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Global change alters the stability of food webs

M A R K E M M E R S O N *, M A R T I J N B E Z E M E R w 1 , M A R K D . H U N T E R z and T . H E F I N J O N E S

*Department of Zoology, University College Cork, Lee Maltings, Prospect Row, Cork, Ireland, wNetherlands Institute of Ecology

(NIOO-KNAW), PO Box 40, 6666 ZG Heteren, The Netherlands, zInstitute of Ecology, University of Georgia, Ecology Building,

Athens, GA 30602-2202, USA, Cardiff School of Biosciences, Cardiff University, PO Box 915, Cardiff, CF10 3TL, UK

Abstract

Recent research has generally shown that a small change in the number of species in afood web can have consequences both for community structure and ecosystem processes.However change is not limited to just the number of species in a community, but mightinclude an alteration to such properties as precipitation, nutrient cycling andtemperature. How such changes might affect species interactions is important, not justthrough the presence or absence of interactions, but also because the patterning ofinteraction strengths among species is intimately associated with community stability.Interaction strengths encompass such properties as feeding rates and assimilationefficiencies, and encapsulate functionally important information with regard to

ecosystem processes. Interaction strengths represent the pathways and transfer ofenergy through an ecosystem. We review the best empirical data available detailing thefrequency distribution of interaction strengths in communities. We present theunderlying (but consistent) pattern of species interactions and discuss the implicationsof this patterning. We then examine how such a basic pattern might be affected givenvarious scenarios of change and discuss the consequences for community stability andecosystem functioning.

Keywords: community, ecosystems, food webs, herbivore, persistence, plant, predators, prey, resilience,

stability

Received 9 September 2004; and accepted 15 October 2004

Introduction

Pioneering early ecologists (for example Odum, 1953;

MacArthur, 1955; Elton, 1958) held that the dynamics of

complex ecological systems should be more stable than

simpler systems. They based this view on the premise

that those ecological systems, which were more species

rich, have more interspecies pathways along which

energy can flow. These pathways can be depicted

graphically as food web diagrams and energy flow

may be characterized by the trophic interactions that

take place among species within the food web. The

biological strength of these trophic interactions details

the conversion of abundance or biomass from one

trophic level to another. In this paper, we will explore

how increased productivity brought about by rising CO2

levels and global warming might effect change in speciesinteractions at local scales. This is important because a

disruption to the strength or arrangement of interspecific

interactions in food webs can have consequences for the

stability of those same systems. For instance, in the early

1970s, Rosenzweig (1971) warned against the artificial

enrichment of ecosystems (to increase productivity) in

terms of increased nutrient and energy supply. He

showed that for two trophic-level systems, either

scenario would destroy steady states in simple models.

Throughout this paper, we describe a scenario of global

change (increased productivity) brought about by

coincident global increases in temperature and CO2concentration. We explore this scenario of change using

a simple three-species food web consisting of a basal

resource (plant), primary consumer (herbivore) and

secondary consumer (omnivorous predator).

Although the study of species interactions has tradi-

tionally been concerned with their effects on community

stability (sensu May, 1973), recent research has focused

more on how ecosystem processes such as productivity,

nutrient flux and nutrient retention are affected by the

Correspondence: Mark Emmerson, tel. 1 353 (0)21 490 4190.

fax: 1353 (0)21 427 0562, e-mail: [email protected] address: Nature Conservation and Plant Ecology Group,

Wageningen University and Research Centre, Bornsesteeg 69, 6708

PD Wageningen, The Netherlands.

Global Change Biology (2005) 11, 490501, doi: 10.1111/j.1365-2486.2005.00919.x

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loss of species (Chapin et al., 2000; McCann, 2000; Loreau

et al., 2001; Madritch & Hunter, 2002). The transfer of

energy (or carbon) through an ecosystem represents one

such ecosystem process and interaction strengths provide

a direct link between the diversity and ecosystem

functioning of a community and its ecological stability

(Duffy, 2002). Interaction strengths have been quantified

in a number of ways and this general term has been usedvariously to describe both (i) the biological flux of

material or energy from one trophic level to another and

(ii) per-capita effects of a species on the growth rate of

another. In this paper, we will focus on the latter use of

the term interaction strength.

