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Keywords:

  • stream ecosystem;
  • population models;
  • predation impact;
  • indirect interactions

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

1. Population models based on Lotka–Volterra-type differential equations with logistic prey were made for a simple stream community including two stonefly prey Leuctra nigra Olivier and Nemurella pictetii Klàpalek, and two predators, the caddisfly Plectrocnemia conspersa (Curtis) and the alderfly Sialis fuliginosa Pictet. In order to assess the importance of predation in this system, we constructed both an explicit four-species model and a simplified model with two functional groups which was more amenable to analytical treatment.

2. The models were parameterized using new data on adult emergence and recruitment combined with previously published data on larval densities and prey uptake. The models were falsified if parameterizations led either to negative prey carrying capacities or to unstable dynamics.

3. Both the functional group and four-species models predict asymptotically stable interactions, with feasible carrying capacities. The models are consistent in predicting that the observed prey are in excess of 70% of their carrying capacities. The four-species model indicates that predation impact is not evenly shared between the two prey, with L. nigra being depressed further from its carrying capacity than N. pictetii.

4. Sensitivity analysis shows that the results of the full four-species model remain very robust to realistic levels of stochastic variation in the input data.

5. The four-species model is used to predict the outcome of an ongoing large-scale field experiment involving the transfer of all S. fuliginosa eggs from one stretch of the stream to another. Although the equilibrial prey populations are barely affected by the manipulation, the model predicts marked transient prey-release and prey-depression of L. nigra in the predator addition and removal areas, respectively.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

Macroinvertebrate predators frequently form a conspicuous component of freshwater food-webs (Peckarsky 1984; Hildrew 1992). As a consequence, predation has been one of the most intensively studied of the various processes proposed as structuring agents in benthic stream communities (Hildrew & Giller 1994). However, in contrast with results from lentic communities (Kerfoot & Sih 1987), clear-cut overall patterns have been elusive and the general importance of predation in the stream benthos remains an area of debate (Cooper et al. 1998).

Predation may have demonstrable effects in a number of independent, but not necessarily mutually exclusive, ways. For example, it may produce short-term depletion of local prey density in regions where predator density is especially high or prey recruitment is particularly low, or it may reduce broad-scale average prey densities relative to the predator-free equilibrium value. Alternatively it may act indirectly, for example by mediating the coexistence of prey that would otherwise eliminate each other through competitive interactions.

Most of the literature on predation in benthic stream communities has focused on the first of these possibilities. Although some studies have used gut contents analysis as a direct measure of predation rates (Townsend & Hildrew 1977; Hildrew & Townsend 1982; Lancaster 1996), the majority have utilized enclosure–exclosure techniques (Walde & Davies 1984; Peckarsky 1985; Cooper, Walde & Peckarsky 1990; Lancaster, Hildrew & Townsend 1990; 1991; Sih & Wooster 1994; England & Olsson 1996; England 1997). In these experiments local areas are set up in which predator density is manipulated, but the (generally smaller) prey have free exchange with the surroundings. Although the results from such studies have been varied, the majority of those involving invertebrates appear to demonstrate detectable local prey depletion in the presence of predators (Sih & Wooster 1994).

Although enclosure studies provide evidence of high predation rates in lotic ecosystems, their short duration compared to the relatively long development times of aquatic insects necessarily restricts the light they shed on the long-term balance between predation and recruitment. Further understanding of the processes that shape stream insect communities at intergenerational time-scales requires detailed knowledge of recruitment dynamics which have been much less intensively studied (Hildrew & Giller 1994).

The first aim of the work reported here was to close this gap for Broadstone stream, a well-studied lotic ecosystem in southern England whose macro-fauna is dominated by insect larvae. Our second aim was to use simple population models to generate a consistent description of the main demographic components of the system, which could then be used to make a general assessment of the dynamic impact of predation in the community as well as predicting the outcome of a projected large-scale experiment in which one of the main predators will be removed from a large area of the system.

In the following sections, we begin by outlining the basic demographic data that were used to parameterize the models. This falls into four subsections: the benthic densities of the larvae of the two prey species and the two predator species that constitute the four main macroinvertebrates in the system; the rates of adult insect emergence; the rates of egg laying; and the predation rates. Next we outline our general modelling approach, before detailing a simplified (one-predator/one-prey) model suitable for analytical treatment and then a full four-species model which we treat by largely numerical methods.

In both cases we are able to show that despite the apparent efficiency of the predators, the long-term average prey abundance is relatively close to carrying capacity – suggesting that in this case resource limitation or prey density dependence is a more likely determinant of community structure than predation pressure. Since there is considerable statistical variability associated with most of the demographic data, we then investigate the robustness of this result to measurement error in the parameters.

