Ecological Applications, 23(4), 2013, pp. 815–828
Ó 2013 by the Ecological Society of America
Mechanistic models for the spatial spread of species
under climate change
SHAWN J. LEROUX,1,4 MAXIM LARRIVÉE,1,5 VÉRONIQUE BOUCHER-LALONDE,1 AMY HURFORD,2,6 JUAN ZULOAGA,1
JEREMY T. KERR,1 AND FRITHJOF LUTSCHER3
1
Canadian Facility for Ecoinformatics Research, Department of Biology, University of Ottawa, 30 Marie Curie,
Ottawa, Ontario K1N 6N5 Canada
2
MPrime Centre for Disease Modelling, YIHR 5021 TEL Building, York University, 4700 Keele Street,
Toronto, Ontario M3J 1P3 Canada
3
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa, Ontario K1N 6N5 Canada
Abstract. Global climate change is a major threat to biodiversity. The most common
methods for predicting the response of biodiversity to changing climate do not explicitly
incorporate fundamental evolutionary and ecological processes that determine species
responses to changing climate, such as reproduction, dispersal, and adaptation. We provide
an overview of an emerging mechanistic spatial theory of species range shifts under climate
change. This theoretical framework explicitly defines the ecological processes that contribute
to species range shifts via biologically meaningful dispersal, reproductive, and climate
envelope parameters. We present methods for estimating the parameters of the model with
widely available species occurrence and abundance data and then apply these methods to
empirical data for 12 North American butterfly species to illustrate the potential use of the
theory for global change biology. The model predicts species persistence in light of current
climate change and habitat loss. On average, we estimate that the climate envelopes of our
study species are shifting north at a rate of 3.25 6 1.36 km/yr (mean 6 SD) and that our study
species produce 3.46 6 1.39 (mean 6 SD) viable offspring per individual per year. Based on
our parameter estimates, we are able to predict the relative risk of our 12 study species for
lagging behind changing climate. This theoretical framework improves predictions of global
change outcomes by facilitating the development and testing of hypotheses, providing
mechanistic predictions of current and future range dynamics, and encouraging the adaptive
integration of theory and data. The theory is ripe for future developments such as the
incorporation of biotic interactions and evolution of adaptations to novel climatic conditions,
and it has the potential to be a catalyst for the development of more effective conservation
strategies to mitigate losses of biodiversity from global climate change.
Key words: butterflies; climate change; climate envelope; climate velocity; dispersal; global change;
intrinsic growth rate; invasive species; mathematical model; mechanistic model; range shift; reaction–
diffusion.
INTRODUCTION
Anthropogenic global changes including habitat loss
and fragmentation, pollution, exotic species invasions,
and climate change threaten biodiversity and associated
ecosystem services (Vitousek et al. 1997, Foley et al.
2005, Kerr et al. 2007). Predicting the response of
biodiversity to climate change, in particular, has become
a burgeoning field of study (Bellard et al. 2012) because
Manuscript received 15 August 2012; revised 28 November
2012; accepted 2 January 2013. Corresponding Editor: J.
Franklin.
4 Present address: Department of Biology, Memorial
University of Newfoundland, 232 Elizabeth Ave, St John’s,
Newfoundland A1B 3X9 Canada. E-mail: [email protected]
5 Present address: Montreal Insectarium, 4581 Rue Sherbrooke Est, Montreal, Quebec H1X 2B2 Canada.
6 Present address: Department of Biology, Memorial
University of Newfoundland, 232 Elizabeth Ave., St John’s,
Newfoundland A1B 3X9 Canada.
climate change is emerging as a major threat to
biodiversity in the next few decades (Thomas et al.
2004, Leadley et al. 2010). The distribution and
persistence of many species is constrained by climate
(Bryant et al. 1997, Hill et al. 2001), and recent species
range expansions and shifts show patterns consistent
with contemporary climate warming (e.g., Parmesan et
al. 1999, Parmesan and Yohe 2003, Root et al. 2003,
Chen et al. 2011, Devictor et al. 2012). In the face of
changing climate, species may persist by moving or
dispersing to track preferred conditions (Hickling et al.
2006, Parmesan 2006), demonstrating in situ plastic or
acclimatory responses to changing climate (Nussey et al.
2005, Durant et al. 2007), or evolving adaptations to
novel climatic conditions (Visser 2008, Gardner et al.
2009). For example, European bird and butterfly
communities are moving northward (Devictor et al.
2012) and Dutch Great Tits (Parus major) show
plasticity in the timing of reproduction over a 32-year
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SHAWN J. LEROUX ET AL.
period that is consistent with climate change (Nussey et
al. 2005). A mechanistic framework for disentangling
the role of these three main strategies for species
responses to climate change will be an invaluable
predictive tool for global change biology.