Concerns over increased global rates of species

extinction are driven by the urgent need to understand

and predict the consequences of species loss for the

stable and reliable provision of ecosystem services

(Duarte, 2000; Engelhardt & Ritchie, 2001; Loreau

et al., 2001; Lerdau & Slobodkin, 2002). To understand

how communities might respond locally to global

change (changes in productivity driven by increasedtemperature and CO2 concentrations), requires that we

identify the basic arrangement of species interaction

strengths in communities and investigate how one

measure of global change might affect that pattern at

the local ecological scale. To achieve this, we now

review and detail measures of interaction strength, and

present what we consider to be some of the best data

currently available detailing the patterns of interaction

strengths in ecological systems; this is necessary for our

exploration of change. We then define some simple and

biologically reasonable scenarios of ecological change

and explore how the patterning of interaction strengthsmight be affected by such changes using a Lotka

Volterra modelling framework. Our aim in using Lotka

Volterra dynamics is to make qualitative forecasts

regarding the consequences of change, not to make

quantitative predictions. Our goal is to make a heuristic

exploration of these scenarios using this simple model-

ling approach. Finally, we will discuss the consequences

of change for community and ecosystem stability.

What is the pattern of interaction strengths incommunities?

The effects of species interactions on community stability

were first explored by May (1973) using linear stability

analyses. May used LotkaVolterra models of the form:

dXidt

Xi ri Xnj1

aijXj

0@

1A; 1

where Xi and ri define the population density and

intrinsic rate of increase of species i, to explore the

dynamical behaviour of model communities close to

equilibrium. The interaction between a predator and its

prey is defined by the coefficient aij and represents the

negative per-capita effect of a predator species j on a prey

species i. The negative effects of species i on itself are

denoted aii, while the positive effects of the prey species

i on the predator j are given by aji. A system of such n

linear equations describing the dynamics of a set ofspecies can be represented in matrix algebra terms so that

the interaction coefficients (aij terms) are the elements of

an n n matrix (here called A). Using matrix algebra, (1)

above can be rewritten as:

dX

dt X r AX;

where Xand rare vectors containing the densities (Xi,n)

and intrinsic rates (ri,n) of those species present in the

community or food web. In fact, May (1973) explored

the stability of these model systems at equilibrium; he

explored the dynamics of a matrix known as the

Jacobian (C) whose elements (cij terms) were theproduct of the interaction coefficients contained in A

and an n n matrix containing the vector of nontrivial

equilibrium population densities (Xi where Xi does not

equal zero) on the diagonal and zeros elsewhere (so

that cij aijXi ). Essentially, the Jacobian matrix de-

scribes the dynamics of a community close to an

equilibrium point and details population-level interac-

tions. May (1973) explored randomly constructed

model communities by assigning interaction strengths

(cij) from a uniform distribution to the elements of the

Jacobian (C). He found that in systems with randomly

assigned trophic interactions, an increased speciesrichness tended to decrease the stability of model

communities. Pimm & Lawton (1977, 1978) investigat-

ing simple model food webs subsequently showed that

the patterning of species interactions (that is the

presence or absence of omnivorous links and therefore

food web structure alone) could have dramatic effects

on the stability of such systems. Yodzis (1981) working

with a compiled set of 40 real, as opposed to model,

food webs found that the arrangement of species

interaction strengths was essential for food web

stability. When the detailed pattern of interaction

strengths in stable food webs was disrupted the

resulting webs were, on average, less stable.

