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Ideas and Perspectives
127 Limits on ecosystem trophic complexity: insights from
ecological network analysis
Robert E. Ulanowicz, Robert D. Holt & Michael Barfield
137 Route optimisation and solving Zermelo’s navigation problem
during long distance migration in cross flows
Graeme C. Hays, Asbjørn Christensen, Sabrina Fossette,
Gail Schofield, Julian Talbot & Patrizio Mariani
Letters
144 The ecological consequences of megafaunal loss: giant tortoises
and wetland biodiversity
Cynthia A. Froyd, Emily E. D. Coffey, Willem O. van der Knaap,
Jacqueline F. N. van Leeuwen, Alan Tye & Katherine J. Willis
155 Climate change alters ecological strategies of soil bacteria
Sarah E. Evans & Matthew D. Wallenstein
165 Ecology drives intragenomic conflict over menopause
Francisco Úbeda, Hisashi Ohtsuki & Andy Gardner
175 General relationships between consumer dispersal, resource
dispersal and metacommunity diversity
Bart Haegeman & Michel Loreau
185 Partner manipulation stabilises a horizontally transmitted
mutualism
Martin Heil, Alejandro Barajas-Barron,
Domancar Orona-Tamayo, Natalie Wielsch & Ales Svatos
193 Facilitative plant interactions and climate simultaneously drive
alpine plant diversity
Lohengrin A. Cavieres, Rob W. Brooker, Bradley J. Butterfield,
Bradley J. Cook, Zaal Kikvidze, Christopher J. Lortie,
Richard Michalet, Francisco I. Pugnaire, Christian Schöb, Sa Xiao,
Fabien Anthelme, Robert G. Björk, Katharine J. M. Dickinson,
Brittany H. Cranston, Rosario Gavilán, Alba Gutiérrez-Girón,
Robert Kanka, Jean-Paul Maalouf, Alan F. Mark, Jalik Noroozi,
Rabindra Parajuli, Gareth K. Phoenix, Anya M. Reid,
Wendy M. Ridenour, Christian Rixen, Sonja Wipf, Liang Zhao,
Adrián Escudero, Benjamin F. Zaitchik, Emanuele Lingua,
Erik T. Aschehoug & Ragan M. Callaway
203 Trees to treehoppers: genetic variation in host plants
contributes to variation in the mating signals of a plant-feeding
insect
Darren Rebar & Rafael L. Rodríguez
211 Body mass evolution and diversification within horses (family
Equidae)
Lauren Shoemaker & Aaron Clauset
221 The ecology of sexual conflict: ecologically dependent parallel
evolution of male harm and female resistance in Drosophila
melanogaster
Devin Arbuthnott, Emily M. Dutton, Aneil F. Agrawal &
Howard D. Rundle
229 Simultaneous inbreeding modifies inbreeding depression in a
plant–herbivore interaction
Aino Kalske, Pia Mutikainen, Anne Muola, J. F. Scheepens,
Liisa Laukkanen, Juha-Pekka Salminen & Roosa Leimu
239 Rescaling the trophic structure of marine food webs
Nigel E. Hussey, M. Aaron MacNeil, Bailey C. McMeans,
Jill A. Olin, Sheldon F. J. Dudley, Geremy Cliff, Sabine P. Wintner,
Sean T. Fennessy & Aaron T. Fisk
251 Predicting the process of extinction in experimental
microcosms and accounting for interspecific interactions in
single-species time series
Jake M. Ferguson & José M. Ponciano
Corrigendum
260 Corrigendum to Elmendorf et al. (2012)
Sarah C. Elmendorf
Cove r Caption:
Multi species interactions: Rescaling the trophic structure of marine food webs
Photo credit: Graham Fenwick
From: Hussey et al., p. 239
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Volume 17 Number 2
February 2014
Ecology Letters
ECOLOGY
LETTERS
Volume 17 Number 2 February 2014 127–260
ISSN 1461-023X www.ecologyletters.com
Volume 17 Number 2 | February 2014
This journal is a member of and subscribes to the principles of the
Committee on Publication Ethics.