Finally, we examine the full model’s predictions for a field experiment in which all the eggs of one of the predators are removed from a stretch of stream thereby halting recruitment there, and translocated to increase recruitment further downstream. We show that the removal of one predator eventually results in a significant increase in the abundance of its competitor, accompanied by a change in prey abundance small enough to render it essentially unobservable (< 3%). However, these changes occur over a time-scale of almost 10 years. During a 2-year experiment, the likely results are slightly different – with no observable change in the abundance of the unmanipulated predator and a (transient) 8% increase in the abundance of the most commonly taken prey.

Demographic data

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

Broadstone stream

Broadstone stream is a small, acidic, iron-rich stream in the Ashdown forest in Sussex, SE England (0°03′00″E, 51°04′30″N). The mean depth and width at base flow are 0·14 m and 1·38 m, respectively, and the mean flow is 0·01 m–3s–1 (Lancaster & Hildrew 1993). The surrounding riparian woodland consists mainly of birch (Betula pubescens Ehrh.), alder (Alnus glutinosa (L.) Gaertner), oak (Quercus robur L.), hazel (Corylus avellana L.), elder (Sambucus nigra L.), and sweet chestnut (Castanea sativa Miller).

Fish are absent, but the stream contains a relatively simple and well-documented macroinvertebrate community consisting principally of aquatic insect larvae (Hildrew & Townsend 1976; Townsend & Hildrew 1979a, 1980, 1982). Two predators are common as larvae all the year round; the caddisfly Plectrocnemia conspersa, and the alderfly Sialis fuliginosa. Their predominant prey throughout the year are detritivorous stoneflies, of which Leuctra nigra, and Nemurella pictetii are by far the most abundant. Chironomid larvae of various species are also numerically important during the summer (Townsend & Hildrew 1979a,b).

The larvae of P. conspersa develop in the stream for a year, before pupating, eclosing as short-lived adults, mating, and laying egg-masses on emerging stones in the stream. By contrast S. fuliginosa takes 2 years from egg to adult. Final instar S. fuliginosa crawl out of the stream before pupating in the soil of the bank. Their eggs are layed in conspicuous clumps on the underside of leaves of trees overhanging the stream. Adult stoneflies of both species emerge directly from the stream, but adult N. pictetii attach jelly egg masses to stones while the egg masses of L. nigra disperse on contact with water.

The impact of predation by P. conspersa in Broadstone stream has been assessed from estimates of consumption rates obtained by gut contents analysis (Townsend & Hildrew 1977; Hildrew & Townsend 1982). They found considerable seasonal variation, with rates between 0·2 and 2·8% of the standing biomass per day. Lancaster et al. (1990, 1991) have shown that the presence of P. conspersa can reduce prey numbers during the colonization of enclosures, as well as reducing prey density in established enclosures when prey exchange with the surrounding habitat is sufficiently restricted. However, until the present study, no information was available on the adult stages of either predators or prey in this system.

Larval benthic densities

Random sampling of the benthos using a standard protocol (e.g. Hildrew & Townsend 1982) over a period of almost 25 years has produced a large amount of information on the larval densities of the four main species. Samples were taken over a 500-m stretch of the stream using modified Surber samplers (area 0·00625 m2, 330 µm net mesh) placed randomly on the stream bed. The substratum was removed to a depth of about 5 cm and washed into the downstream net.

Although sampling has been maintained over a prolonged period, there are many gaps in the resulting time-series (Fig. 1), which contains few samples taken in consecutive years. All the species have highly variable abundances, due in part to the temporally restricted period during which recruitment takes place, but there is no clear evidence of secular trends in the density of any of them.

image

Figure 1. Time-series of population densities of major Broadstone species, showing the monthly sample means with 95% confidence intervals. The markers on the ordinate indicate January of each year.

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As an approximate check for long-term changes that takes seasonality into account, we calculated the deviation between the abundance obtained from each Surber sample and the long-term mean (over all years) for the month in which the sample was taken and regressed the result against time (the number of months elapsed since the first sample was taken). No significant trend was apparent for either prey species at the P < 0·05 level. While there was a significant decrease in the abundance of P. conspersa, and a significant increase in S. fuliginosa, the magnitude of both trends was very small (implying changes only of the order of 1% per year). We therefore remain confident that the yearly averaged system is essentially equilibrial on a medium-term time-scale. Over the entire dataset all months are adequately represented, so the benthic density over the year is reasonably characterized by the arithmetic mean and standard error of the densities of all the samples (Table 1).