These classic studies using linear stability analyses, all

made use of the Jacobian matrix for the determination of

food web stability (May, 1973; Pimm & Lawton, 1977,

1978; Yodzis, 1981). From these theoretical studies, there

are two quantities that emerge as characterizing the

interactions among species, aij the interaction coefficient

and aijXi the element of the Jacobian matrix at equili-

brium (it should be noted that the Jacobian exists

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Predictions suggest that atmospheric CO2 levels will

rise from around 350700 ppm by the end of this

century (IPCC, 2001). Associated with this increase in

CO2 levels are: (1) direct effects of atmospheric carbon

on plant photosynthesis and primary production; (2)

increases in average global temperature (global average

surface temperature has already increased by 0.6 1C in

the 20th century, IPCC, 2001), and (3) changing patternsof precipitation (Aber & Melillo, 2001). Coupled atmo-

sphereocean general circulation models provide us

with the best predictions on how patterns of tempera-

ture and precipitation will change as CO2 levels rise

(IPCC, 2001). Most terrestrial regions will experience an

increase in mean annual temperature between 3 and

10 1C whereas rates of precipitation may either decrease

or increase by up to 2 mm day1, depending on region.

To begin integrating these climatic predictions into an

understanding of species interactions, we begin our

analysis with anticipated effects upon primary produc-

tion. Then we will suggest how changes in primary

production might influence the strength of trophicinteractions. Critical to this last step is distinguishing

between changes in the rate of primary production and

changes in the quality of the plant tissue that is

produced. Whether plants react in a qualitative or

quantitative way, or even both, is likely to have very

different effects on species interactions.

Direct effects of CO2 on primary production. It is generally

accepted that photosynthetic rates are limited primarily

by carbon (Drake et al., 1997; Norby et al., 1999; Aber &

Melillo, 2001). Hence, most models predict that net

primary production will increase as atmosphericconcentrations of CO2 increase. Various models

predict changes that range from 0.7% to 1 32.4%

(VEMAP, 1995). Overall, the average predicted increase

in net primary production is around 20%. In many

plant communities, primary production will increase

below ground as well as above ground (Tate & Ross,

1997) and elevated CO2 may increase fine root

production of some forest trees by as much as 96%

(King et al., 2001). For most plant species, increased

rates of primary production are associated with

increases in relative growth rate (Saxe et al., 1998). For

example, in their study of 10 Acacia species, Atkin et al.

(1999) reported that the relative growth rates of Acacia

trees increased by an average of 10% over a 12-week

period under elevated CO2 conditions.

CO2-mediated changes in temperature and precipitation.

While the prediction of increased productivity in

response to rising concentrations of CO2 appears to be a

relatively robust generalization, the effects of concomitant

changes in temperature and precipitation are harder to

predict (Aber & Melillo, 2001). Temperature effects are

particularly difficult to predict because while photo-

synthesis has a bell-shaped response to temperature,

respiration increases exponentially (Larcher, 1995).

Carbon gain relative to loss may therefore decline at

high temperatures, resulting in a decline in production.

The responses of various ecosystems to the

interactive effects of changes in CO2, temperature andprecipitation will depend, in part, on their relative

sensitivities to those environmental factors. Simply put,

an ecosystem whose productivity is limited by

precipitation is more likely to respond to a change in

precipitation than one that is limited by temperature

(Schloss et al., 1999). Broad generalizations that operate

across many different ecosystems therefore, become less

robust as the number of climatic variables considered is

increased. Our ability to make predictions is also limited

because models that include multiple interactive effects

of global change are still relatively rare (Aber & Melillo,

2001). One noteworthy study is that by Yu et al. (2002).

This study incorporates predicted changes in CO2concentration, precipitation and temperature from

seven general circulation models into predictions of

ecosystem change in a forest transect in east China.

While all seven general circulation models predicted

increases in net primary productivity under elevated

CO2, effects of precipitation were relatively weak

whereas productivity was negatively correlated with

temperature. Broadleaf forests were predicted to

increase while conifer forests, shrubs and grasses were

predicted to decrease. Overall, however, primary

production was still seen to increase under all

scenarios of elevated CO2 (Yu et al., 2002).Despite Yu et al. (2002) and predictions made from the

bell-shaped relationship between photosynthesis and

temperature (Aber & Melillo, 2001), increases in

temperature under rising CO2 levels may still have an

overall positive impact on net primary production,

particularly if growing seasons are extended. A meta-

analysis of available studies suggests that ecosystem

warming should increase plant productivity by an

average of 19%, with a 95% confidence interval of 15

23% (Rustad et al., 2001). If true, then the effects of global

warming on productivity may operate in exactly the

same direction as the direct effects of CO2 on

productivity.