A number of different approaches have been developed for predicting the response of species to global
change. Many of these approaches, however, do not
explicitly incorporate fundamental evolutionary and
ecological processes that may determine the ability of
a species to respond to changing climate, such as rates of
reproduction, dispersal, and adaptation (Keith et al.
2008, Kearney and Porter 2009, Buckley et al. 2010,
Chevin et al. 2010, Zhou and Kot 2011). Correlative
species distribution models, for example Maxent (Phillips et al. 2006) and BIOMOD (Thuiller 2003) relate
species occurrence records to environmental conditions
to infer abiotic correlates of a species’ realized niche.
Mechanistic distribution models (reviewed in Kearney
and Porter 2009, Buckley et al. 2010) or habitat
suitability models coupled with stochastic population
models (Keith et al. 2008, Araújo and Peterson 2012) are
alternatives to correlative models as they relate species
processes (e.g., activity levels, survivorship, fecundity,
and so forth) to environmental conditions. But, these
models require more detailed data than correlative
models (Keith et al. 2008, Thuiller et al. 2008, Buckley
et al. 2010), and it remains unclear whether current
mechanistic distribution models perform better than
correlative models in predicting the current and future
distribution of species (Kearney and Porter 2009, Morin
and Thuiller 2009, Buckley et al. 2010). We present a
theoretical framework for improving our predictions of
global change outcomes. This framework explicitly
defines the ecological processes that contribute to species
range shifts via biologically meaningful dispersal,
reproductive, and climate envelope parameters.
Mathematical biologists have developed spatial theory that has been widely used to predict the spread of
species invasions (reviewed in Shigesada and Kawasaki
1997, Hastings et al. 2005). For example, reaction–
diffusion and integro-difference models have been
applied to predict the spatial spread of a range of taxa
including House Finches (Veit and Lewis 1996), gray
squirrels (Okubo et al. 1989), muskrat (Andow et al.
1990), wolves (Hurford et al. 2006), and cabbage white
butterflies (Andow et al. 1990). Recognizing that the
spatial spread of invasive species is a mathematical
problem similar to that of the spatial spread of species in
response to changing climate, Potapov and Lewis (2004)
developed a general mathematical theory of species
range shifts under changing climate. Their analytical
model relates the velocity of a species’ specific climate
envelope to basic species processes of reproduction and
dispersal. Dispersal is a fundamental process that can
facilitate (or restrict) a species’ range by enabling (or
preventing) a species to reach suitable sites (Stevens et
al. 2010, Boulangeat et al. 2012). Having high vagility,
however, is not sufficient to guarantee species persistence, because persistence also is dependent on speciesspecific growth rates and the speed at which the suitable
climate zone is moving. Since the initial derivation by
Potapov and Lewis (2004), there has been some
theoretical development (see Roques et al. 2008,
Berestycki et al. 2009, Zhou and Kot 2011), but the
theory has yet to be confronted with empirical data, and
methods for empirically estimating parameters of the
models have not been developed. Our goal is to bridge
the gap between the simple analytical predictions of
Potapov and Lewis (2004) and empirical observations of
species spread under climate change. We set out to make
this theory accessible to ecologists because we believe
this framework will help to organize the current research
agenda, inform data needs and best-use practices, and
disentangle the multiple ways that biodiversity may
respond to changing climate.
Here we provide a brief primer on the use of reaction–
diffusion equations in spatial ecology and on recent
theoretical developments to include climate change in
these models. Then we present methods for estimating
parameters of a simple reaction–diffusion model with
changing climate and apply these methods to 12 North
American butterfly species. We compare our mechanistic, species-specific mobility predictions to realized
mobility estimates for our study species in order to
determine the relative risk that these species may lag
behind climate change. We end by discussing the
advantages and future directions in the development
and application of this theory to improve predictions of
global change outcomes.
OVERVIEW
OF
SPATIAL THEORY OF SPECIES SPREAD
CLIMATE CHANGE
UNDER
Reaction–diffusion equations have been used extensively in spatial ecology since the seminal papers of
Skellam (1951) and Kierstead and Slobodkin (1953).
When applied to climate change, reaction–diffusion
equations allow us to derive conditions for a species to
keep pace with changing climate (Pease et al. 1989,
Potapov and Lewis 2004, Berestycki et al. 2009, Chevin
et al. 2010). These conditions depend on the speed at
which a species can move and the minimum patch size
necessary for it to persist. We will present both of these
properties.