Table 1.  Basic demographic information available for the four main species. Yearly means are given for each category together with the standard errors in parentheses
Species Mean benthic density (m–2)Adult emergence rate (m–2 year–1)Egg production (m–2 year–1)
L. nigra1825 (85)419 (70)11653 (2680)
N. pictetii429 (27)330 (50)8780 (2019)
P. conspersa119 (6)123 (59)
S. fuliginosa44 (2)1·41 (0·2)232 (70)

Adult emergence

The numbers of eclosing adult stoneflies were estimated during 1996 using 15 pyramidal emergence traps with basal area of 0·15 m2 which were placed randomly in the stream over a 200-m stretch between 7 March to 1 November – thus encompassing the entire the flight period of the adults (Petersen et al. 1999). Emerging adults crawled up the sides of the trap, were caught in a collecting head containing cotton wool soaked in 70% isomethylated spirits and glycerol, and counted approximately weekly. Although an individual P. conspersa was also caught using this method, we were unable to estimate emergence rates for this species.

For S. fuliginosa emerging final instar larvae were sampled using pitfall traps of length 0·5 m that were embedded parallel to the stream at random over 200 m of the bank. In 1995 20 traps were thus set, while in 1996 the number was increased to 50. The traps were checked every 4 days over the pupation period from March to May. The yearly rate at which individuals emerge from the stream can therefore be estimated by the mean total individuals per trap over each year divided by the area of stream from which the trap collects (i.e. the length of the trap multiplied by half the stream width). The yearly average rates of emergence for all the species except P. conspersa are summarized in Table 1.

Egg production

For the three species that lay egg masses (N. pictetii, P. conspersa and S. fuliginosa) the egg production rate was estimated from weekly counts of the egg masses found over a 200-m stretch of the stream. Both the predators, P. conspersa and S. fuliginosa, lay conspicuous clumps of eggs and it was possible on each occasion to record all newly appeared egg masses. These counts were converted to numbers of individual eggs using empirically derived relationships between the dimensions of the egg masses and egg number per mass. Table 1 shows the resulting values for annual average egg production per unit area of stream.

For N. pictetii, new egg masses were counted from a series of 20 1-m2 quadrats placed every 10 m along the stretch. In this case only the mean number of eggs per mass was known, so the specific egg production rate shown in Table 1 was inferred by multiplying the mean rate of egg mass production by the mean egg content per mass. For L. nigra we know from dissections of newly emerged adults that the mean number of eggs per female is 398 (J.H. Winterbottom, unpublished data). We assume that the proportion of eclosing females who produce an egg complement is the same as the proportion of females of the other stonefly species (N. pictetii) who lay an egg mass (0·12). Combining this with a breakdown by sex of adult stoneflies emerging from pyramidal emergence traps (Petersen et al. 1999) which gives an emergence rate of female N. pictetii of 226 m–2 year–1, we can infer the annual average egg production rate shown in Table 1.

Prey uptake rates

To evaluate the predator feeding rates we reanalysed the counts of number of prey, classified by species, in the guts of a random subset of the predator individuals collected in bimonthly Surber samples by Hildrew & Townsend (1976, 1982) and Townsend & Hildrew (1977, 1979a,b). For S. fuliginosa we had additional samples, collected using the same protocol, from three months in 1995.

The number of prey items in a predator’s gut at time t, n(t), is determined by the balance between prey uptake rate, φ(t) and digestion, which we characterize by a residence time τ. It must thus change at a rate

  • image(eqn 1)

If φ changes from one value to another, the gut contents come into equilibrium with the new uptake rate over a time comparable with the gut residence time. We expect average prey abundance in the environment, and hence the medium-term average uptake rate, φ, to change on a time-scale of weeks or months. Since this is inevitably much longer than the time-scale for digestion, we can safely relate φ to the observed gut contents by

  • image(eqn 2)

We calculate the gut residence time, τ, from the observations of Townsend & Hildrew (1977) who estimated the mean values for stonefly and chironomid prey as 9·8 h and 7·3 h, respectively, at 13 °C, with a Q10 of 2·3.

We now proceed by calculating the residence time values for each month in which samples are taken using the mean stream temperature for that month given in Hildrew & Townsend (1982). We then use eqn 2 to infer the prey uptake rates for each individual predator sampled, and hence calculate the average uptake rate for that month for each predator and prey combination. The resulting monthly values are averaged over the year to produce the yearly average values shown in Table 2.