ECOLOGY
LETTERS
ECOLOGY
LETTERS
ele_17_2_oc_Layout 1 12/19/2013 9:11 AM Page 1
IDEA AND
PERSPECTIVE Route optimisation and solving Zermelo’s navigation problem
during long distance migration in cross flows
Graeme C. Hays,
1,2*†
Asbjørn
Christensen,
3
Sabrina Fossette,
2,4
Gail Schofield,
1,2
Julian Talbot
5
and Patrizio Mariani
3†
Abstract
The optimum path to follow when subjected to cross flows was first considered over 80 years ago
by the German mathematician Ernst Zermelo, in the context of a boat being displaced by ocean
currents, and has become known as the ‘Zermelo navigation problem’. However, the ability of
migrating animals to solve this problem has received limited consideration, even though wind and
ocean currents cause the lateral displacement of flyers and swimmers, respectively, particularly dur-
ing long-distance journeys of 1000s of kilometres. Here, we examine this problem by combining
long-distance, open-ocean marine turtle movements (obtained via long-term GPS tracking of sea
turtles moving 1000s of km), with a high resolution basin-wide physical ocean model to estimate
ocean currents. We provide a robust mathematical framework to demonstrate that, while turtles
eventually arrive at their target site, they do not follow the optimum (Zermelo’s) route. Even though
adult marine turtles regularly complete incredible long-distance migrations, these vertebrates primar-
ily rely on course corrections when entering neritic waters during the final stages of migration. Our
work introduces a new perspective in the analysis of wildlife tracking datasets, with different animal
groups potentially exhibiting different levels of complexity in goal attainment during migration.
Keywords
Evolution, migration, navigation, optimal route finding, telemetry.
Ecology Letters (2014) 17: 137–143
INTRODUCTION
The optimal movement strategies animals should adopt under
various circumstances is both a long-standing and topical
issue generating considerable recent interest (Alerstam &
Lindstr€
om 1964; Chapman et al. 2011). In some cases, there
may be relatively simple solutions to optimise movement, such
as migrating insects and birds timing departures to those
times when air flows are in a suitable direction (Chapman
et al. 2011). In other cases, there may be complex mathemati-
cal solutions to describe optimal movement patterns. For
example, in the case of foraging animals, intense controversy
exists over how widely Levy-like movement patterns occur in
which there is specific pattern of short movement steps inter-
spersed with much longer relocation steps (Viswanathan et al.
1999; Edwards et al. 2007; Travis 2007; Reynolds 2012). Yet,
there are other situations in which different mathematical
solutions optimise animal movement. For example, often ani-
mals might not be searching for resources but, instead, are
simply trying to optimise travel between two locations. This
situation occurs over a variety of spatial scales from a few
metres to thousands of kilometres. At a local scale, animals
may perform short daily commuting trips (from several
metres to tens of kilometres), from the place where they rest
to the place where they feed. At the other extreme, many ani-
mals, including birds, mammals, fish, reptiles and insects, per-
form very long annual migrations, which are often associated
with the availability of resources (e.g. specific conditions
required for foraging or breeding, such as food and tempera-
ture), varying spatially and seasonally across the globe. Find-
ing the optimum path to follow when subjected to cross
flows, which is termed the ‘Zermelo navigation problem’ (Zer-
melo 1931), has prompted much reflection from mathemati-
cians and theoretical physicists into calculating its various
solutions, given different environmental (e.g. wind or ocean
currents) flow patterns (Zermelo 1931; Talbot 2010; Techy
2011). Although there has been a lot of published work look-
ing at the ability of flying and swimming animals to deal with
cross flows to reach their goal (e.g. Krupczynski & Schuster
2008; Chapman et al. 2011), in the context of the Zermelo
navigation problem limited focus has been given as to
whether travelling animals are able to find the optimum solu-
tion (the ‘Zermelo solution’). There is an a priori expectation
that it may be difficult for animals migrating at sea to assess
1
Centre for Integrative Ecology, School of Life and Environmental Sciences,
Deakin University, Warrnambool, Vic, 3280, Australia
2
Department of Biosciences, Swansea University, Swansea, SA2 8PP, UK
3
Centre for Ocean Life, National Institute of Aquatic Resources, Technical
University of Denmark, Jægersborg Alle 1, 2920, Charlottenlund, Denmark
4
Environmental Research Division, SWFSC, NOAA, Pacific Grove, CA, 93950,
USA
5
Laboratoire de Physique Th
eorique de la Mati
ere Condens
ee, UPMC, CNRS
UMR 7600, 4, place Jussieu, 75252 Paris Cedex 05 France
*Correspondence: E-mail: g.hays@deakin.edu.au
†These authors contributed equally to this work.
