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The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.
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DOI: 10.1126/science.1215747
, 918 (2012);335 Science , et al.Vincent A. A. Jansen
Movement and Environmental Complexity''
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Comment on Lévy Walks Evolve
Through Interaction Between Movement
and Environmental Complexity
Vincent A. A. Jansen,
1
*Alla Mashanova,
1
Sergei Petrovskii
2
de Jager et al. (Reports, 24 June 2011, p. 1551) concluded that mussels Lévy walk. We confronted
a larger model set with these data and found that mussels do not Lévy walk: Their movement is
best described by a composite Brownian walk. This shows how model selection based on an
impoverished set of candidate models can lead to incorrect inferences.
ALévy walk is a form of movement in
which small steps are interspersed with
very long ones, in such a manner that
the step length distribution follows a power law.
Movement characterized by a Lévy walk has no
characteristic scale, and dispersal is superdiffu-
sive so that individuals can cover distance much
quicker than in standard diffusion models. de Jager
et al.(1) studied the movements of individual
mussels and concluded that mussels move
according to a Lévy walk.
The argument of (1) is based on model se-
lection, a statistical methodology that compares
a number of modelsin this case, different step
length distributionsand selects the model that
describes the data best as the most likely model to
explain the data (2). This methodology is used to
infer types of movements of animals (3) and has
led to a number of studies that claim Lévy walks
are often encountered in the movement of ani-
mals. The methodology in (1) contrasts a power-
law distribution, which is indicative of a Lévy
walk, with an exponential distribution, which
indicates a simple random walk. If one has to
choose between these alternatives, the power-law
distribution gives the best description. However,
if a wider set of alternatives is considered, this
conclusion does not follow.
Heterogeneity in individual movement be-
havior can create the impression of a power law
(46). Musselsmovement is heterogeneous as
they switch between moving very little or not at
all, and moving much farther (1,7). If mussels
switch between different modes, and in each mode
display Brownian motion, this suggests the use of
a composite Brownian walk, which describes the
movement as a sum of weighted exponential dis-
tributions. We confronted this plausible model
with the mussel movement data (8).
Visual inspection of the data shows that the
cumulative distribution of step lengths has a humped
pattern that is indicative of a sum of exponentials
(Fig. 1A). We applied a model selection pro-
cedure based on the Akaike information criterion
(AIC) (2,3). We compared six different step
length distributions: an exponential distribution,
a power law, a truncated power law, and three
hyperexponential distributions (a sum of two,
three, or four exponentials to describe composite
Brownian walks). We did this for the data trun-
cated as in (1) (Fig. 1A) as well as all the full,
untruncated data set (Fig. 1B). In both cases, we
found that the composite Brownian walk con-
sisting of the sum of three exponentials was the
best model (Fig. 1 and Table 1). This convinc-
ingly shows that the mussels described in (1)do
not do a Lévy walk. Only when we did not take
the composite Brownian walk models into
account did the truncated power law model
perform best and could we reproduce the result
in (1).
Mussel movement is best described by a
composite Brownian walk with three modes of
movement with different characteristic scales be-
tween which the mussels switch. The mean move-
ment in these modes is robust to truncation of the
data set, in contrast to the parameters of the power
law, which were sensitive to truncation [Table 1;
TECHNICAL COMMENT
1
School of Biological Sciences, Royal Holloway, University of
London,SurreyTW200EX,UK.
2
Department of Mathematics,
University of Leicester, Leicester LE1 7RH, UK.
*To whom correspondence should be addressed. E-mail:
vincent.jansen@rhul.ac.uk
Fig. 1. The step length
distribution for mussel
movement [as in (10)]
and curves depicting some
of the models. The circles
represent the inverse cu-
mulative frequency of step
lengths, The curves repre-
sent Brownian motion
(blue), a truncated power
law (red), and a composite
Brownian walk consisting
of a mixture of three ex-
ponentials (blue-green). (A)
Data as truncated in Fig.
1in(1,10) (2029 steps).
(B) The full untruncated
data set (3584 steps).
24 FEBRUARY 2012 VOL 335 SCIENCE www.sciencemag.org918-c
on February 27, 2012www.sciencemag.orgDownloaded from
also see supporting online material (SOM)]. This
analysis does not tell us what these modes are,but
we speculate that it relates to the stop-move
behavior that mussels show, even in homoge-
neous environments (1). We speculate that the
mode with the smallest average movement
(~0.4 mm) is related to nonmovement, combined
with observational error. The next mode (average
movement ~1.5 mm) is related to mussels moving
their shells but not displacing, and the mode with
the largest movements (on average 14 mm, about the
size of a small mussel) is related to actual dis-
placement. This suggests that in a homogeneous
environment, mussels are mostly stationary, and
if they move, they either wobble or move about
randomly. Indeed, if we remove movements smaller
than half the size of a small mussel (7.5 mm), the
remaining data points are best described by
Brownian motion. This shows that mussel move-
ment is not scale invariant and not superdiffusive.
de Jager et al.sanalysis(1) does show that
mussels do not perform a simple random walk
and that they intersperse relatively long displace-
ments with virtually no displacement. However,
one should not infer from that analysis that the
movement distribution therefore follows a power
law or that mussels move according to a Lévy
walk, and there is no need to suggest that mussels
must possess some form of memory to produce a
power lawlike distribution (9). Having included
the option of a composite Brownian walk, which
was discussed in (1) but not included in the set of
models tested, one finds that this describes
musselsmovement extremely well.
