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LETTERS
Highamplitude fluctuations and alternative
dynamical states of midges in Lake Myvatn
Anthony R. Ives
1
,A
´
rni Einarsson
2
, Vincent A. A. Jansen
3
& Arnthor Gardarsson
2
Complex dynamics are often shown by simple ecological models
1,2
and have been clearly demonstrated in laboratory
3,4
and natural
systems
5–9
. Yet many classes of theoretically possible dynamics are
still poorly documented in nature. Here we study longterm
timeseries data of a midge, Tanytarsus gracilentus (Diptera:
Chironomidae), in Lake Myvatn, Iceland. The midge undergoes
density fluctuations of almost six orders of magnitude. Rather
than regular cycles, however, these fluctuations have irregular
periods of 4–7 years, indicating complex dynamics. We fit three
consumer–resource models capable of qualitatively distinct
dynamics to the data. Of these, the bestfitting model shows
alternative dynamical states in the absence of environmental vari
ability; depending on the initial midge densities, the model shows
either fluctuations around a fixed point or highamplitude cycles.
This explains the observed complex population dynamics: high
amplitude but irregular fluctuations occur because stochastic
variability causes the dynamics to switch between domains of
attraction to the alternative states. In the model, the amplitude
of fluctuations depends strongly on minute resource subsidies into
the midge habitat. These resource subsidies may be sensitive to
humancaused changes in the hydrology of the lake, with human
impacts such as dredging leading to higheramplitude fluctua
tions. Tanytarsus gracilentus is a key component of the Myvatn
ecosystem, representing twothirds of the secondary productivity
of the lake
10
and providing vital food resources to fish and to
breeding bird populations
11,12
. Therefore the highamplitude,
irregular fluctuations in midge densities generated by alternative
dynamical states dominate much of the ecology of the lake.
Although the possibility of alternative states in ecological systems
has been recognized for several decades
13,14
, only recently have good
empirical examples been established
9,15,16
. The most familiar type of
alternative states is alternative stable states in which a system has two
(or more) stable equilibria, with the system settling to one or the
other depending on initial conditions
17
. Alternative stable states lead
to the possibility that a system may be shifted from one state to
another, less favourable, state by a sudden shock or other distur
bance, with unfortunate ecological consequences. Once trapped in
the new state, undoing the disturbance will not return the system to
its original (desirable) state, because the system will remain trapped
in the domain of attraction of its new state.
Alternative states, however, need not be stable equilibrium points;
they may instead be dynamical structures such as cycles
18–22
. Here we
investigate the possibility of alternative dynamical states, in which
one state is an equilibrium point and the other is a highamplitude
stable cycle. Data on the longterm dynamics of the midge Tanytarsus
gracilentus suggest these alternative states, because they show high
amplitude fluctuations that are not regularly periodic. In most
populations in nature and in most simple models, if highamplitude
fluctuations occur, they occur as fairly regular cycles, with the strong
ecological forces that drive the high amplitudes also entraining the
dynamics into a stable limit cycle
23
.
Tanytarsus gracilentus is the dominant herbivore/detritivore in
Myvatn, comprising roughly 75% of the secondary consumers and
66% of secondary production in this shallow, naturally eutrophic
lake in northern Iceland
10
. As larvae, T. gracilentus individuals feed
from tubes they construct in the benthic sediment, grazing on both
benthic diatoms (algae) and detritus
24
consisting largely of dead
benthic and planktonic algae, and midge frass. They have two non
overlapping generations per year, with adults forming large swarms
around the perimeter of the lake over two 1–2week mating periods,
the first in May and the second in July and early August. In genera
tions with high midge abundance, larvae are limited by food, and
adult size decreases for several generations before the population
crashes. Detailed statistical evaluation of data on population density,
body size and predator abundance suggests that fluctuations in
T. gracilentus populations are driven by consumer–resource inter
actions, with midges being the consumers and algae/detritus the
resources, as opposed to predator–prey interactions with midges
being the prey
25
.
We have collected data on the abundance of adult midges since
1977 by using window traps at two locations on the shore of the
lake
26
. We have fitted these data to a model constructed to describe
the fundamental interactions among midges, algae and detritus (Box
1). In the model, the midge population growth is dependent on
density and is limited by the availability of food. Food consists of
algae and detritus, which may differ in quality for midges. Algae have
densitydependent growth, and detritus is formed from dead algae.