©2013 John Wiley & Sons Ltd/CNRS
Ecology Letters, (2014) 17: 137–143 doi: 10.1111/ele.12219
current flows due to the general absence of fixed reference
points (Chapman et al. 2011) and hence achieving the Zer-
melo solution may be very challenging. Here, for a paradig-
matic group of long distance migrators, marine turtles, we
provide the first test of whether the Zermelo solution is
achieved or instead how ‘suboptimal’ real tracks are relative
to the time and energy optimal Zermelo solution. We GPS-
tracked marine turtles travelling many hundreds of kilometres
across the open ocean in a known current field and then cal-
culated the optimum route that solved Zermelo’s navigation
problem (‘Zermelo’s route’, Zermelo 1931). Calculating this
optimum route needs some elaborate modelling since cross
flows move in variable directions and speeds, both of which
change with time and location. So our models involve semi-
continuous adjustments in direction through the oceanic
crossing of turtles. In addition, we compared the actual
routes to simpler models that assumed individuals adjusted
their heading, so that they always oriented towards their final
goal, or maintained a heading that was goal orientated at the
start of their journey, i.e. a route that showed no compensa-
tion for current drift. In this way, we were able to estimate
the degree to which these long distance migrants use optimal
paths.
MATERIAL AND METHODS
Turtle tracking
During May 2010 and 2011, we attached Fastloc GPS-Argos
transmitters to male loggerhead turtles (Caretta caretta) at the
Greek island of Zakynthos (37°43′N, 20°52′E), which hosts
the largest breeding population of this species in the Mediter-
ranean. Turtles were captured at sea, and the transmitters
were attached using established methods (Schofield et al.
2013). The transmitters acquired GPS ephemeris during brief
surfacing periods by the turtles. This ephemeris was part-
processed on-board the transmitters, and then relayed via the
Argos satellite system to ground stations. Fastloc GPS
positions were then determined by post-processing. Fastloc
GPS positions generally have an accuracy of a few tens of
metres. Foraging sites were identified by turtles remaining in
fixed areas for extended periods (many weeks) after extended
periods of travel (see Schofield et al. 2013 for details), i.e. for-
aging sites were located at the end of migration routes and
not along the way during migration. Some turtles remained
resident in Greek waters, whereas others moved to foraging
sites in the north (along the shore up the Adriatic) or south
(North African coast) Mediterranean. Here, we consider those
individuals that travelled south across the open ocean to for-
aging sites on the North African coast. Typically, several
Fastloc GPS locations were obtained each day during migra-
tion (mean 8.4 GPS locations per day, n=8 turtles, SD =6.5)
so we were able to accurately reconstruct the routes followed
to the foraging grounds. We then performed linear interpola-
tion between successive locations to produce a position every
4 h during this migration phase, i.e., the same density of loca-
tions was then available for each turtle for all subsequent
analyses. So, there were 282–400 interpolated points per
migration track depending on the length of each track. The
distance travelled was determined by summing the distances
between these interpolated positions.
Ocean current determination and route simulation
Ocean current data were obtained from the Mediterranean
Ocean Forecasting System (Pinardi et al. 2003; Tonani et al.