Our analysis illustrates why one has to be
cautious with inferring that animals move accord-
ing to a Lévy walk based on too narrow a set of
candidate models: If one has to choose between a
powerlawandBrownianmotion,oftenthepower
law is best, but this could simply reflect the
absence of a better model. To make defensible
inferences about animal movement, model selec-
tion should start with a set of carefully chosen
models based on biologically relevant alterna-
tives (2). Heterogeneous random movement often
provides such an alternative and has the addition-
al advantage that it can suggest a simple mech-
anism for the observed behavior.
References and Notes
1. M. de Jager, F. J. Weissing, P. M. J. Herman, B. A. Nolet,
J. van de Koppel, Science 332, 1551 (2011).
2. K. P. Burnham, D. R. Anderson, Model Selection and
Multimodel Inference: A Practical Information-Theoretic
Approach (Springer-Verlag, New York, ed. 2, 2002).
3. A. M. Edwards et al., Nature 449, 1044 (2007).
4. S. Benhamou, Ecology 88, 1962 (2007).
5. S. V. Petrovskii, A. Y. Morozov, Am. Nat. 173, 278
(2009).
6. S. Petrovskii, A. Mashanova, V. A. A. Jansen, Proc. Natl.
Acad. Sci. U.S.A. 108, 8704 (2011).
7. J. van de Koppel et al., Science 322, 739 (2008).
8. We found that the results published in (1) were
based on a corrupted data set and that there were
errors in the statistical analysis. [For details, see our
SOM and the correction to the de Jager paper (10).]
Here, we analyzed a corrected and untruncated data
set provided to us by M. de Jager on 20 October 2011.
This data set has 3584 data points, of which 2029
remain after truncation. Since doing our analysis, an
amended figure has been published (10), which
appears to be based on ~7000 data points
after truncation.
9. D. Grünbaum, Science 332, 1514 (2011).
10. M. de Jager, F. J. Weissing, P. M. J. Herman, B. A. Nolet,
J. van de Koppel Science 334, 1641 (2011).
Acknowledgments: We thank M. de Jager for supplying
the data to do this analysis and the authors of (1)for
their constructive comments. This work was funded by
Biotechnology and Biological Sciences Research Council
Grant BB/G007934/1 (to V.A.A.J.) and Leverhulme Trust
Grant F/00 568/X (to S.P.).
Supporting Online Material
www.sciencemag.org/cgi/content/full/335/6071/918-c/DC1
Materials and Methods
SOM Text
References
25 October 2011; accepted 13 January 2012
10.1126/science.1215747
Table 1. Model parameters and Akaike weights. The maximum likelihood parameter
estimates, log maximum likelihoods (ML), AIC values, and Akaike weights are
calculated (for details, see SOM) for the datashowninFig.1,AandB.TheAkaike
weights without the composite Brownian walks are given in brackets. We analyzed the
full data set (*) with x
min
= 0.02236 mm, and the data set truncated as in (1)()with
x
min
= 0.21095 mm. For x
max
, the longest observed step length (103.9mm) was
used. The mix of four exponentials is not the best model according to the AIC
weights. It gives a marginally, but not significantly, better fit and is overfitted.
Models Formula Parameters* ParametersML AIC Weight
Exponential
(Brownian
motion)
P(X=x)=lel(xxmin)l= 1.133 l= 0.770 3136.89*
2558.67
6275.78*
5119.37
0 (0)*
0(0)
Power law
(Lévy walk)
P(X=x)= m1
x1m
min
xmm= 1.397 m= 1.975 2290.10*
1002.32
4582.20*
2006.64
0(0)*
0 (0.006)
Truncated
power law
(Lévy walk)
P(X=x)= m1
x1m
min
x1m
max
xmm= 1.320 m= 1.960 2119.55*
997.29
4241.10*
1996.58
0(1)*
0 (0.994)
Mix of two
exponentials
(Composite
Brownian walk)
P(X=x)=X
i=1
2
pilieli(xxmin)
with X
i=1
2
pi=1
p = 0.073,
l
1
= 0.122,
l
2
= 3.238
p = 0.127,
l
1
= 0.123,
l
2
= 3.275
906.15*
1022.44
1818.31*
2050.87
0*
0
Mix of three
exponentials
(Composite
Brownian walk)
P(X=x)=X
i=1
3
pilieli(xxmin)
with X
i=1
3
pi=1
p
1
= 0.034,
p
2
= 0.099,
l
1
= 0.069,
l
2
= 0.652,
l
3
= 3.613
p
1
= 0.063,
p
2
= 0.210,
l
1
= 0.072,
l
2
= 0.832,
l
3
= 4.309
861.55*
966.70
1733.11*
1943.40
0.881*
0.873
Mix of four
exponentials
(Composite
Brownian walk)
P(X=x)=X
i=1
4
pilieli(xxmin)
with X
i=1
4
pi=1
p
1
= 0.014,
p
2
=0.034,
p
3
= 0.085,
l
1
= 0.656,
l
2
= 0.069,
l
3
= 0.652
l
4
= 3.613
p
1
= 0.017,
p
2
= 0.060,
p
3
= 0.202,
l
1
= 0.377,
l
2
= 0.070,
l
3
= 0.902,
l
4
= 4.345
861.55*
966.63
1737.11*
1947.26
0.119*
0.127
www.sciencemag.org SCIENCE VOL 335 24 FEBRUARY 2012 918-c
TECHNICAL COMMENT
on February 27, 2012www.sciencemag.orgDownloaded from