In the model, midge populations are allowed to reach densities at
which all algae are consumed, at which point the midge population
crashes, with the rate of crash being moderated by the presence of
detritus, which serves as an alternative food source. A feature crucial
to the model is that if all algae are consumed, algal populations can
recover through the input of small subsidies from outside the midge–
algae–detritus system. These subsidies represent small influxes of
algae and detritus into the muddy midge habitat from hardbottom
areas where midges are few. Although we have no direct measure
ment of this input, much of the algae and detritus in the lake occurs
in areas inaccessible to midge larvae, and the hydrological mixing of
the shallow lake
27
makes influxes of small amounts of this material
into the midge habitat a certainty. We added environmental
stochasticity to the model as random variation in per capita changes
in abundances of midges, algae and detritus. Finally, we fitted the
data by using a statespace version of the model
28
to incorporate the
measurement error that we knew to be significant (Supplementary
Methods). Predictions by the fitted model about changes in log
(midge populations) from one generation to the next explain 74%
1
Department of Zoology, University of Wisconsin
–
Madison, Madison, Wisconsin 53706, USA.
2
Myvatn Research Station and Institute of Biology, University of Iceland, Sturlugata 7,
IS101 Reykjavik, Iceland.
3
School of Biological Sciences, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK.
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of the variance in generationtogeneration population fluctuations
(Fig. 1 and Table 1).
When stripped of environmental stochasticity, the deterministic
skeleton of the model shows alternative dynamical states (Fig. 2a).
There is a relatively high stable point surrounded by a stable cycle of
very high amplitude. The existence of alternative dynamical states
pervades the biologically plausible range of parameter values (white
areas in Fig. 3a), demonstrating that they are a common feature
caused by the general structure of the model rather than phenomena
requiring unlikely parameter values. In the fully stochastic model,
produced by including the level of environmental stochasticity esti
mated in the fitted model, the population trajectory skips between
the domains of attraction of the two alternative states (Supplemen
tary Figs 1–9), for some stretches of time fluctuating around the
stable point and for other stretches showing cycles (Fig. 2b). The
amplitude of fluctuations is highly sensitive to the rate of influx of
resources into the system; as the influx of algae and detritus, c,
decreases from 10
23
to 10
29
, the amplitude increases from less than
three orders of magnitude to more than ten orders of magnitude
(Fig. 3b). This occurs because lower subsidies (lower values of c)
allow midge populations to crash to lower levels before they are saved
from extinction by the recolonization of algae. The amount of sub
sidy needed to save the population is low. The value of c in the model
fitted to Myvatn data, c 5 10
26.4
, implies that inputs are six orders of
magnitude lower than the abundance of algae at the stable equilib
rium point. A full pictorial analysis of the deterministic and stoch
astic behaviours of the model is given in Supplementary Figs 1–9.
Our midge–algae–detritus model is firmly anchored in biology
and fits the data well. The model displays alternative dynamical states
and highamplitude fluctuations over a broad range of parameter
values governing the influxes of resources into the system (Fig. 3).
This strongly suggests the existence of alternative dynamical states in
the real midge system. We obtained further statistical support for
alternative dynamical states in two ways. First, the model contains
a parameter, q, that dictates the strength of density dependence
Box 1

The midge
–
algae
–
detritus model and alternatives
We constructed a midge
–
algae
–
detritus model to give a basic
description of their interactions, attempting to have a minimum
number of parameters that must be estimated from the data. The
midge dynamics are
x(tz1)~r
1
x(t) 1z
x(t)
R(t)
{q
e
e
1
(t)
ð1Þ
where x(t) is the abundance of midges in generation t, r
1
is the intrinsic
population growth rate, larger values of q produce stronger density
dependence, and e
1
(t) is a normal random variable representing
stochastic environmental variability. The dimensionless measure of
resource abundance in generation t, R(t) 5 y(t) 1 pz(t), is composed of
algae, y(t), and detritus, z(t), with the parameter p giving the quality of
detritus for midge population growth relative to algae. Because we were
interested in dynamics rather than mean abundance, we ‘non
dimensionalized’ midge densities to produce equation (1) and used a
separate scaling parameter K when fitting the modelsothat the observed
log(adult midge density) equalled K 1 log(x(t)) (Supplementary
Methods). Furthermore, the data showed a distinct seasonal pattern in
which spring generations had mean densities 3.4 times higher than
summer densities. This might reflect either true differences in survival
and/or fecundity between generations or sampling bias due to
differences in weather conditions and hence flight activity and
catchability. Because we were interested in longterm, multi
generational dynamics, we factored out this consistent seasonal pattern
by multiplying summer midge densities by 3.4 before statistical analyses.