2009). The current data are based on a state-of-the-art three
dimensional ocean model (Oddo et al. 2009) that appears to
be accurate in reproducing most of the surface mesoscale
dynamic in the study area (Nilsson et al. 2012). It is known
that migrating turtles travel near the ocean surface and so
deeper currents are unlikely to impact the turtle trajectories
(Hays et al. 2001). Surface daily data at 1/16
o
spatial resolu-
tion (6.9 km latitude, 5.7 km longitude) were used to integrate
the kinematic equation for an individual particle:
v¼dx
dt ¼uþwð1Þ
where uis the ocean velocity vector obtained by linear interpo-
lation of the ocean currents (both speed and direction) to the
particle position and wis the swimming velocity with constant
speed and changing heading rules. The value of the speed
(w=0.46 m s
1
) is derived from the median value calculated
in the empirical observations, when currents were subtracted
from the observed turtle tracks (see Fig. S1). It is known from
tracking studies that individual turtles generally each have fidelity
to a particular foraging site that they use across different years
(Broderick et al. 2007; Schofield et al. 2010). Hence, individuals
disperse widely from their breeding areas with each individual
travelling to a pre-determined target location. So assuming that
the turtle foraging sites were the final goal of migration, we tested
two rules for swimming direction: heading adjusted to the goal
and heading constant to the goal. In the former, the direction
was dynamically calculated as the vector connecting the position
of a given turtle to the foraging site. In the latter, the direction
was calculated once, and represented the vector connecting the
breeding ground to the final goal. Details of the numerical inte-
gration of Eqn 1 are provided in the Supplementary Material.
Algorithm for migration time optimisation
Using the realistic ocean currents described in the previous
section, we explored swimming path optimisation to determine
which path the turtles should have travelled to minimise the
travel time from the breeding ground to the foraging site.
During their oceanic crossing hard-shelled turtles are thought
not to feed, with long fasting facilitated by their low ectother-
mic metabolic rate, and to swim at a speed that minimises
their cost of transport (Southwood & Avens 2010; Hays &
Scott 2013). Hence, time spent conducting their migration to
foraging grounds will likely equate with energy expended. The
time required to travel from given starting and ending points
along a path x(t) is given by the integral:
E½x¼Z1
vðu;x;b
dx
dt ;tÞ
dx ð2Þ
where vis the effective migration velocity (including the effect
of cross-currents) and measured in the Earth Coordinate Sys-
©2013 John Wiley & Sons Ltd/CNRS
138 G. C. Hays et al. Idea and Perspective
tem, expressed as the function of the local current vector uat
position xat time t, and with d
dx=dt being the path unit tan-
gent vector. The biological meaning of the integrand is that
since dx/dt =v, then dx/vis the time required to travel dx
which is then summed over the path x(t) in the integral. In
the supplementary material, we demonstrate how to find the
minimal travel time by finding the path that minimises this
integral. To avoid becoming trapped in a local minimum, sev-
eral initial paths were used as the starting point of travel time
minimisation. A numerical algorithm was used to minimise
the travel time equation including schemes to avoid numerical
instabilities and artificial land-crossings of turtle tracks. All
technical and numerical issues are explained in the supple-
mentary material.
RESULTS AND DISCUSSION
The mathematical solution to Zermelo’s navigation problem
is, perhaps, not intuitively obvious. In a cross flow, the opti-
mum route to minimise the crossing time is not to travel in a
straight-line to the goal but rather to travel upstream of the
goal (Fig. 1a). The optimum upstream angle to travel along,
depends both on how fast a migrator can move and the
strength of the cross flow and this angle can change pro-
foundly with the distance to the target. Note that the same
considerations apply whether the medium of travel is air (fly-
ers) or water (swimmers).
If a migrator adjusted its heading to travel in a straight-line
to the destination, more time is always required compared to
the optimum trajectory. The difference in crossing time
between the straight-line and optimum trajectory increases as
the lateral current increases (Fig. 1b). These examples illus-
trate solutions to Zermelo’s navigation problem in the sim-
plest case, where the swimming speed of an animal is constant
and the pattern of the currents is simple. Here, we expand on
this simple case, considering solutions with real currents that
vary non-predictably through space and time with average
speed 0.1 m s
1
(range 0.001–0.6 m s
1
) and in which the
swimmer (turtle) has the potential to adjust its heading while
moving at a constant speed (w=0.46 m s
1
).