A reaction–diffusion equation describes the change in
the density of a population (u(t, x)) through time (t) and
space (x). In the simplest case, individuals move
randomly in one-dimensional space with diffusion rate
D and reproduce at a constant per capita rate r. The
corresponding equation (for variables, parameters, and
their units, see Table 1) is as follows:
]
]2
u ¼ D 2 u þ ru:
]t
]x
ð1Þ
In invasion ecology, this equation can be used to predict
the speed of spatial spread of a locally introduced
June 2013
SPECIES RANGE SHIFT UNDER CLIMATE CHANGE
817
TABLE 1. Reaction–diffusion model variables/parameters, definitions, and units for the spatial spread of species under changing
climate.
Variable or parameter
u
x
t
D
r
q
L
Lc
Lc(q)
c*
Dc
Definition
population density
space
time
diffusion rate
per capita growth rate
climate envelope movement rate
bounded habitat domain
pffiffiffiffiffiffiffiffi
critical patch size; p D=r
ffi!1
pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
critical patch size with moving temperature isoclines; p D=r
1 pffiffiffiffiffiffi
2 Dr
pffiffiffiffiffiffi
speed of spatial spread of a locally introduced species; 2 Dr
threshold value of D for a species to keep pace with changing climate; q2/4r
*
species
pffiffiffiffiffiffi in an unbounded homogeneous landscape as c ¼
2 Dr (reviewed in Shigesada and Kawasaki 1997,
Okubo and Levin 2001, Hastings et al. 2005).
In conservation biology, one can predict the minimum
size required for a certain habitat to support a given
species. One assumes that Eq. 1 holds on a bounded
domain of length L, and, as a worst case, that the
surroundings are completely hostile.
pffiffiffiffiffiffiffiffiThis setup gives a
critical patch size of Lc ¼ p D=r (Skellam 1951,
Kierstead and Slobodkin 1953). Dispersal can induce
loss from a given patch. If the patch is small and the
surroundings are hostile, then this dispersal-induced loss
can cause population decline and eventual extinction
(Perry 2005, Kenkre and Kumar 2008).
Recently, theoreticians have begun to consider the
effect of global change on the critical patch size in this
reaction–diffusion framework (Pease et al. 1989, Potapov and Lewis 2004, Berestycki et al. 2009, Chevin et al.
2010). Potapov and Lewis (2004) implemented the
effects of a latitudinal shift of temperature isoclines by
considering the x-axis as a north–south section through
the landscape. They assumed that the species’ growth
rate is positive in some patch [x1, x2] of length L, and is
negative outside the patch. They furthermore assumed
that the boundaries of the favorable patch move
northward with constant speed q, i.e., xi,t ¼ xi,0 þ qt
(Fig. 1), so that the size of the patch remains constant
over time. Parameter q represents the rate of movement
of a species’ climate envelope. In the special case that the
environment outside the favorable patch is completely
hostile, following Potapov and Lewis (2004), the critical
patch size with moving temperature isoclines is
ffi1
pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
p
ffiffiffiffiffi
ffi
1
ð2Þ
Lc ðqÞ ¼ p D=r
2 Dr
provided
pffiffiffiffiffiffi
q , c ¼ 2 Dr :
ð3Þ
For q ¼ 0, we recover the critical patch size from above.
If the climate envelope moves more quickly, as q
approaches c* from below, the critical patch size
Units
no. individuals/km2
km
years
km2/year
no. individuals/year
km/year
km
km
km
km/year
km2/year
increases nonlinearly to infinity. When the speed of the
temperature isoclines (q) is faster than the spread rate of
the population in a homogeneous landscape (c*), the
population will not persist in any patch of finite size.
Formally, a population
not keep up with changing
pffiffiffiffiffiwill
ffi
climate if q . c* ¼ 2 Dr.
When the conditions outside of the favorable patch
are not completely hostile (i.e., the population has a
finite death rate), then the explicit expression for the
critical patch size is more cumbersome (see Potapov and
Lewis 2004). However, the key model prediction still
holds that the population cannot keep up with changing
climate on any patch of finite size when q . c*.
There is a parallel body of literature on mathematical
models for the spread of populations with discrete,
nonoverlapping generations, so-called integro-difference
equations (e.g., Kot and Schaffer 1986, Kot et al. 1996).
Integro-difference equations can accommodate more
detailed distributions of dispersal distances, a key trait
when dealing with species with frequent long-distance
dispersal events. Zhou and Kot (2011) investigated the
effects of shifting climate zones on population persistence in these models and arrived at qualitatively similar
results.
pffiffiffiffiffiffi
In summary, this theory predicts that if q . 2 Dr, a
population cannot keep up with changing
and
pffiffiffiffifficlimate
ffi
will eventually go extinct. If q , 2 Dr , then the
population can persist, provided its favorable habitat is
large enough.