Table 2.  Estimated yearly per capita uptake rates of each of the two prey species by each of the two predator species. Standard errors of the estimates are given in parenthesis. The combined values are obtained by summing the prey uptake of each predator species and then forming an average uptake rate weighted by the abundance of the two predators
PredatorPreyUptake rate (year–1)Prey weight (µg)
P. conspersaL. nigra18·4 (3·8)184 (24)
N. pictetii12·4 (2·6)175 (44)
Chironomidae113·6 (49·4)41 (6)
S. fuliginosaL. nigra12·7 (6·0)194 (38)
N. pictetii4·5 (2·7)313 (36)
Chironomidae109·7 (32·7)230 (44)
CombinedStoneflies27·1 (4·7)192 (19)
Chironomidae112·5 (5·8)92 (13)

General modelling approach

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

Given the relatively limited information at our disposal, we seek the most parsimonious characterization of the principal elements of system. We begin by observing three key features. First, adult dispersal is typically of the order of a few tens of metres (Petersen et al. 1999) and there are no nearby streams, so demographic exchange between the stream and its environment is limited to wash-out of larval stages from the lower reaches. Secondly, although predator and prey larvae are patchily distributed (Hildrew & Townsend 1982), they are relatively long-lived and highly mobile (Townsend & Hildrew 1976; Winterbottom et al. 1997). Thus, although fine-grained heterogeneity is undoubtedly important in determining individual behaviour, population interactions can reasonably be regarded as spatially homogeneous over the approximately yearly time-scale of individual development. Thirdly, most developmental stages are present all the year round and adult emergence is not strongly synchronized; for example, adult N. pictetii are found continuously between March and October.

We thus feel confident that the yearly demographic balance of the system can be captured by a continuous-time closed system model with continuous breeding, and we choose a model framework based on coupled differential equations of Lotka–Volterra type. Since the benthic densities of the aquatic immature stages of all the species are the main census units we chose these as our state variables – an approach which has the additional benefit of not requiring the respiration rate information which is central to biomass-based modelling.

Since we have chosen a modelling framework which cannot properly represent within-year variations in abundance, and there is no evidence of multiyear cycles in the time-series of benthic density, we shall regard our models as being falsified if they predict either extinction of any of the component species or unstable equilibria. Since we have no evidence that allows unequivocal identification of the control mechanism which acts to stabilize the system, we must make an arbitrary choice from among the reasonable alternatives. We thus assume that the prey are resource limited and that this is expressed by their growing logistically in the absence of predation. Both the prey species are leaf-shredding detritivores, and experiments involving increasing leaf litter have resulted in increased detritivore densities in other streams, although such attempts have failed in Broadstone because of its already high retentiveness (Dobson & Hildrew 1992; Dobson et al. 1992).

In the absence of any evidence to the contrary, we further assume complete resource partitioning so that there is no direct competition between the prey species. Although the similar feeding habits combined with our assumption of resource limitation might imply competition, we note that partitioning of a resource as varied and complex as leaf litter still remains a distinct possibility. Furthermore, without empirical detection and quantification of competition, including it into the model would mean adding (at least) two free parameters, which we resist on the grounds of parsimony. Our assumptions about resource-based limitation and the lack of competition therefore remain plausible, but untested.

Finally, we characterize food uptake by the predators by species-specific functional responses. Although the functional response of P. conspersa has been studied both from laboratory experiments (Hildrew & Townsend 1977) and from field observations (Townsend & Hildrew 1977; Hildrew & Townsend 1982), we cannot use the results from these studies directly. The laboratory measurements were obtained over periods of c. 3 h and are thus likely to over-estimate long-term uptake rates. They do, however, suggest a saturating form of response. Equally, the relationships between the per capita biomass of prey dissected from P. conspersa guts and the local benthic prey biomass are problematic because no systematic pattern is apparent – possibly because feeding and digestion occur on approximately the same time-scale as prey movement.

Since chironomids are present in significant numbers in predator gut contents, these must be introduced into the model predator uptake. We do not, however, include them as a dynamic element of the model, but rather simply treat chironomid consumption as an additional constant per capita uptake term obtainable from the data (Table 2). Lastly, we assume that consumed food is converted into new predators with a constant conversion efficiency, and that predators have a density-independent death rate.

Within this formulation we use the relatively limited demographic information available to us to find self-consistent parameter values. Since we do not have enough information to extract all the parameters directly, our general strategy is to assume that the field system is at equilibrium so that the rates of change of all the populations are zero. We can then substitute the known benthic densities for the equilibrium values and solve for the unknown parameters as described in more detail below. We then have two requirements which must be satisfied for the model to be consistent with the data. First, we require that the predicted prey carrying capacities shall be positive. Secondly, we need the parameterization to result in an equilibrium that is an asymptotically stable attractor.

Interaction between trophic groups

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

For simplicity, we begin by considering the interaction between two trophic groups: a predator group composed of P. conspersa and S. fuliginosa, and a prey group comprising the two stoneflies, L. nigra and N. pictetii. We denote the total stonefly and predator densities by S and P, respectively. Following the reasoning outlined in the previous section, we described their rates of change by

  • image(eqn 3)
  • image(eqn 4)

where r is the intrinsic growth rate of the prey, K is their carrying capacity, α is the predator search rate, Uc is the (per predator) uptake rate of chironomids, hs and hc are the time a predator spends handling an individual stonefly and an individual chironomid, respectively, εs and εc are the prey to predator conversion rates for stoneflies and chironomids, respectively, and µ is the per capita death rate of the predators. We take the units of S and P as m–2, and measure time in years.