Algae dynamics are
ytz1ðÞ~ r
2
y(t) 1zy(t)ðÞ
{1
{
y(t)
R(t)
x(tz1)zc
e
e
2
(t)
ð2Þ
where r
2
is the algae intrinsic population growth rate and c is the influx
of algae from outside the midge habitat. Because we have no data on
algae abundance available to midges, y(t) is not observed; therefore, in
the model the mean value of y(t) need not be included, and y(t)is
dimensionless. The term [y(t)/R(t)]x(t 1 1) is the amount of resource
consumed, x(t 1 1), scaled by the proportion of that resource which is
algae, y(t)/R(t). A key feature of algae dynamics is that midge
populations can build to sufficient abundance to consume all algae.
When the term for the amount of algae consumed, [y(t)/R(t)]x(t 1 1),
is greater than the amount produced, r
2
y(t)[1 1 y(t)]
21
, we assume
that all algae come from influx, so y(t 1 1) 5 c.
The detritus dynamics are
ztz1ðÞ~ dz(t)zy(t){
pz(t)
R(t)
x(tz1)zc
e
e
3
(t)
ð3Þ
where d gives the retention rate of detritus in the midge habitat. We
assume that the influx rate of detritus equals that of algae, and that
detritus is produced in proportion to the quantity of algae in the
previous generation, y(t). As with algae, if all detritus in the midge
habitat is consumed, then z(t 1 1) 5 c. Because both algae and detritus
were not measured, we assumed for estimation purposes that the
standard deviations of e
2
(t) and e
3
(t) are equal: s
2
5 s
3
.
We compared the midge
–
algae
–
detritus model to two additional
models. The multidimensional Gompertz log
–
linear model
30
is
u
1
tz1ðÞ~b
11
u
1
(t)zb
12
u
2
(t)zb
13
u
3
(t)ze
1
(t) ð4Þ
u
2
tz1ðÞ~b
21
u
1
(t)zb
22
u
2
(t)zb
23
u
3
(t)ze
2
(t) ð5Þ
u
3
tz1ðÞ~b
31
u
1
(t)zb
32
u
2
(t)zb
33
u
3
(t)ze
3
(t) ð6Þ
where u
1
(t) 5 log x(t), u
2
(t) 5 log y(t) and u
3
(t) 5 log z(t). The
Lotka
–
Volterra model is
xtz1ðÞ~r
1
x(t) exp {dzb
12
y(t)zb
13
z(t)ze
1
(t)ðÞð7Þ
ytz1ðÞ~r
2
y(t) exp 1zb
21
x(t)zb
22
y(t)zb
23
z(t)ze
2
(t)ðÞð8Þ
ztz1ðÞ~r
3
z(t) exp 1zb
31
x(t)zb
32
y(t)zb
33
z(t)ze
3
(t)ðÞð9 Þ
In equations (7)
–
(9), three parameters can be removed to non
dimensionalize the equations without changing the observed dynamics
of midges; we therefore set b
12
5 1 and b
13
5 1 (assuming that midges
benefit from both resources) and b
21
521 (assuming that midges
reduce algae abundance). As with the midge
–
algae
–
detritus model,
for both alternative models we fitted the data with a scaling parameter
K to factor out mean midge density. Fitting of all three models was
performed with a statespace approach factoring in measurement
error; see Supplementary Methods for details.
0 1020304050
Generations
Midge abundance
10
1
10
2
10
3
10
4
10
5
10
6
Figure 1

Population dynamics of T. gracilentus in Myvatn. The solid line
gives the abundance of midges in each generation, averaged between two
traps. The dashed line gives the predicted ‘true’ (unobserved) abundances
from the model given by Box 1 equations (1)–(3) with parameter values
estimated by maximum likelihood: r
1
5 3.873, r
2
5 11.746, c 5 10
26.435
,
d 5 0.5517, P 5 0.06659, q 5 0.9026, K 5 9.613, s
1
5 0.3491 and
s
2
5 s
3
5 0.7499.