To apply this theory, we must estimate three
parameters (r, D, q). Estimates for dispersal, and D in
particular, are notoriously difficult to come by (Grosholz 1996) because D estimates usually require detailed
multisite mark–recapture studies (for a review of
methods for quantifying butterfly dispersal, see Stevens
et al. 2010). However, we can use existing data to obtain
estimates for the population growth rate, r, and the
climate envelope movement rate, q, and then find the
critical threshold value for D, Dc. A species will be able
to keep pace with climate if D . Dc. After rearranging
Eq. 3, we find Dc ¼ q2/4r. This elegant theoretical
prediction allows empiricists to readily test the influence
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SHAWN J. LEROUX ET AL.
Ecological Applications
Vol. 23, No. 4
FIG. 1. The Potapov and Lewis (2004) model
for the spatial spread of species under climate
change models a suitable climate envelope [x1,t,
x2,t] of length, L moving with a constant speed, q,
which is determined by the velocity of climate
change. The size of the suitable climate envelope
remains constant over time.
of different processes, i.e., reproduction (r), movement
(D), and climate change (q), on species persistence in
light of climate change, depending on the data that is
available to them. Next, we illustrate how this theory
can be used to predict the relative ability of 12 North
American butterfly species to keep pace with changing
climate.
AN APPLICATION
OF THE
THEORY
Empirical evidence shows many species expanding or
shifting their ranges in the direction of changing climate.
However, many of these species are not actually keeping
pace with the velocity of climate change (Loarie et al.
2009, Devictor et al. 2012). The dynamics of species
ranges during a period of climate change are determined
by a suite of ecological and evolutionary processes such
as rates of reproduction and dispersal (Gaston 2009,
Atkins and Travis 2010, Angert et al. 2011), but much of
the empirical evidence of expanding ranges does not
quantify the role of reproduction or dispersal. Predicting
which species are at a higher risk of lagging behind
climate change is critical for identifying future risks,
supporting development of proactive strategies to reduce
climate change impacts on biodiversity, and prioritizing
policy initiatives (Bellard et al. 2012). The body of
theory that we have summarized in the previous section
predicts that D . Dc ¼ q2/4r is necessary for a
population to keep up with climate change. In other
words, a species must have a movement rate above a
certain threshold (i.e., q2/4r) in order to keep pace with
changing climate. Here, we derive methods for estimating q and r parameters for a suite of North American
butterfly species. Then we use these estimates to
calculate the threshold D values for these species, which
enables us to determine the ability of each species to
track climate. Our analysis was conducted on all
ecozones of the Canadian mainland east of the
Canadian Rockies and south of the Northern Arctic
(Fig. 2). We excluded ecozones with high elevations
from our study area because these areas are highly
heterogeneous on small scales and therefore do not
match our model assumptions.
Estimating the climate envelope movement rate, q
We obtained occurrence data from the Canadian
Biodiversity Information Facility for 12 butterfly
species: 3 species of Lycaenidae, 5 species of Nymphalidae, 1 species of Papilionidae, and 3 species of Pieridae
butterflies. This database contains ;300 000 precisely
georeferenced, dated records for 297 Canadian butterfly
species (Layberry et al. 1998) from specimens stored at
one of many museums across Canada. See Kharouba et
al. (2009) for more details on these data. These 12
species were the only ones for which we could obtain
both occurrence and abundance data. We had a mean of
106 (SD ¼ 59) geographically unique occurrence records
for the time period 1960–1970 per study species.
Phenological and range-shift responses for species in
North America are predominantly subsequent to 1970
(Parmesan 2006), as is directional climate change, which
is very likely to be attributable to human activities
(Hansen et al. 1999). Thus the period 1960–1970 was
used as a historical baseline in which to construct species
climate envelope models.
We used Maxent (Phillips et al. 2006) to model a
potential climate envelope for each species based on
1960–1970 occurrence records. Maxent predicts where a
species may be found across geographical space, derived
from its occurrence records relative to environmental
predictors. We used minimum winter temperature, mean
summer temperature, annual precipitation, and seasonality of precipitation as our environmental predictors.