Quality vs. quantity. In a food chain context, the

predicted increases in primary production may

appear to favour herbivore populations; however,

many empirical studies suggest that plant quality for

consumers will decline as CO2 levels rise (Bezemer &

Jones, 1998). This will have implications for the strength

of per-capita trophic interactions between a consumer




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and its plant resource. Specifically, the accumulation of

carbon under elevated CO2 dilutes concentrations of

nitrogen in plant tissues by 1525% (Lincoln et al., 1993;

Lindroth et al., 1995) thereby increasing C : N ratios

(Ceulemans & Mousseau, 1994; Wilsey, 1996; Hughes &

Bazzaz, 1997). Given that nitrogen is considered a

limiting factor for many herbivore species (Mattson,

1980), dilution of nitrogen under elevated CO2 mayplace further constraints on the growth rates of

herbivores. In addition, the concentrations of carbon-

rich secondary metabolites sometimes increase under

elevated CO2 (Lindroth et al., 1995; Agrell et al., 2000).

While there appears to be considerable variation in the

responses of plant chemical defences to rising CO2levels (Hunter, 2001), some classes of tannin, such as

condensed tannins, seem to respond most frequently.

To summarize, in nearly every case examined to date,

foliar nitrogen concentrations decline under elevated

CO2 and, when present, foliar concentrations of

condensed tannins increase (Fajer et al., 1989, 1991;

Johnson & Lincoln, 1991; Lincoln et al., 1993; Lindrothet al., 1995; Jones & Hartley, 1999).

Herbivores confronted with low-quality plant tissue

would be expected to compensate by eating more. This

prediction usually holds true, particularly for chewing

insects (Lincoln et al., 1984, 1986, 1993; Fajer et al., 1989;

Lindroth et al., 1993, 1995; Salt et al., 1995; Docherty

et al., 1996; Kinney et al., 1997; Williams et al., 1997;

Whittaker, 2000). Likewise, the area damaged by leaf-

mining insects may also increase, for example, the area

of leaf mines on Quercus myrtifolia increase by over 25%

under elevated CO2, apparently because nitrogen

concentrations fall by over 11% (Salt et al., 1995;Stiling et al., 2003). Nonchewing insects have been less

well studied and in general patterns are yet to emerge.

Bezemer et al. (1998), for example, showed that peach

potato aphid (Myzus persicae) abundance was enhanced

by elevated CO2 and temperature, but in a companion

study, Bezemer et al. (1999) found that plant and aphid

species significantly influenced the response.

As well as separating quality and quantity effects, it

is also crucial that we distinguish between overall

levels of defoliation and per-capita consumption by

insects. Increases in plant productivity and nitrogen-

mediated declines in insect density can result in lower

levels of defoliation on plants despite increases in per-

capita consumption rates by herbivores (Hughes &

Bazzaz, 1997, Stiling et al., 2003). A 2-year study of

herbivore communities under open-topped CO2chambers in scrub oak forest has shown that all

herbivore species decline in density under elevated

CO2 (Stiling et al., 2003).

While there is little information on how primary

consumers (herbivores) respond, in the longer term, to

changes in temperature and atmospheric CO2concentrations, there is even less information on the

effects of climate change on secondary consumers

(parasitoids and predators). In the few studies

currently available on the direct effects of elevated

CO2 on parasitoids, CO2 did not influence Cotesia

melanoscela, a parasitoid of the gipsy moth (Lymantria

dispar), although pre-enclosure mortality of theparasitoid was slightly increased when CO2 was

elevated (Roth & Lindroth, 1995). Bezemer et al. (1998)

also showed that parasitism rates of M. persicae

remained unchanged in elevated CO2. Using

mathematical models Hassell et al. (1991) have shown

that hostparasitoid relationships are altered by

environmental change; temperature elevation, for

example, may differentially affect developmental rates

of hosts and parasitoids. Such differences can

potentially result in the breakdown of synchronization

between the two populations, which, in turn, may have

major effects on population dynamics.