Parameterization

In order to use the data in Tables 1 and 2 to parameterize the model, we need some further assumptions about the demographic processes that underly the prey intrinsic growth rate (r) and carrying capacity (K). We shall assume that the per capita prey fecundity has a density-independent value, β, and that their per capita mortality rate, δ, is linearly density-dependent, so that δ = δ0 + δ1S. The intrinsic growth rate and the carrying capacity are now related to these new parameters by

  • image(eqn 5)

Next, we assume that the system is in equilibrium, so we can identify the average benthic density of prey and predators obtained from Table 1 with the values (S*,P*) which render [S\dot] = [P\dot] = 0 in eqns 3 and 4. We then equate the average yearly recruitment of stoneflies and predators (denoted by Rs and Rp, respectively) to the yearly egg production values given in Table 1. Identifying the square bracketed term in eqn4 as Rp, and remembering that the system is in equilibrium, then allows us to relate the value of µ to Rp and P*. If we further use eqn 5 to relate r to Rs, S* and the prey background mortality δ0, then we know that

  • image(eqn 6)

We next make use of the data on the (per predator) yearly uptake rate of stoneflies and chironomids given in Table 2. At equilibrium, the round bracketed term in eqns 3 and 4 must be equal to the yearly per capita uptake of stoneflies, Us. Hence, equating the square bracketed term in eqn 4 to Rp tells us that

  • image(eqn 7)

Finally, equating the round bracketed term in equation 3 to Us, shows that

  • image(eqn 8)

The demographic data given in Tables 1 and 2 provide values for all the terms on the right-hand sides of eqns 68 except the handling times hs and hc, the prey-to-predator conversion ratio (εcs) and the background death-rate δ0. We now develop estimates for these quantities from information in the literature, and then summarize our final model parameterization in Table 3.

Table 3.  Parameter values for the two trophic group model
ParameterValueUnitsParameterValueUnits
r8·7year–1hs0·0017years
K2973m–2hc0·0013years
µ2·2year–1εs0·027
α0·0127m2 year–1εc0·013
Uc112·5year–1   

The observations of Townsend & Hildrew (1977) imply typical digestion times for stonefly and chironomid prey as 15 h (0·0017 years) and 11 h (0·0013 years), respectively. We take these as our estimates of hs and hc. To estimate (εcs) we assume that the conversion of prey into new predators occurs with approximately constant efficiency for a unit of biomass. Thus, (εcs) is the ratio of the average weight of a captured chironomid to the average weight of a captured stonefly, which we can estimate from the gut contents analyses given in Townsend & Hildrew (1977). To estimate the background mortality rate, δ0 we note that stoneflies pass from egg to adult in about a year and adults are relatively short-lived. Thus, if we assume that at low densities in the absence of predation all individuals would reach maturity, the average lifetime in these conditions would be approximately a year. Although this picture clearly implies a step-function lifetime distribution, the nearest approximation in an unstructured model is an exponential with the same mean value, which we achieve by setting δ0 = 1 year–1.

Stability

Local stability analysis of the model defined by eqns 3 and 4 (e.g. Gurney & Nisbet 1998) shows that the requirement for the system to have a stably attracting equilibrium is:

  • image(eqn 9)

In Fig. 2, we show the region of the (S*/K) –θ plane in which the model predicts stable coexistence between predator and prey and is thus not falsified. The solid circle in this plot shows the position defined by the parameter set given in Table 3, which we see lies well inside the region of stability.

image

Figure 2. Stability diagrams for the model specified by eqns 3 and 4. The quantity θ is defined in eqn 9. The filled circle indicates the position implied by the best estimate parameters given in Table 3.

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We also see from Fig. 2 that reducing the stonefly handing time, hs, to zero, which is equivalent to reducing the parameter θ to zero, would have virtually no effect on the dynamic properties of the system. This implies that, despite the prey being at over 70% of carrying capacity, the average predator uptake is so far below its potential maximum rate that the functional response is perfectly adequately described by a straight line. In the rest of this paper, we shall therefore simplify our parameterization by assuming that the predator functional response is linear.

Four-species model

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

In this section we extend our model to include explicit representations of both stonefly species, L. nigra and N. pictetii, whose benthic densities we denote by L and N, respectively, and both the predator species, P. conspersa and S. fuliginosa, whose benthic densities we denote by P and S, respectively. Following the general rationale of our previous model, but using our analysis of it to justify an assumption of linear functional responses for both the predators, we write

  • image(eqn 10)
  • image(eqn 11)
  • image(eqn 12)
  • image(eqn 13)

where the rs and Ks are the intrinsic rates of increase and the carrying capacities of the two stonefly species, µ and Uc are the mortalities and chironomid uptake rates of the two predator species, and the αs and εs represent the attack and conversion rates for each predator/prey combination.