These variables reflect previously documented environmental limits for butterflies in this region (Kharouba et
al. 2009). Climate observations were constructed using
ANUSPLIN, a regression splines interpolation, across
all available weather station data for North America
(McKenney et al. 2006). Data are available at 10 arc
minute [1 minute of arc is 1/60 of one degree] resolution
annually from 1961 to 2006. These data were developed
at the Canadian Forest Service and are used in climate
reporting by the Government of Canada. We projected
species climatic envelopes through time based on 5-year
climate normals (i.e., 1971–1975, 1976–1980, and so
forth). A mean projected suitability envelope was
June 2013
SPECIES RANGE SHIFT UNDER CLIMATE CHANGE
819
FIG. 2. Study area in Canada and butterfly species richness (number of species) based on occurrence records and baseline
Maxent model predictions for the period 1960–1970. The study area includes all ecozones of the Canadian mainland east of the
Canadian Rockies and south of the Northern Arctic.
produced based on 10 iterations of the model to derive
the final output.
For each model, probability of occurrence was
converted into a binary map of areas predicted to be
part of the species’ range (i.e., suitable) and outside the
species’ range (i.e., unsuitable). The threshold suitability
value was calculated by taking the average of the lowest
10 predicted suitability values of the true presences used
to test the 10 model iterations for the baseline 1960–
1970 model (see methods in Liu et al. 2005, Kharouba et
al. 2009). This thresholded output provides a speciesspecific estimate of the potential climate envelope for
each 5-year period. This method assumes that the
species is in equilibrium with climate and that data
collected between 1960 and 1970 are representative of
the species’ climate niche prior to significant climate
change.
The extent of our data does not cover the full climate
envelope of each species, but rather the northern portion
of its range. Consequently, we estimate q as the
expansion of the northern climate envelope edge in
Canada. For each species, we extracted the northern
climate envelope edge of contiguous range patches (i.e.,
we excluded range ‘‘islands’’ distant from the main
predicted range) for each 5-year climate envelope period
(Fig. 3). We calculated the mean distance from the full
length (i.e., east–west) of the 1960–1970 pre-climate
change baseline climate envelope edge to the full length
of the climate envelope edge of each 5-year period (i.e.,
distance from 1960–1970 to 1971–1975, from 1960–1970
to 1976–1980, and so forth; Fig. 3). Once the northern
edge of a climate envelope hit a coastal boundary (e.g.,
Hudson Bay), we excluded all further points along this
boundary from our q calculations. We excluded these
points because the climate envelope of terrestrial species
is bounded by such physical boundaries. Their inclusion
would therefore systematically underestimate the true
climatic shift, q, that affects species. We estimated q as
the slope of a linear regression of cumulative distance
between climate envelope edges (km) vs. time.
The hotspots of predicted species range overlap in
1960–1970 for our sample of species occurs in southern
Ontario and Manitoba, southeastern Ontario, and
southwestern Quebec, but some species have a predicted
1960–1970 climate envelope as far north as Inuvik,
Northwest Territories (Fig. 2). For all species, there was
high variability in the cumulative distance between
northern range edges through time, with R 2 ranging
from 0.02 to 0.65 (Table 2). Our estimated q ranged
from 1.14 km/yr (for Papilio canadensis; 95% CI 0–6.12
km/yr) to 5.51 km/yr (for Callophrys niphon; 95% CI
0.75–10.27 km/yr), with a mean q value of 3.25 km/yr
(SD ¼ 1.36; Table 2, Fig. 4). The lower 95% CI estimate
for 10 of 12 species was 0, indicating the case of no
northern shift in the climate envelope of these species.
These results predict that the climate envelopes of our
study species are shifting at a rate of 3.25 6 1.36 km/yr
(mean 6 SD).
Estimating the per capita growth rate, r
We obtained abundance time series data for our 12
butterfly species from Ross Layberry, Canadian butterfly expert and lead author of The Butterflies of Canada
(Layberry et al. 1998). Since 1989, Layberry has
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SHAWN J. LEROUX ET AL.
Ecological Applications
Vol. 23, No. 4
FIG. 3. Methods for estimating the rate of northern shift of the species-specific climate envelope, q. (a) We use occurrence
records for each butterfly species from 1960–1970 to build a baseline map of the species distribution. Data shown here and in other
panels are for Danaus plexippus. (b) Then we project species distributions through time (5-year time period 2001–2005 shown here)
based on a changing climate envelope. (c, d) We extract the northern edge for each time period. (e) To calculate the distance
between the climate envelope in 1960–1970 and the range in 2001–2005, we convert the northern edge of 1960–1970 to points and
calculate the mean distance (di ) between each point from 1960–1970 to the nearest point on the northern edge of 2001–2005.
intensively sampled a 300-ha patch of mixed-wood,
open habitat in Eastern Ontario, Canada several times
during the butterfly flight season and recorded species
identity and abundances. We used the maximum
abundance estimates per season for every species with
at least nine consecutive years of abundance data (mean
13 years) for estimating the population growth rate
parameter, r, of our model. All data were collected
between 1989 and 2009.