Models and methods

Simulating the effects of change

We use a LotkaVolterra framework, within which to

explore the effects of change (defined here as an

increase in productivity) on interaction strength. To

investigate how increased productivity might affect

species population sizes (Xi), per-capita interaction

coefficients (aij) and, in turn, the elements of the

Jacobian matrix (aijXi), we determined the period

doubling bifurcations for a simple three species foodchain featuring omnivory in discrete time. The bifurca-

tions occur when the stability of a system changes as a

model parameter passes through a critical value.

Essentially, the bifurcations describe how the dynamics

of a species population change as a function of the

species intrinsic rate of increase. The bifurcation

diagram describes how a species population will

change from a stable equilibrium, to limit cycles of

varying amplitude and periodicity, to chaotic dynamics

as the intrinsic rate of the basal species increases. In the

simple food chain investigated here, Species 1 is basal,

Species 2 is an herbivore and Species 3 is an omnivore

feeding on Species 1 and 2 (see simple food web

detailed in Fig. 1). Species population dynamics were

determined using a discrete time version of the Lotka

Volterra equation;

X1;t1 X1;t X1;tb1 a11X1;t a12X2;t a13X3;t;

X2;t1 X2;t X2;td2 a21X1;t a22X2;t a23X3;t;

X3;t1 X3;t X3;td3 a31X1;t a32X2;t a33X3;t;

2

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where birth and death processes have slightly different

meanings for different trophic levels. b1 defines the

positive per-capita birth rate of Species 1; basal species

population growth is logistic so that b15 r1/K1, r1 being

the intrinsic rate of increase of Species 1 and K1 the

carrying capacity which is set equal to 1. For nonbasal

species, d2 and d3 define the negative per-capita death

rates of Species 2 and 3, set equal to 0.01 and 0.001,

respectively (in the absence of food these species

populations could not increase as for the basal auto-

trophs. Consequently, they have a negative intrinsic

population growth rate, which is offset by the food that

they eat). The biological justification for this decline

with increasing trophic height being that death rate is

strongly correlated with a species body size. Body sizetends to increase with trophic height and so death rate

would decline. Finally, t is time. Intraspecific terms also

have different meanings for the different species

present in the system. For the basal species a115 r1/K1,

while for nonbasal species intraspecific competition is

set equal to 0.1 (a22 and a33). Therefore, as r1 increases,

the strength of basal species intraspecific competition

increases relative to nonbasal species intraspecific

competition.

In this simple food chain, the secondary consumer

(omnivorous predator) is capable of feeding on both the

primary consumer (herbivore) and the basal resource

(plant). A gradient of productivity was established byincrementing the intrinsic rate of increase (0.01 incre-

ments), of the basal species r1 in the food chain. r1 was

incremented over the interval [0,4]. To produce the

bifurcations, at each value ofr1, the three discrete time

equations of system (2) above, were iterated for 40 000

time steps. Initial conditions for each run were set at

X151, X25 0.5, X350.01, reflecting a biologically

plausible pyramidal population structure (population

density tends to decrease with increasing trophic height

and is negatively correlated with increasing body size).

The population density over the last 500 time steps of

this series is represented in one dimension as a functionof the basal species intrinsic rate. Over this time interval,

the system of equations converged either to an equili-

brium value or onto stable attractors in the system.

We explored three different scenarios to investigate

how the persistence of this simple three species food

chain might be affected given an increase in basal

species productivity. These differing scenarios repre-

sent situations where:

(i) Interspecific and nonbasal species intraspecific

interactions (a22 and a33) are considered constant over

the range of basal species intrinsic rates. The per-capita

effects of prey on predators (aji) are related to the per-

capita effects of predators on prey (aij) by a predators

conversion efficiency so that aji5 e aij, where e is 0.1.