Parameterization

We proceed in a manner directly analogous to that employed in the previous section. We use i = l,n and j = p,s to denote the prey and predator species, i* = L*,N* and j*=P*,S* to represent equilibrium abundances, Ri and Rj to represent average recruitment rates, Uij to represent the per capita uptake rate of prey i by predator j, and δi to represent the background mortality rate of prey i. The equilibrium conditions now show that

  • image(eqn 14)

By defining ωnjεnjlj and ωcjεcjlj, we can show that predator j’s conversion rate from L. nigra is

  • image(eqn 15)

Finally, we can show that the carrying capacity for prey i is

  • image(eqn 16)

As before, all the terms on the right-hand side of eqns 1416 are determined by the field observations except the conversion ratios (ωnp, ωns, ωcp and ωcs) and the background mortality rates (δl and δn). To determine the conversion ratios we argue that they must be approximately equal to the corresponding ratios of prey weights from the gut contents analyses (Table 2). Finally, we assume that both background mortality rates have the value used in the trophic interaction section (δl = δn = 1 year–1), and hence arrive at the overall parameterization given in Table 4.

Table 4.  Parameter values used for the four-species model
ParameterValueUnitsParameterValueUnits
rl5·3year–1rn19·5year–1
Kl2533m–2Kn537m–2
µp1·03year–1µs5·22year–1
εlp0·0186εls0·0352
εnp0·0177εns0·0571
εcp0·0043εcs0·0417
Ucp113·6year–1Ucs109·7year–1
αlp0·0101m2 year–1αls0·0070m2 year–1
αnp0·0289m2 year–1αns0·0105m2 year–1

Stability

Comparing Tables 1 and 4, shows that the long-term average benthic abundances of L. nigra and N. pictetii are at 72% and 80% of their respective carrying capacities. This confirms our earlier conclusion from the simplified trophic interaction model, and leads us to expect that the extended model will have similar local stability properties to those of its strategic cousin.

To confirm this, we note that in the four-species model, small perturbations from a steady state (L*,N*,P*,S*) will decay with time provided that all the eigenvalues of the Jacobian matrix

  • image(eqn 17)

have negative real parts. Although analytical results are possible in this case, they are too cumbersome to be helpful and we resort instead to numerical evaluation. With the parameters given in Table 4 and the average abundances (i.e. steady-state values) given in Table 1, the eigenvalue (λm) with the maximum real part has &Âgr;{λm} = – 4·34 × 10–4, thus proving the local stability of the steady state.

Sensitivity

In order to assess the robustness of our conclusions, we have assessed their sensitivity to the experimental uncertainty in the data (Table 1) from which our best estimate parameter set (Table 4) is derived. To do this, we generated 10 000 sets of randomly perturbed data with each quantity drawn from a normal distribution with mean equal to the best estimate value and standard deviation equal to the relevant standard error given in Table 1. In the absence of uncertainty information for the background mortalities, we arbitrarily assumed that these were subject to a 20% standard error.

The vast majority of data sets (9937) resulted in positive carrying capacities, with 57 instances of Kl < 0 and six of Kn < 0. We conclude that over essentially the whole range of experimental uncertainty, the model can be fitted to the data using biologically feasible parameter values.

For the 9937 feasible parameter combinations, we re-evaluated the local stability of the coexistence steady state by computing the eigenvalues of the Jacobian matrix (eqn 17). The results are displayed in Fig. 3, in the form of a plot of the probability density distribution for &Âgr;{λm}. This shows that although the best estimate data values result in a parameter set implying stable coexistence of all four species, only 74% of the randomly perturbed data-sets imply the same conclusion.

image

Figure 3. Distribution of maximum real eigenvalues of the four-species system.

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In seeking to understand the source of this possible instability we note that our assumption of linear functional responses in both predator species precludes the prey-escape cycles, which were the only potential source of instability in the trophic interaction model. Thus, in the four-species model destabilization must be associated with exploitation competition between the two predators. To investigate this possibility we computed the local stability of the steady states in which one or other predator is absent from the system. We find these states to be local attractors over the entire range of our random parameter sets. We thus speculate that when the steady state with both predator species present becomes unstable, the system will tend asymptotically to one one-predator equilibrium or the other dependent upon the initial condition – a pattern that is confirmed in explicit numerical realizations.

We thus conclude that over about 26% of the possible range of data, the model fails to predict the stable coexistence of predators and prey that is observed in the field and is therefore falsified. However, over the great majority of the data range the model predicts stable coexistence. For these cases it is instructive to examine the relation between the assumed benthic prey abundance and the inferred carrying capacity. We show the probability density for the ratio of these quantities in Fig. 4.

image

Figure 4. Distribution of the ratio of the prey equilibria to their carrying capacities.