We used the Ricker model to estimate the population
growth rate, r, of the 12 butterfly species (Ricker 1954).
The Ricker model is a widely used phenomenological
model of population dynamics that incorporates density
dependence as the mechanism preventing unbounded
growth (Clark et al. 2010). We used a density-dependent
model because there is evidence for density-dependent
population dynamics in a range of taxonomic groups,
including insects (Brook and Bradshaw 2006). The
Ricker model can be written formally as
2
Ntþ1 ¼ Nt erð1Nt =KÞþeð0;r
Þ
ð4Þ
where Nt is population abundance at the current time t.
The per capita growth rate at low abundance is er, and
the population carrying capacity is K. Following Brook
and Bradshaw (2006) and Clark et al. (2010), we model
process error, e, as normally distributed with zero mean
and variance, r2.
June 2013
SPECIES RANGE SHIFT UNDER CLIMATE CHANGE
TABLE 2. Results of linear regression models of cumulative
spread (km) vs. time with q as the slope of this linear
regression, for 12 North American butterfly species.
Species, by family
q
Lower
95% CL
Upper
95% CL
R2
Lycaenidae
Callophrys niphon
Celastrina lucia
Glaucopsyche lygdamus
5.51
1.24
4.22
0.75
0
0.62
10.27
10.47
7.82
0.64
0.02
0.65
Nymphalidae
Danaus plexippus
Enodia anthedon
Limenitis arthemis
Phyciodes cocyta
Polygonia comma
2.97
3.57
1.79
3.12
3.70
0
0
0
0
0
7.22
7.64
5.51
6.41
9.07
0.39
0.50
0.24
0.54
0.39
Papilionidae
Papilio canadensis
1.14
0
6.12
0.07
Pieridae
Colias philodice
Pieris oleracea
Pieris rapae
3.62
3.13
5.04
0
0
0
10.91
10.76
11.31
0.25
0.18
0.46
Notes: Negative values for lower 95% confidence limits (CL)
were replaced with zero because zero represents the case of no
northern shift in the climate envelope. Cumulative spread is the
distance between northern range edges in 1960–1970 (baseline)
to successive five-year periods (i.e., baseline to 1971–1975,
baseline to 1976–1980, and so forth). See Fig. 3 for an
illustration of methods for calculating q, and Fig. 4 for
representation of data and regression line fits.
821
We estimated r, K, and r with maximum likelihood
implemented with the bbmle (Bolker 2008) package in R
v.2.14.1 (R Development Core Team 2011). We
rearranged Eq. 4 for fitting as follows:
Ntþ1
Nt
þ eð0; r2 Þ:
¼r 1
ð5Þ
ln
Nt
K
We calculated 95% confidence intervals for r directly
from the likelihood profiles using the confint function
in R.
For all 12 species, the maximum likelihood Ricker
model r estimates were identifiable (Table 3). These r
estimates ranged from 0.69 (Polygonia comma, 95% CI
0.23–1.14) to 1.72 (Glaucopsyche lygdamus, 95% CI 0.9–
2.54), with a mean r value of 1.24 (SD ¼ 0.33; Table 3).
These results suggest that our study species produce 3.46
6 1.39 viable offspring per individual per year (mean 6
SD).
Estimating the diffusion rate required to keep pace
with climate change, Dc
We used our estimates of q and r to calculate a
threshold value for D for each species, the minimum
value of D required for a species to track climate change
(i.e., Dc ¼ q2/4r). We calculated Dc for our mean
estimates of r and q values. An upper confidence limit
FIG. 4. Mean distance (km) between northern range edges 1960–1970 (baseline) to 1971–1975, 1960–1970 to 1976–1980, 1960–
1970 to 1981–1985, and so forth (this is cumulative spread) with regression line of cumulative spread vs. time (in 5-year periods) fit
for 12 species of butterflies organized in four families. Values in parenthesis are the slopes of the regression lines for each species.
Negative cumulative spread values represent a northern range retraction. See Table 2 for full scientific names.
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SHAWN J. LEROUX ET AL.
TABLE 3. Number of years of continuous abundance data (n)
and maximum-likelihood estimates for population-growthrate parameter r (with 95% confidence limits) for 12 North
American butterfly species.