For the simulations presented here, the interaction

coefficient matrix Awas parameterized in the following

way:

A

rK 0:5 0:05

0:05 0:1 0:30:0005 0:03 0:1

24

35:

Fig1 Period doubling bifurcations for a three species omnivor-

ous food web featuring (a) primary producer, (b) primary

consumer and (c) omnivorous secondary consumer (for the foodweb diagram also shown filled circles indicate the position of the

species in the food web). The omnivorous secondary consumer

feeds on both the primary producer and the primary consumer.

Three scenarios are represented: (i) Interspecific and consumer

species intraspecific interactions are considered constant. (ii)

Herbivorous interactions (aij) are considered a function of basal

species intrinsic rates (r1), because as plant growth increases,

plant quality declines and plant consumers must ingest more to

compensate. (iii) Both herbivorous interactions and consumer

assimilation efficiencies are considered to be functions of the

primary producers intrinsic rate. This simulates increased plant

quality at low levels of productivity and decreased quality at

high levels of plant productivity (see accompanying text for

further details).




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(ii) As the intrinsic rate of a plant increases, plant

quality may decline. To compensate for this a herbivore

must compensate and ingest more plant material to

fulfil its nutritional requirements. Therefore, as the

basal species intrinsic rate increases so too will the

herbivores per-capita effects (aPH), where P and H refer

to plant and herbivore, respectively. To simulate this

scenario, we assumed herbivorous interactions (a12 anda13) in the simple food chain to be functions of the basal

species intrinsic rate so that:

aPH Fr1; 3

where

Fr1 l ur1

b r1: 4

Here, l and u are constants, approximately defining the

lower and upper bounds to the interaction coefficient

and were set at 0.2 and 0.8 for the primary consumer

and 0.002 and 0.008 for the secondary consumer,

respectively, b is a constant, set equal to unity. Forinteraction coefficients, this choice of l and u give a

range from 0.2 to 1 for the primary consumer, and

0.002 to 0.01 for the secondary consumer. Using this

function, the interaction coefficient asymptotes as the

intrinsic rate of the basal species increases. The

biological justification for choosing this function is that

herbivores are only capable of handling food at some

maximal rate. This is essentially a Holling Type II

functional response, but in this study, rather than

ingestion being a function of prey density, it is a

function of prey growth rate.

(iii) At low levels of productivity, plant quality

should be high, when prey intrinsic growth rates are

low, the per-capita effects of herbivores on prey are

consequently small. However, the benefits to predators

should be large, because plant quality is high. It is

difficult to represent this if it is assumed that the

ecological efficiency of the predator remains constant.

To simulate an increase in plant quality at low levels of

plant productivity we assume that the ecological

efficiency of a predator is also a function of plant

intrinsic growth rate so that:

e Fr1; 5

where

Fr1 u lr1

b r1: 6

As plant productivity increases, so the ability of a

predator to convert plant biomass into predator

biomass decreases asymptotically reflecting the fact

that plant quality declines. Here, we set l and u both at

0.3 this provides a biologically plausible range of 0.3

0.06 for the ecological efficiency used in the herbivorous

interactions of the primary and secondary consumers

(Jonsson & Ebenman, 1998).

Results

Population size

We constructed the period doubling bifurcations foreach of the scenarios ((i)(iii)) (Fig. 1). When interac-

tion coefficients are independent of basal species

intrinsic rate (Fig. 1, ac, i) the abundance of both

primary and secondary consumer is low, being largely

unresponsive to the behaviour of basal species popula-

tion dynamics. However, the simple food web is not

feasible when the intrinsic rate of the basal species is

43 (there are no positive population densities). When

herbivorous interaction coefficients are considered to be

a function of basal species intrinsic rate (Fig. 1, ac, ii),

basal species population dynamics remain largely

unaffected. Nonbasal species populations, on the other

hand, increase monotonically until the first basalspecies period doubling occurs. Subsequent increases

in basal species intrinsic rate result in a slight decline in

nonbasal species population size and an increased

variability of each species population. When both,

predator assimilation efficiency and interaction coeffi-

cient are considered functions of basal species intrinsic

rate (Fig. 1, ac, iii), the population size of nonbasal

species both declines substantially and becomes more

variable as basal species intrinsic rate increases. The

decline in average population size and coincident

increase in variability mean, that each population of

these species may well be more prone to stochasticenvironmental perturbations as productivity increases.