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For L. nigra the median equilibrium prey density is over 72% of carrying capacity, with about 75% of the parameter sets implying that predation depresses the prey by less than 40% of carrying capacity. For N. pictetii the distribution of results is tighter, with a median equilibrium density of almost 80% of carrying capacity, and with about 75% of parameter sets resulting in less than 25% depression. Thus, although there are clear-cut differences in the impact of predators between the two prey species, the broad conclusion is that over the majority of the plausible range of input data, the fitted parameters place both prey relatively close to their carrying capacity. This confirms that the conclusions drawn from our simplified trophic interaction model are robust both against extension to include all four of the main players in the system and against the relatively large uncertainties in the input data.

A Sialis manipulation experiment

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

In this section we use our four-species model to predict the effects of an ongoing large-scale experimental manipulation of S. fuliginosa recruitment in the field. Since S. fuliginosa lays its eggs in prominent clumps on the underside of leaves, it is possible to remove these from a large stretch of the stream and translocate them to a similar downstream stretch. Because of the small size of Broadstone it is also possible to halt recruitment from the entire length upstream of the experimental manipulation site in order to minimize drift-based recruitment. Thus, the experiment involves the halting of recruitment in one stretch, and the increase of recruitment in another.

To examine the effects of halting recruitment of S. fuliginosa and hence eventually removing it from the system, we examine the properties of the S. fuliginosa-free stationary state, which we denote by (L^, N^, P^, 0). Routine algebra eventually enables us to show that

  • image(eqn 18)
  • image(eqn 19)

where ψ≡(rnαlpKl)/(rlαnpKn) and ξ≡ (εcpUcp)/(εnpαnp). The stability of this steady state is determined by the eigenvalues of the Jacobian given by the first three rows and columns of eqn 17. We have evaluated these for our (7398) stable random parameter sets and find that the S. fuliginosa-free steady state is invariably locally stable – as we would expect since instability in the four-species model can only result from exploitation competition between the two predators.

Further algebraic labour now lets us prove that the S. fuliginosa-free steady state (L^, N^, P^, 0) is related to the coexistence steady state (L*, N*,P*,S*) by

  • image(eqn 20)

and

  • image(eqn 21)

Equation 20 tells us that removing S. fuliginosa from the system will cause the abundance of the two prey species to change in opposite senses. Equation 21 tells us that removal of S. fuliginosa will always cause the abundance of P. conspersa to increase. Indeed, if the search rates of the two predator species are roughly equal (as in our best estimate parameter set) the long-term increase in P. conspersa abundance is approximately equal to the (steady state) abundance of S. fuliginosa prior to its removal.

To illustrate the likely size of the changes in prey and predator abundance, Fig. 5 shows the probability distribution of the fractional changes produced by removal of S. fuliginosa, calculated over our 7398 stable parameter sets. As we expect, these plots show that the abundance of the remaining predator, P. conspersa, always increases, with the most probable size of the change being about 20% of its abundance prior to removal. Again as expected, the changes in prey abundance have opposite signs, with the most probable outcome being an increase in the abundance of L. nigra and a corresponding decrease in N. pictetii. However, the most striking feature of the changes in prey abundance is their small size, with the most probable outcome being a change in prey abundance of less than 2% of abundance prior to removal.

image

Figure 5. Probability distributions of the fractional change in each of the three remaining populations after the removal of S. fuliginosa: (a) L. nigra; (b) N. pictetii; (c) P. conspersa.

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Although the foregoing analysis makes robust predictions about the long-term effects of preventing the recruitment of S. fuliginosa, it gives no indication of the time-scale over which the new equilibria are established or the nature of the intervening transient. To elucidate these matters, Fig. 6 shows trajectories calculated numerically from eqns 10 to 13 using the best-estimate parameter set. The solid lines show the trajectories predicted for a stretch in which recruitment of S.fuliginosa is halted at the end of year 1. The dotted lines show trajectories for a stretch to which the removed S. fuliginosa eggs are added.

image

Figure 6. Predicted population density trajectories for each of the four main species for an experiment involving the translocation of all S. fuliginosa recruitment from one segment of the stream to another. The solid trajectories represent population densities in the region from which S. fuliginosa eggs have been removed, while the dashed lines represent the region to which the removed eggs have been added. The horizontal broken line represents the P.conspersa-only equilibrium. The manipulation of recruitment begins at t = 1 year.