Species, by family
n
r
Lower
95% CL
Upper
95% CL
Lycaenidae
Callophrys niphon
Celastrina lucia
Glaucopsyche lygdamus
13
20
9
0.86
0.95
1.72
0.12
0.26
0.90
1.59
1.63
2.54
Nymphalidae
Danaus plexippus
Enodia anthedon
Limenitis arthemis
Phyciodes cocyta
Polygonia comma
9
12
10
21
12
1.35
1.40
1.15
1.06
0.69
0.13
0.15
0.27
0.34
0.23
2.59
2.66
2.04
1.78
1.14
Papilionidae
Papilio canadensis
13
1.56
0.67
2.45
Pieridae
Colias philodice
Pieris oleracea
Pieris rapae
21
17
12
1.48
0.99
1.62
0.71
0.36
0.92
2.25
1.62
2.32
for Dc was calculated with our mean estimate of q and
lower 95% CI estimate of r, and a lower confidence limit
for Dc was calculated with our mean estimate of q and
upper 95% CI estimate of r (see Tables 2 and 3 for
parameter estimates). We ranked species according to
their mean calculated Dc values to determine their
relative movement rate required to keep pace with
climate change.
Our estimates of Dc only provide us with a prediction
for how mobile a species must be to track climate
change; they do not tell us anything about the actual
mobility of a species. To determine the relative ability of
each species to track climate change, we must compare
the predicted Dc to an actual measure of species
mobility. An estimate of actual mobility for our study
species was obtained from Burke et al. (2011), who
asked 51 North American lepidopterists to score
Canadian butterfly species on their mobility from 0
(sedentary) to 10 (extremely mobile). They summarized
the mean scores for the group of experts into a relative
mobility index for 297 butterfly species in Canada.
Expert opinions may reflect the migration propensity of
butterflies instead of realized dispersal (Stevens et al.
2010), but these data represent the best available
mobility data for our study species. We ranked our
species according to their mobility index score and
calculated the difference between the mobility index
score rank and the critical Dc estimate rank. We present
this simple rank difference method as a first pass at
comparing predicted vs. observed butterfly dispersal
abilities. Future comparisons should use empirical
dispersal data where available.
Mean Dc ranged from 0.21 km2/yr (for Papilio
canadensis; CI 0.13–0.48 km2/yr) to 8.83 km2/yr (for
Callophrys niphon; CI 4.77–63.25 km2/yr), with a grand
mean Dc value across all species of 2.71 km2/yr (SD ¼
2.36; Table 4, Fig. 5a). Species in the family Pieridae
require relatively high Dc to keep pace with climate
change. Burke et al. (2011) mobility index scores ranged
from 3.71 (Celastrina lucia) to 9.50 (Danaus plexippus)
for our study species (Table 4), with a mean mobility
index score of 6.25 (SD ¼ 1.67; Table 4).
The difference in the relative ranks of the mobility
index and the Dc value ranged from 10 to 10 (median ¼
1; Fig. 5b). Five species (Callophrys niphon, Glaucopsyche lygdamus, Polygonia comma, Pieris oleracea,
Enodia anthedon) differed in their relative rank by 3
or less; three species (Celastrina lucia, Phyciodes cocyta,
Pieris rapae) differed in their relative ranking by 1 to 1;
and four species (Colias philodice, Limenitis arthemis,
Danaus plexippus, Papilio canadensis) ranked relatively
higher on the mobility index than on the critical Dc scale
(Fig. 5b).
Andow et al. (1990) estimated a diffusion coefficient
for Pieris rapae between 4.8 and 129 km2/yr, based on
mark–recapture data collected by Jones et al. (1980).
Our Dc estimate for Pieris rapae ranges between 2.74
and 6.90 km2/yr, which falls in the lower range of the
realized mobility estimate from Andow et al. (1990).
IMPROVING PREDICTIONS
OF
GLOBAL CHANGE OUTCOMES
Potapov and Lewis (2004) developed a framework for
a general mathematical theory of species range shifts
under changing climate, based on reaction–diffusion
models for invasive species. The main prediction of this
theory relates the velocity of climate change (q) to
species reproduction (r) and diffusion (D); if D , Dc ¼
q2/4r, a population is at risk of not keeping track with
changing climate and will eventually go extinct. We
provide a road map for the application of this theory by
presenting methods for estimating the parameters of this
model and applying these methods to parameterize the
model for 12 North American butterfly species. The
application of this theory allowed us to identify the
relative risk of 12 butterfly species not keeping pace with
climate change.
Global change biologists have assembled extensive
large-scale data on species distribution (e.g., Global
Biodiversity Information Facility)7 and abundance (e.g.,
Global Population Dynamics Database)8 as well as
global climate (e.g., WorldClim)9 and land cover data
(e.g., Global Landcover 2000).10 As we have shown here,
these data correspond to parameters that have been
defined in theoretical ecology and can be put to good use
testing theoretical predictions on the spatial spread of
species under changing climate.
