities (x:*, x3**), the condition for species 4 to colonize is a4,x:*

+ ad3x:* >

ht.

Assume that each predator gains, on the average, the same advantage from each prey ( a3, = a3* = ad, = a,,), and that each predator declines, on the average, at the same rate in the absence of its prey (b, = b4). Then whether

190

W. M. POST AND

S. L. PIMM

Equation (Al) or (A2) is easier to satisfy will depend on the initial rnagnitudes of x:, x;, xi+*, and x;*. Consider the x, terms first. For species 2 to be driven to extinction by species 3, the derivative kz must be less than zero when x, and x3 are at their equilibrium densities. This requires - b, + u2,x;* - az3x$* -C 0.

(A3)

The term in x;* will often be relatively small because a predator’s equilibrium density is usually much lower than that of its prey (x:*). If this is true, then Equation (A3) may be simplified. Even if it is not, the following ensures that (A3) will be true: - b, + u2,x;* < 0. Now bz/a,, is species l’s equilibrium i.e., XT. So Equation (A4) becomes

(A4)

density in the presence

x;* < x:.

of species 2,

(AS)

Simply, a decrease in the equilibrium density of x, will usually accompany the pattern of assembly suggested by Figure 5. Returning to Equations (Al) and (A2), the decline in the equilibrium density of species 1 makes it more difficult to satisfy (A2) than (Al). But what about the relative sizes of x; and x $*? The equilibrium density of any predator, x7, in a two-species model is

q = h - a,,~;

646)

alj

Thus, for the simple Lotka-Volterra models, when XT decreases, x,‘! increases. This would make it easier for (Al) to be satisfied. Whether (Al) or (A2) is easier to satisfy depends on the relative changes of the prey’s and the predator’s equilibrium densities with the invasion of the predator. This is the term a, ,/a,i. It is likely to be small; it is the effect that one prey has on its own growth rate divided by the effect that one predator has on the prey’s growth rate. When this ratio is small, the increase from xf to xd* will be small relative to the decrease from XT to x:*. Simply, it should be harder for species 4 to invade than species 3. APPENDIX

B

In the two species Lotka-Volterra predator-prey are calculated as the roots to the equation det

- allxl *-x aj,x,?

- a,jx : _A

system, the eigenvalues

=O, I

(Bl)

FOOD WEB STABILITY

191

or

The real part of the maximum eigenvalue is plotted in Figure of x;. The real part of the larger eigenvalue will decrease increase) as x; decreases from its maximum value ( - b, /a, value of x: for which the real part of the larger eigenvalue is call this value min( h ,,), and it occurs when

7 as a function (resilience will I or K,) to the a minimum. We

(B3) The parameter ai, is the effect of the predator (speciesj) on the prey (species 1). When x; is depressed below this value, the system will become less resilient and the populations will begin to oscillate as they return to equilibrium. We can determine whether increasing or decreasing return time is more likely by calculating the approximate size of the elements in Equation (B3). Suppose we fix the values of aj, for each successive predator but allow the predators’ rates of decrease, bj, to vary. For a predator xj to replace the existing predator xj, it must have a smaller rate of decrease. Now, XT = b,/a. JI'

(B4)

so decreasing b, means decreasing x,* . For the return time to increase with assembly, XT must exceed the value for which Xmaxis a minimum. This yields the inequality bj >

4afl b,

a:, +4a,,ajl

’

@5)

The effect of prey on the predator growth rate, ajr, is likely to be very small, even compared with the other parameters. Thus, the inequality (B5) is likely to be met. Alternatively, we assume that bj is fixed and each successive predator has a larger effect on its prey species; that is, ajl increases during assembly. In this case, the value for x: for which A,, is a minimum is shifted toward higher values of x:. This occurs while x: is shifting simultaneously toward lower values. Thus x; will likely become less than the value for which the return time is minimum, and the return time will increase with food web assembly.

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REFERENCES 1 2

J. E. Cohen, Food Webs and Niche Space, Princeton U.P., Princeton, N.J., 1978. D. L. DeAngelis, Stability and connectance in food web models, Eco/ogy 56:238-243 (1975).

3

Energy

flow, nutrient

cycling,

and ecosystem

resilience,

Ecology

61:764-771

(1980). 4

5 6 7 8

L. E. Hurd, M. V. Mellinger, L. L. Wolf, and S. J. McNaughton, at three trophic levels in terrestrial successional ecosystems, (1971). M. E. Gilpin and T. Case, Multiple Nature 282:838-839 (1979).

domains

of attraction

Stability and diversity Science 173: 1134- I136

in competition

communities,

J. H. Lawton and S. L. Pimm, Population dynamics and the length of food chains, Nature 272:189-190 (1978). 0. L. Loucks, Evolution of diversity, efficiency, and community stability, Amer. Zool. 10:189-190 (1970). N. MacDonald,

Simple aspects

of food web complexity,

J. Theoref. Biol. 80:577-588

(1979). 9 10 II 12 13 14 15 16 17 18 19 20 21

R. Margalef, Perspectives in Ecological Theoty, Univ. of Chicago Press, Chicago, 1968. E. P. Odum, The strategy of ecosystem development, Science 164:262-270 (1969). S. L. Pimm, Complexity and stability: another look at MacArthur’s original hypothesis, Oikos 33:351-357 (1979). The structure of food webs, Theoret. Population Bio/. 16: 114-158 (1979). -, Bounds on food web connectance, Nature 284:591 (1980). -. _,

Food web design and the effect of species deletion, Oikos 35: 139- 149 (1980). Properties of food webs, Ecology 61:219-225 (1980). Food Webs, Chapman and Hall, London, 1982. -1 S. L. Pimm and J. H. Lawton, The number of trophic levels in ecological communities, -,

Nature 268:329-331 (1977). S. L. Pimm and J. H. Lawton, 275~542-544 (1978).

On feeding

on more

than one trophic

S. L. Pimm and J. H. Lawton, Are food webs compartmented?, 49:879-898 (1980). M. Rejmanek and P. Stary, Connectance in real biotic communities for stability of model systems, Nafure 280:3 1 l-3 12 (1979). A. Roberts

and K. Tregonmng,

The robustness

of natural

systems,

level, Nature

J. Anim. and critical

Ecol. values

Nature 288265-266

26

(1980). J. V. Robinson and W. D. Valentine, The concepts of elasticity, invulnerability, and invadability, J. Theoret. Biol. 8 I:91 - 104 (1979). H. H. Shugart and J. M. Hett, Succession: similarities of species turnover rates, Science 180:1379-1381 (1973). G. Sugihara, Niche hierarchy: Structure, organization and assembly in natural communities, PhD Dissertation, Princeton Univ., 198 1. K. Tregonmng and A. Roberts, Ecosystem-like behavior of a random-interaction model, Bull. Math. Biol. 40:513-524 (1978). Complex systems which evolve towards homeostasis, Nature 28 1:563-564

27 28

(1979). P. Yodzis, The connectance of real ecosystems, Nafure 284:544-545 (1980). P. Yodzis, The structure of assembled communities, J. Theoret. Biol. 92:103-l

22 23 24 25

(1981).

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