For eight turtles tracked to foraging areas on the coast of
North Africa (Fig. 2), the total distance travelled between the
breeding area and the foraging sites averaged 1150 km (range:
940–1340 km; n=8 turtles). The turtles generally followed
fairly straight-line routes during the section of ocean crossing
(800–1000 km) on these journeys, with the calculated straight-
ness index (the total distance travelled divided by the straight
line distance) falling in the range 0.65–0.94 (mean 0.83,
n=8). We compared the routes of tracked turtles to various
models. Assuming a constant swimming speed during migra-
tion, we examined whether turtles adjusted their heading to
solve Zermelo’s navigation problem for the optimum route;
whether the turtles initially left the breeding ground with a
heading orientated to their foraging site, and then kept this
heading constant, even if they were advected off-course; or
whether they constantly adjusted their heading so that they
were orientated to the goal. Finally, a fourth model was
tested: assuming the turtles made a single major course correc-
tion (essentially ‘turn left or turn right’) as they approached
land to take account of making landfall far from their goal.
All of the modelled scenarios produced routes that were fairly
well oriented to the goal (Fig. 3). However, often the actual
tracks of the turtles were very different to all of these mod-
elled scenarios (Fig. 3b,c,f). For example, turtles ‘b’, ‘c’ and
‘f’ all travelled far to the east of the modelled routes, before
changing direction near the end of migration, when they were
156–314 km off the modelled courses. Individual ‘a’ travelled
(a)
(b)
Figure 1 (a) Potential routes of an animal travelling between two sites
assuming linear current flow u=cy and two current gradients: low c
(0.3) (solid lines) and large c(0.7) (dashed lines). Y-axis, proportion of
the completed journey. X-axis, lateral displacement from the direct route
to the target; up-current and down-current movement is represented by
negative and positive values respectively. Blue lines in x<0 show the
solution to Zermelo’s navigation problem with increasing flow rates. The
optimal trajectory is upstream and curved. Red lines in x>0 show the
model for an animal continuously adjusting its heading to orient to the
goal. Green lines show the model using a constant heading. (b) Assuming
linear current flow, the crossing time as a function of current gradient c
for a straight line trajectory (red) vs. the optimal trajectory (blue). The
difference is smaller for weaker currents, with the optimum trajectory
being more efficient for stronger currents.
©2013 John Wiley & Sons Ltd/CNRS
Idea and Perspective Route optimisation during migration 139
north of all the modelled courses, before turning as it
approached the coast of Sicily. In this case, the track seems to
follow the optimal track at the beginning of travel, with the
Zermelo solution describing the observed track better than a
model in which the turtle kept the heading constant towards
the goal (Fig. 4 and Fig. S3a). However, in all other cases
(turtles ‘b’ to ‘h’), the optimal (Zermelo) track performs worse
at reconstructing the observed track compared to the other
models (Fig. 4 and Fig. S3b–h). In other words the observed
tracks were not as time efficient as Zermelo’s solution with
the turtles taking 4–150% (mean 57%) longer to complete
migration compared to the Zermelo route (Fig. 4b). As we
have argued previously, this extra time to complete migration
probably also equates with an increased energy expenditure
compared to the Zermelo route.
In some cases, turtles exhibited a major change in course
direction when approaching a coastline. For example, there
were distinct changes in the course of turtle ‘c’ when it was
within 5 km of the North African coast. However, in other
cases, the major change in course occurred when the turtles
were further from land. For example, turtle ‘b’ showed a
major change in course direction when it was around 60 km
from the mainland coast of North Africa, and still in waters
that were >200 m deep. When we compared the actual tracks
of turtles ‘b’, ‘c’ and ‘f’ against the model that assumed a
major course correction as individuals approached land (at
the location indicated by the star in Fig. 3b,c,f), this model
performed better than all other models (Fig. S3b,c,f), and the
calculated travel time was the most similar to the observed
migration time (Fig. 4c).
Our results show that migrating sea turtles often fail to
solve Zermelo’s navigation problem, with migration routes
tending to fall far from optimal; however, the actual routes
are nonetheless sufficient to eventually arrive at target sites.