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The results for the stretch from which S. fuliginosa are removed, show that the establishment of the new equilibrium takes over 10 years, and that the transient leading to that equilibrium produces quite large short-term disturbances in abundance. The reason for the slow establishment of the new equilibrium is the slow build-up of P. conspersa to its new equilibrium value. This implies that following the removal of S. fuliginosa both prey species show a rapid increase in abundance. This fuels a steady increase in P. conspersa, which eventually cuts the prey back to roughly its original abundance. However, since P. conspersa takes N. pictetii at a higher per capita rate than S. fuliginosa, the eventual equilibrium abundance of N. pictetii is lower than before S. fuliginosa was removed.

Figure 6 confirms that the only experimentally detectable long-term effect of removing S. fuliginosa will be an increase of some 20% in the abundance of P. conspersa. However, it also indicates that in an experiment lasting only 3 years, the likely increase in P. conspersa abundance would be little more than 5% and is thus close to the likely limit of observability. Of the two transient prey-release effects, the 8% increase in L. nigra may perhaps be observable, while the 3% increase in N. pictetii will not.

The most clearly observable effect of the removal/addition experiment is shown by the predicted trajectories for the stretch to which the removed S. fuliginosa eggs are added. Here the long-term effect should be negligible, because the rate of egg removal from the companion stretch (and hence the rate of addition to the ‘addition’ stretch) rapidly becomes small as S. fuliginosa adults are eliminated from the ‘removal’ stretch. However, although the predicted transient behaviour in this case shows effects on the other players of very comparable size to those in the ‘removal’ stretch, the transient effect on the abundance of S. fuliginosa itself is both observably large and very long-lived.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

Although Broadstone stream is one of the most extensively documented small lotic systems in the world, the available data are barely sufficient to parameterize even the very simple models reported in this paper. Moreover, since all the available data have been used in parameterization, the only tests at our disposal to falsify the models are three qualitative requirements: that the parameters should take biologically feasible values; that all four explicitly represented species should coexist permanently; and that the system should not exhibit multiyear cycles in abundance. Despite this restriction, by using models which incorporate a minimalist picture of the food web (essentially who eats whom, and how often) we have been able to draw a series of robust conclusions about the dynamics of the system, and make a number of testable predictions about the outcome of an ongoing field manipulation experiment.

We began by considering a strategic prey–predator model whose extreme simplicity permitted almost complete determination of its properties by analytical means. Our analysis of this model suggested (i) that the prey are surprisingly close to their carrying capacity, and (ii) that predator uptake rates are well below their potential saturation value.

The second of these conclusions allowed us to make a significant increase in the simplicity of our detailed four-species model, and a commensurate reduction in its parameter count, by assuming that both predator species had a linear functional response. Our (mainly numerical) analysis of this model confirmed that the long-term average abundance of both prey species exceeds 70% of their respective carrying capacities, thus indicating that prey abundance in this system is determined by a complex of factors including implicit or explicit density dependence in the prey dynamics as well as predation loads.

This model also highlights some important differences between the two prey species. Although N. pictetii has a much higher intrinsic rate of increase than L. nigra, it is considerably less abundant because its carrying capacity is much lower. However, because of its high rate of increase, the impact of predation produces less depression of N. pictetii abundance below carrying capacity than is the case for L. nigra.

We then examined the effects of an ongoing manipulation experiment in the Broadstone system, in which recruitment of S. fuliginosa is prevented in one stretch and enhanced in another. We predict that, although such a manipulation would produce an observable change in P. conspersa abundance in the removal stretch over a 10-year period, the effects over a 2–3 years run of experiments will be below the natural level of variability. The (very) long-term effects on the stretch to which removed eggs are added must be zero, because elimination of adults from the removal stretch will eventually drive the egg production there to zero. However, we predict a measurable short-term increase in the abundance of S. fuliginosa.

In the broader context we note that observations of high predation rates in stream ecosystems have traditionally been regarded as evidence that predation is the principal determinant of prey abundance. Our results emphasize that drawing such conclusions in the absence of full demographic data including the emergence, behaviour and reproductive efficiency of adult stages may be potentially misleading. Predation rates in Broadstone stream are large (Hildrew & Townsend 1982) but form a relatively small proportion of prey reproduction, leading to prey abundances close to carrying capacity. We conclude that the study of the adult stages of stream insects may be more rewarding than has been thought hitherto, and suggest that such studies may usefully be combined with broad-brush demographic modelling of the type reported in this paper.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References

We wish to thank Stuart Orton and the many other people who helped with the fieldwork. The work was funded by grants to W.S.C.G. and A.G.H. by the Natural Environment Research Council (UK) in the Testable Models in Aquatic Ecosystems Special Topic, and we also thank those involved with the Special Topic who contributed to numerous useful discussions.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Demographic data
  5. General modelling approach
  6. Interaction between trophic groups
  7. Four-species model
  8. A Sialis manipulation experiment
  9. Discussion
  10. Acknowledgements
  11. References
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Received 18 November 1998;revisionreceived 5 March 1999