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affecting midge growth and reproduction (Box 1). As q decreases and
density dependence weakens in the model, the stable point is lost,
leaving only the highamplitude cycle. For the model fitted to the
data, the value of q is 0.903, yet the value below which only the high
amplitude cycle remains is 0.737 (Supplementary Fig. 1). We refitted
the model to the data constraining q to be small enough for only
the highamplitude cycle to occur, and the fit of the resulting model
was statistically significantly worse than the fit with q 5 0.903 (like
lihood ratio test, x
2
5 6.34, d.f. 5 1, P , 0.012; see Supplementary
Methods). This represents a conservative test because, even for values
of q low enough to rule out alternative dynamical states, the stoch
astic dynamics nevertheless show many of the same characteristics;
although the deterministic skeleton of the model does not have
alternative states, there is a residual ‘ghost’ that is still detected in
the region surrounding the formerly stable point (Supplementary
Fig. 9).
As a second line of statistical support, we fitted the data to two
additional models and compared the fits with our midge–algae–
detritus model (Box 1). We selected the additional models to have
flexibility in fitting the Myvatn midge dynamics and yet to be
incapable of producing alterative dynamical states. The first is a
threevariable Gompertz (log–linear) model. This model has nine
parameters governing the midge dynamics, in contrast with six in
the midge–algae–detritus model. Furthermore, we did not constrain
the sign of the parameters, so the interactions between the three
variables could be positive or negative. Thus, the threevariable
Gompertz model represents the most general threedimensional
log–linear model possible, yet because it is log–linear it cannot pro
duce either stable limit cycles or alternative dynamical states. Our
second additional model is a tworesource, oneconsumer Lotka–
Volterra model. Like the Gompertz model, it contains nine para
meters governing midge dynamics, and these are fitted only with
constraints to guarantee that midges are consumers of the two
resource variables. The Lotka–Volterra model can produce stable
limit cycles, although it cannot have alternative states. Our strategy
was to select additional models that are overparameterized (nine
parameters) and thus should have an advantage over the midge–
algae–detritus model yet cannot produce alternative dynamical
states.
Despite the advantages of the additional models, the midge–algae–
detritus model outperformed both of them (Table 1), giving evidence
for the plausibility of alternative dynamical states underlying midge
population dynamics. Further support for our model comes by
applying it to a shorter data set from another shallow eutrophic lake
nearby, Lake Vikingavatn (Supplementary Methods). The model fits
well, and the parameter estimates are similar to those from Myvatn,
with exceptions being explained by characteristics such as lake size.
A striking biological conclusion from the model is the sensitivity
of the amplitude of midge fluctuations to very small amounts of
resource input, c (Fig. 3); the resource input sets the lower boundary
of midge abundance and hence the severity of population crashes.
Thus, even though resource input might be six orders of magnitude
less than the abundance of resources in the lake in most years, this
vanishingly small source of resources is nevertheless critical in setting
the depth of the midge population nadir and the subsequent rate of
recovery. This sensitivity to resource subsidies might explain changes
in midge dynamics that have apparently occurred over the last dec
ades. Although Myvatn has supported a local charr (salmonid) fish
ery for centuries
29
, this fishery collapsed in the 1980s, coincident with
particularly severe midge population crashes
11
. Over the same period,
waterbird reproduction in Myvatn was also greatly reduced during
the crash years
12
. These changes might have been caused by dredging
in one of the two basins in the lake that started in 1967 to extract
diatomite from the sediment. Hydrological studies
27
indicate that
dredging produces depressions that act as effective traps of organic
particles, hence reducing algae and detritus inputs to the midge
habitat. Our model predicts that even a slight reduction in subsidies
can markedly increase the magnitude of midge fluctuations. Such
slight environmental changes can then have seriously negative con
sequences for fish and bird populations.
Midges are central to the functioning of Myvatn, not only provid
ing food for fish and birds but also representing most of the second
ary production in the lake. Our analyses show that the marked,
complex midge population dynamics can be explained by alternative
Table 1

Goodnessoffit measures for the midge
–
algae
–
detritus and alternative models
Goodness of fit Model Description
Midge
–
algae
–
detritus Gompertz Lotka
–
Volterra
Number of parameters* 69 9Parameters included in the model deterministic skeleton
–
2 LL 156.2 174.7 185.5 22 3 log likelihood function
Total R
2
0.98 0.98 0.97 1 2 var E(t)/var X(t){
Prediction R
2
for
^
XX(t 1 1)
0.74 0.57 0.38
1 2 var
^
EE( t)/var [
^
XX(t 1 1) 2
^
XX(t)]{
Prediction R
2
for X(t 1 1) 0.53 0.39 0.25 1 2 var E(t)/var [X(t 1 1) 2 X(t)]
See Supplementary Methods for descriptions of measures, and Box 1 for descriptions of the models.