Arrival at these small target sites was facilitated by a change
in track direction during the later stages of migration. This
observation supports experimental evidence that long distance
migrators may use a system of general ‘signposts’ to arrive at
goals, rather than perform detailed real-time continuous
course correction (Lohmann et al. 2008). The most parsimoni-
ous explanation of these findings is that turtles, in general,
only discern an approximate heading to their goal, and are
not able to constantly assess their position in relation to their
goal; consequently, they do not correct their course accord-
ingly. These conclusions fit with the general view that it is dif-
ficult for individuals to assess flow direction in the open
ocean, because of the lack of fixed reference points; hence, the
current often causes marine migrants to drift (Lohmann et al.
2008; Sale & Luschi 2009). For example, tidal action causes
migrating cod to drift when maintaining a fixed compass
heading across a tidal stream (Arnold et al. 1994). Along the
same lines it may be difficult for marine organisms like turtles
to cash in on a potential advantage by choosing an optimal
swimming route, because they are likely unable to predict
ocean currents sufficiently in open waters; however, when
ocean current variability is predictable marine organisms may
adapt behaviours to exploit the potential, e.g. vertical behav-
iour of crabs to achieve selective tidal transport (Queiroga
1998). This shallow-water situation resembles that experienced
by flying animals (e.g. bats, birds and insects), which are often
able to assess wind flow direction during flight, particularly
day-flying species travelling over solid terrain (Riley et al.
1999; Klaassen et al. 2011).
It is unlikely that other explanations, such as feeding en
route or predator avoidance, can explain the suboptimal circu-
itous migration routes taken by loggerhead turtles. First, it is
unlikely that there was extensive foraging en route, as evi-
denced by the fact that individuals did not show any periods
of localised residence in the open ocean, in contrast to other
species that are known to forage pelagically (Fossette et al.
2010). Second, predators of adult loggerhead turtles, such as
the tiger shark (Galeocerdo cuvier), are rare in the Mediterra-
nean with no known concentration of occurrence that turtles
might try to avoid.
12 oE 15 oE 18 oE 21 oE
32 oN
35 oN
38 oN
100 km
12 oE 15 oE 18 oE 21 oE
32 oN
35 oN
38 oN
100 km
0 0.1 0.2 0.3 0.4
(a) (b)
Figure 2 The routes of eight loggerhead turtles tracked in (a) 2010 and (b) 2011 from a single breeding ground (Zakynthos) in Greece to their foraging sites. The
eight turtles travelled south to sites off the North African coast (Tunisia and Libya). The tracks are super-imposed on averaged current speed and direction (shades
of green in m s
1
and blue arrows which show current vectors), which were extracted from the model for the periods during which the tracks were observed (a) 11
May –23 June 2010 and (b) 17 May –29 June 2011. Note that these are average current speeds and directions, but in the calculation of the optimal tracks we have
used time-dependent velocity vectors and not mean values.
©2013 John Wiley & Sons Ltd/CNRS
140 G. C. Hays et al. Idea and Perspective
A key conclusion from our work is that course correction
during the final stages of long distance travel seems key to
reaching the goal. Similarly, course correction during the final
stages of migration might be necessary for birds to eventually
arrive at the target site (Alerstam 1979; Klaassen et al. 2011).
Course correction may occur in a number of ways. Perhaps,
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(a)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(b)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(c)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(d)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(e)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(f)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(g)
100 km
12
o
E 15
o
E 18
o
E 21
o
E
32
o
N
35
o
N
38
o
N
(h)
100 km
Figure 3 The migratory routes of eight loggerhead turtles (black lines) tracked in 2010 (a–d) and 2011 (e–h) compared to the modelled track, assuming: (i)
heading to goal (red lines), where the heading was continuously adjusted towards the goal; (ii) single heading (green lines), where the heading was fixed
towards the goal at the beginning of migration; and (iii) optimal track (blue lines), where Zermelo’s problem was solved. When simulating the migrations,
the starting point (red square) represented the breeding ground (Zakynthos, Greece), while the final target (red triangles) represented the foraging sites.
Speed was fixed at 0.46 m s
1
, which was equal to the median of the observed values. Note that at this speed the heading to goal track simulated in h (red
line) did not reach the final target by 3 km. The points where turtles were observed to change course (red stars) are shown in panels b, c and f.