This is especially so when the scenarios that lead to a

suggested increase in productivity also predict in-

creased occurrence of extreme climatic events. Despite

this, both scenarios ((ii) and (iii)) result in consumer

population sizes that are still on average much larger

than for Scenario (i) where interaction coefficients and

assimilation efficiencies are independent of basal

species productivity.

Patterns of stability

The mechanisms detailed above such as increased

consumption with increased productivity, underlie

herbivorous interactions and have implications for the

stability of the simple three species food chain

examined. Stability in the present context refers to the

persistence of species in the food chain. For all three

scenarios, persistence is determined largely by basal

species population dynamics at high levels of produc-

tivity. When interaction coefficients are insensitive to




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basal species productivity the chain does not persist

when the basal species intrinsic rate exceeds 3 (r143).

When only interaction coefficients are considered to be

a function of basal species intrinsic rate, no species are

lost from the food chain up to an intrinsic rate of 3.16

(and the food chain persists). When both interaction

coefficient and assimilation efficiency are considered as

functions of r1, the chain does not persist abover15 3.11. At no point in the simulations do either

species 2 or 3 become extinct, affecting the persistence

of the entire food chain. This potentially could occur

with either the primary consumer becoming extinct,

leaving the omnivorous secondary consumer or the

omnivorous secondary consumer itself could become

extinct, leaving only the basal species and the primary

consumer. Despite the three differing scenarios, we find

that there is little qualitative difference in terms of

species persistence. However, the differing scenarios do

affect consumer population sizes differently. What,

then, are the implications for the patterning of Jacobian

elements (aijXi)?

Patterning of interaction strengths

To examine the distribution of predatory interaction

coefficients or Jacobian elements for a simple three-

species food chain is inappropriate, as the three

predatory food chain interactions detailed here do not

constitute a comprehensive distribution. Instead, we

consider what happens to the Jacobian elements in the

food chain as basal species productivity increases. Put

simply, the patterning of Jacobian elements (here a12 see

Fig. 2a) mirrors the patterning of each species popula-tion size. As basal species productivities increase, and

each species population undergoes bifurcations, so the

Jacobian elements become more variable. For Scenario

(ii), Jacobian elements become increasingly negative

(stronger) with increases in productivity. As interaction

coefficients increase as a function of basal species

intrinsic rate and assimilation efficiency remains con-

stant, each consumer population size increases. For

Scenario (iii), the Jacobian elements become weaker but

still increase in variability. This decline in the strength

of Jacobian elements is a consequence of the predatory

interaction coefficients increasing as basal species

productivities increase while the predators assimila-

tion efficiency declines, resulting in a smaller predator

population size overall. Given the scenarios we explore

here, the patterning of the Jacobian elements seems to

have little consequence for the persistence of the food

chain, which remains feasible over the gradient of basal

species productivities. At high productivities, the

simple food chain and its constituent species popula-

tions do not exist at a simple equilibrium (Fig. 1ac),

but rather exist in proximity to some form of cyclic or

chaotic attractor.

Continuous time

Until now, we have only considered the discrete time

version of the LotkaVolterra equation ((2) above) and

we have seen that the system converges to an

equilibrium point or stable attractor in phase space. If

we use the same parameter values for the interaction

coefficients and birth and death processes using the

continuous form of the LotkaVolterra equation (Eqn

(1)), then a local stability analysis of the omnivorous

food chain can be carried out. A stability analysis

determines the ability of a community to return to

equilibrium following a small perturbation that moves

a community away from the equilibrium point. Such a

stability analysis reveals that in close proximity to an

attractor the food chain is always stable (see Fig. 2b),

although if a perturbation moves a community a

large distance from the attractor (which defines the

Fig. 2 The consequences of Scenarios (i)(iii) (see text) for (a)

Patterning of Jacobian elements (c12) as a function of primary

producers intrinsic rate (r1) using LotkaVolterra dynamics in

discrete time (Eqn (2)), and (b) Stability of three species food

web measured as return time (1/Re(l1)) to equilibrium. For the

determination of return time the model parameterizations

suggested by Scenarios (i)(iii) are used to parameterize

LotkaVolterra dynamics in continuous time (see Eqn (1)).