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Semiarid ecosystems are threatened by global warming due to longer dehydration times and increasing soil degradation. Mounting evidence indicates that, given the current trends, drylands are likely to expand and possibly experience catastrophic shifts from vegetated to desert states. Here, we explore a recent suggestion based on the concept of ecosystem terraformation, where a synthetic organism is used to counterbalance some of the nonlinear effects causing the presence of such tipping points. Using an explicit spatial model incorporating facilitation and considering a simplification of states found in semiarid ecosystems including vegetation, fertile and desert soil, we investigate how engineered microorganisms can shape the fate of these ecosystems. Specifically, two different, but complementary, terraformation strategies are proposed: Cooperation -based: C -terraformation; and Dispersion -based: D -terraformation. The first strategy involves the use of soil synthetic microorganisms to introduce cooperative loops (facilitation) with the vegetation. The second one involves the introduction of engineered microorganisms improving their dispersal capacity, thus facilitating the transition from desert to fertile soil. We show that small modifications enhancing cooperative loops can effectively modify the aridity level of the critical transition found at increasing soil degradation rates, also identifying a stronger protection against soil degradation by using the D -terraformation strategy. The same results are found in a mean-field model providing insights into the transitions and dynamics tied to these terraformation strategies. The potential consequences and extensions of these models are discussed.
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Ecologists have fiercely debated for many decades whether populations are self-regulated by density-dependent biological mechanisms or are controlled by exogenous environmental forces. Here, a stochastic mechanistic model is used to show that the interaction of these two forces can explain observed large fluctuations in Dungeness crab (Cancer magister) numbers. Relatively small environmental perturbations interact with realistic nonlinear (density dependent) biological mechanisms, to produce dynamics that are similar to observations. This finding has implications throughout population biology, suggesting both that the study of deterministic density-dependent models is highly problematic and that stochastic models must include biologically relevant nonlinear mechanisms.
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(1) This paper documents the changes in (a) the distribution of bracken (Pteridium aquilinum (L.) Kuhn) and Scots pine (Pinus sylvestris L.) at Lakenheath Warren using a historical series of vegetation maps produced from aerial photography, and (b) the vegetation of A. S. Watt's bracken v. Calluna (Calluna vulgaris (L.) Hull) plot. (2) The size of the bracken patch at Lakenheath Warren has increased by only 24 ha (<0.2% per year) between 1918-22 and 1984, although ebbs and flows were found around the boundary. Since 1968 however, the centre of the bracken patch has degenerated from dense bracken to a sparse bracken/grass heath community. (3) There has also been a rapid invasion of the Warren by Scots pine, and no evidence was found to suggest that dense bracken inhibited invasion at this site. (4) At the "bracken v. Calluna area" the apparent stability reported by Watt, where there was interdigitation between bracken and the four phases of Calluna, has broken down. The Calluna has now almost disappeared, and bracken now covers most of the former Calluna area. (5) The results of this study were used to test some of Watt's hypotheses on bracken dynamics, and the relationship between bracken and other species. (6) The relevance of these results for the practical management of vegetation was also discussed, namely (a) the potential interference of bracken degeneration when making reliable estimates of both bracken encroachment and long-term sustainable yield, and (b) the management required to maintain both Calluna and bracken along an invading bracken front.
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Preface Part I. Unstructured Population Models Section A. Single Species Models: 1. Exponential, logistic and Gompertz growth 2. Harvest models - bifurcations and breakpoints 3. Stochastic birth and death processes 4. Discrete-time models 5. Delay models 6. Branching processes Section B. Interacting Populations: 7. A classical predator-prey model 8. To cycle or not to cycle 9. Global bifurcations in predator-prey models 10. Chemosts models 11. Discrete-time predator-prey models 12. Competition models 13. Mutualism models Section C. Dynamics of Exploited Populations: 14. Harvest models and optimal control theory Part II. Structured Population Models Section D. Spatially-Structured Models: 15. Spatially-structured models 16. Spatial steady states: linear problems 17. Spatial steady states: nonlinear problems 18. Models of spread Section E. Age-Structured Models: 19. An overview of linear age-structured models 20. The Lokta integral equation 21. The difference equation 22. The Leslie matrix 23. The McKendrick-von Foerster PDE 24. Some simple nonlinear models Section F. Gender-Structured Models: 25. Two-sex models References Index.