©2013 John Wiley & Sons Ltd/CNRS
Idea and Perspective Route optimisation during migration 141
the simplest is when individuals reach a mainland coast, at
which point they have a binary choice to make: ‘turn left or
right’. Examination of published turtle tracks from other stud-
ies indicates that this type of course correction upon arrival at
land is common for marine turtles, supporting the general
applicability of our results (Luschi et al. 2003). In compari-
son, it is more difficult to explain instances where turtles
change direction when in deep water far from shore; however,
such instances might potentially be explained by an accumu-
lated deviation in magnetic geo-coordinates that become suffi-
cient for a turtle to perceive that it is off-course (Lohmann
et al. 2008). Finally, it is possible that course correction
occurs during the final stages of migration (i.e. the final few
tens of kilometres) using cues emanating from the target site.
For example, green turtles crossing the Atlantic to their
nesting beaches at Ascension Island perform final course
correction when downwind of the island, indicating the use of
certain cues (e.g. olfactory) (Hays et al. 2003). Wind-borne
cues might allow land to be perceived by sea turtles, even
when far out at sea.
The level of individual variability in the straightness of
routes by migrating turtles is intriguing. We do not know why
some turtles travelled fairly directly to their goal along routes
that were little different from optimum solutions, whereas
others made wide detours. For migrating birds, it has been
demonstrated that migration performance improves with the
number of times an individual completes a trip. Therefore,
juveniles frequently head off course compared to experienced
migrators (Thorup et al. 2003). Whether this ontogeny of
migration performance applies to marine turtles requires the
repeat tracking of individuals through their first and subse-
quent migrations.
The results of this study help clarify the processes underly-
ing long distance migration to specific targets, by suggesting
that these incredible journeys might be achieved without
migrants having the ability to know their precise position in
relation to the goal during much of the journey. Instead,
migrants simply follow a heading that is approximately goal
orientated, and then make track corrections during the final
stages of the journey. The long distance migrations of many
marine turtle populations (Hays & Scott 2013) point to this
suboptimal solution to long-distance migration being a suc-
cessful strategy.
In conclusion, our work introduces a new perspective in the
analysis of wildlife tracking datasets, through which it might
be observed that different animal groups potentially exhibit dif-
ferent levels of complexity in goal attainment during migration.
ACKNOWLEDGEMENTS
We thank the National Marine Park of Zakynthos (NMPZ)
for the permission to conduct this research, and the many
people who provided assistance with in-water capture of tur-
tles. We wish to thank Giulio Giunta for suggesting reading
material on the Zermelo navigation problem.
AUTHORSHIP
GCH and PM conceived the study with contributions from
SF. GS and SF organised the fieldwork and filtered the track-
ing data sets. AC developed the optimisation algorithm and
PM and AC ran the simulation of movement in current flows.
GCH and PM wrote the manuscript with contributions from
all authors.
e h g a d f c b
15
20
25
30
35
40
45
50
Migration time (day)
(a)
Dscore = 0.46
e h g a d f c b
15
20
25
30
35
40
45
50
Migration time (day)
(b)
Dscore = 0.64
e h g a d f c b
20
25
30
35
40
45
50
Migration time (day)
(c)
Dscore = 0.28
Turtle id
Figure 4 Performance of the behavioural models at simulating the eight
observed turtle migratory tracks: duration of migration based on
observed turtle tracks (black circles) and two of the models heading to
goal (a, red triangles), optimal Zermelo route (b, blue diamonds), heading
course correction (c, pink squares). Note, the tracks from the single
heading model do not reach the final target; therefore, the migration time
is not shown. Also shown for each behavioural model the average value
of the non-dimensional distance score as derived using the sum of the
three metrics: straightness index, Euclidean average distance, Hausdorff
distance.
©2013 John Wiley & Sons Ltd/CNRS
142 G. C. Hays et al. Idea and Perspective
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SUPPORTING INFORMATION
Additional Supporting Information may be downloaded via
the online version of this article at Wiley Online Library
(www.ecologyletters.com).
Editor, John Fryxell
Manuscript received 16 August 2013
First decision made 20 September 2013
Manuscript accepted 17 October 2013
©2013 John Wiley & Sons Ltd/CNRS
Idea and Perspective Route optimisation during migration 143