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temperature dependent and clearly scenarios of global

warming also have implications for the strength of

trophic interactions via a quite different mechanism

than those we envisage here. The temperature depen-

dence of basal metabolic rate clearly has implications

for food web dynamics and ecosystem functioning.

Linking the study of ecosystem dynamics through the

study of interacting populations, metabolic theory andecological stoichiometry would be a fruitful area of

research.

In the present study, we chose a simple food web and

a range of parameter values that produced feasible

communities (where all species populations have

positive densities) and have explored how changes to

these parameter values affect community feasibility. At

first, this might seem trivial but feasibility is of

paramount importance. The exploration of complex

food web dynamics is problematic mainly because

finding parameter values that result in feasible com-

munities is very difficult. We explored three scenarios

that might affect a communitys coordinates in para-meter space and investigated whether predicted sce-

narios of change might lead a community along some

trajectory into an unfeasible region, where one or more

species populations would decline or become extinct. In

general for the scenarios we have explored, increased

productivity does not result in decreased community

persistence. However, we have investigated just one

possible food web structure. There are many possible

(simple) food web scenarios that remain to be explored,

for example those that include apparent competition or

intraguild predation.

Our discussion of change has been limited to theeffects of increased productivity on the strength of a

class of consumer resource interactions (plantherbi-

vore). We have not considered other forms of change

for example; changes in the size distribution of

harvested populations, and how this might affect the

strength or existence of size-based feeding relation-

ships. We have not considered the effects of increased

O3 on ultraviolet light radiation levels and how this

might affect species in food webs detrimentally. There

is a range of climatic factors that we have not

considered and importantly, these remain to be ex-

plored. We have not considered how species might

adapt or respond evolutionarily to possible changes in

the global biosphere and such scenarios remain an area

ripe for research. Here, we show that simplistic models

can provide important mechanistic insights into the

causes of empirically observed patterns in real com-

munities and the types of patterns we might expect to

see as the global biosphere changes.

Even within the limited scope of this study, we can

suggest a number of consequences of change for

communities. When change is envisaged as an

increase in productivity: (1) herbivorous interaction

coefficients (per-capita effects) will become stronger as

consumers fulfil their dietary requirements, faced with

a fast-growing poor-quality food resource. While per-

capita interactions could increase, the strength of

population-level effects (described by the Jacobian

matrix) might decline if population sizes coincidentallydecrease. Neutel et al. (2002) using complex soil food

webs have shown that the pattern of weak interactions

in omnivorous loops is vital for food web stability. A

disruption to the patterning of such interactions in

more complex food webs than the simplistic web we

explore here could have dramatic consequences for

food web and ecosystem stability. (2) Average popula-

tion size of consumers will likely decline as a

consequence of poor-quality food resources; (3) Jaco-

bian elements will become weaker as a consequence of

reduced population size; and (4) species populations

will become more variable and may take longer to

recover from environmental or anthropogenic perturbations. This last point (4), is important as extreme

stochastic environmental effects are predicted to in-

crease with predicted scenarios of global warming. The

combined effect of small population size, increased

population variability and increased occurrence of

stochastic environmental effects will likely result in an

increased chance of extinction for the resulting popula-

tions. In conclusion, empirical validation of our

simulations is necessary but on the basis of this study

communities may become inherently less stable, that is

less resilient, smaller, more variable and hence more

prone to extinction as they respond to predicted globalenvironmental change.

Acknowledgements

Special thanks to Bob Paine, Tim Wootton, Bill Fagan, DaveRaffaelli and Peter deRuiter for the provision of raw data. JohnPitchford for valuable comments. We thank all those thatparticipated in discussion during the Food webs in a changingworld meeting, Texel, the Netherlands. We also thank threeanonymous reviewers for constructive and useful comments.

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