* Number of parameters in the model determining the dynamics. There are six additional parameters in each model for the scaling term K, process variation for midges (s
1
) and algae/detritus
(s
2
5 s
3
), and initial densities for midges, algae and detritus.
{ E(t) 5 X(t) 2
^
XX
p
(t), where
^
XX
p
(t) is the onestepahead prediction of log(midge abundance) made by models in Box 1.
{
^
EE(t) 5
^
XX(t) 2
^
XX
p
(t), where
^
XX(t) is the onestepahead prediction of log(midge abundance) after being updated by the observed value of X(t) to account for measurement error.
0 1020304050
Generations
Midge abundance
10
1
10
2
10
3
10
4
10
5
10
6
10
1
10
2
10
3
10
4
10
5
10
6
b
a
Figure 2

Simulated dynamics of the model given by Box 1 equations
(1)
–
(3) for 50 generations. a, Dynamics in the absence of environmental
stochasticity (e
1
(t) 5 e
2
(t) 5 e
3
(t) 5 0). b, Dynamics in the presence of
environmental stochasticity. In
a, two midge population trajectories starting
from different initial values are illustrated. Parameter values are equal to
those estimated from the data: r
1
5 3.873, r
2
5 11.746, c 5 10
26.435
,
d 5 0.5517, P 5 0.06659, q 5 0.9026, K 5 9.613; in
b, s
1
5 0.3491 and
s
2
5 s
3
5 0.7499.
LETTERS NATURE

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states, with one state a stable point and the other a stable cycle.
Alternative dynamical states mean that the character of the dynamics
(relatively constant versus cyclic) may change abruptly yet naturally.
Moreover, the amplitude of the cycle is highly sensitive to small
subsidies of resources into the midge habitat that rescue crashing
midge populations. From a conservation perspective, this represents
a challenge. Not only are midge dynamics inherently unpredictable,
they may also be extremely and unexpectedly vulnerable to small
disturbances to the lake.
Received 9 August; accepted 19 December 2007.
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330 (2003).
Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank K. C. Abbott, M. Duffy, K. J. Forbes, R. T. Gilman,
J. P. Harmon and members of Zoo/Ent 540, Theoretical Ecology, University of
Wisconsin
–
Madison, for comments on the manuscript. V.A.A.J. thanks
R. A. JansenSpence for the time to do this research. This work was funded in part
by National Science Foundation grants to A.R.I., and grants from the Icelandic
Research Council and the University of Iceland Research Fund to A.E. and A.G.
Author Contributions A.E. and A.G. oversaw the data collection and are
responsible for the longterm study on midge dynamics in Myvatn. A.E. and A.R.I.
conceived the midge
–
algae
–
detritus model, and A.R.I. performed statistical
analyses. V.A.A.J. and A.R.I. performed the mathematical analyses of the
midge
–
algae
–
detritus model.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. Correspondence and requests for materials should be
addressed to A.R.I. (arives@wisc.edu).
Resource input, c
0 0.5 1.0
Detritus retention, d
0 0.5 1.0
a
b
7
8
9
10
11
12
13
14
8
10
18
16
9
15
17
19
x
x
10
–9
10
–7
10
–5
10
–3
10
–9
10
–7
10
–5
10
–3
Figure 3

Dynamics of the midge
–
algae
–
detritus model depending on
resource input rate, c, and detritus retention, d.a
, Alternative dynamical
states in the deterministic skeleton (black, only a single stable point;
white, stable point and stable cycle with integer period labelled).
b, Amplitude of fluctuations in the stochastic model, with lighter shading
corresponding to higher amplitude (black, 10
3
, white, 10
10
). The crosses
mark values estimated from the data. Parameters are r
1
5 3.873, r
2
5 11.746,
c 5 10
26.435
, d 5 0.5517, P 5 0.06659, q 5 0.9026, K 5 9.613; in
b, s
1
5 0.3491 and s
2
5 s
3
5 0.7499.
NATURE

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6 March 2008 LETTERS
87
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