The advantages of adopting a theoretical framework
in global change biology are many. First, a formal
7
http://www.gbif.org
http://www3.imperial.ac.uk/cpb/databases/gpdd
9 http://www.worldclim.org/
10 http://bioval.jrc.ec.europa.eu/products/glc2000/glc2000.
php
8
June 2013
SPECIES RANGE SHIFT UNDER CLIMATE CHANGE
823
TABLE 4. Dc estimates and mean mobility index scores (Burke et al. 2011) for 12 North American
butterfly species.
Our critical Dc estimate
Burke mobility index
Species, by family
Dc
Lower Dc
Upper Dc
Rank
Mean
Rank
Lycaenidae
Callophrys niphon
Celastrina lucia
Glaucopsyche lygdamus
8.83
0.40
2.59
4.77
0.24
1.75
63.25
1.48
4.95
12
2
9
4.20
3.71
5.37
2
1
5
Nymphalidae
Danaus plexippus
Enodia anthedon
Limenitis arthemis
Phyciodes cocyta
Polygonia comma
1.63
2.28
0.70
2.30
4.96
0.85
1.20
0.39
1.37
3.00
16.96
21.24
2.97
7.16
14.88
4
6
3
7
11
9.50
5.12
6.97
5.43
6.64
12
3
8
6
7
Papilionidae
Papilio canadensis
0.21
0.13
0.48
1
7.79
11
Pieridae
Colias philodice
Pieris oleracea
Pieris rapae
2.21
2.47
3.92
1.46
1.51
2.74
4.61
6.80
6.90
5
8
10
7.33
5.36
7.56
9
4
10
Notes: Mean Dc estimates are based on our mean estimates of q and r, whereas the upper and
lower Dc estimates are for our mean CI estimate of q and 95% lower and upper CI estimates of r,
respectively (see Tables 2 and 3 for parameter estimates). Mean mobility index scores for our 12
butterfly species are derived from Burke et al. (2011). We report the relative ranking of each species
according to our estimates of Dc and the mobility index scores of Burke et al. (2011). Species that
rank relatively higher on the Dc scale than on the mobility index may be most at risk of not keeping
pace with changing climate.
mathematical theory will facilitate testing existing
hypotheses and generating novel ones on the conditions
that allow biodiversity to persist in the face of
environmental change. For example, we might derive
competing models of species dynamics in light of the
processes of habitat loss and climate change and
confront the models with empirical data to determine
the relative role of habitat loss and changing climate on
species persistence (Warren et al. 2001, Thuiller et al.
2008). In fact, Potapov and Lewis (2004) organize and
relate species extinction risk due to habitat loss and
climate change through their common dependence on
species reproduction, dispersal, and climate. The results
of Potapov and Lewis (2004) emphasize that tracking
climate change alone does not guarantee that a species
will thrive, because persistence also depends on the size
of available habitat (Eq. 2; Fig. 6). With sufficient data,
our theoretical framework allows one to quantify the
relative risk of species extinction due to insufficient
habitat and/or inability to keep pace with climate
change (see Fig. 6 for an example for Phyciodes cocyta).
In essence, the theory can become an organizing and
predictive framework in the quickly emerging field of
global change biology.
Second, mechanistic mathematical models incorporate key ecological and evolutionary processes (e.g.,
dispersal) that determine the ability of a species to
respond to environmental change a priori. Consequently, these models should predict range dynamics under
future environmental projections better than would a
purely correlative model (Keith et al. 2008, Buckley et
al. 2010, Chevin et al. 2010, Araújo and Peterson 2012).
Approaches that neglect dispersal (e.g., correlative
species distribution models) may overestimate species
persistence under changing climate (Zhou and Kot
2011). Consequently, explicitly stating how the processes
of reproduction and dispersal combine to determine
species persistence may be critical for accurate prediction.
Third, a formal mathematical framework provides
analytical solutions and thresholds that can be used to
predict past, present, and future species responses to
changing climate. Analytical solutions identify key
variables to empirically measure, which encourages
feedback between the model formulation and data.
Continually confronting our models with empirical data
in an adaptive process is a necessary reality check to
identify competing hypotheses that are most consistent
with empirical data, highlight data needs and future
theoretical developments, and ultimately lead to better
predictions of range shifts and extinction risks under
environmental change (Sexton et al. 2009).
LIMITATIONS
OF THE
THEORY
Current models of species spread under climate
change require a number of simplifying assumptions.
The theoretical predictions of these models should be
confronted with empirical data from a range of
ecosystems and taxa in order to determine to what
extent species spread rates under climate change can be
captured by the simple mechanisms currently incorporated in the models. As previously stated, global change
824
SHAWN J. LEROUX ET AL.
Ecological Applications
Vol. 23, No. 4
FIG. 5. (a) Estimates of log-transformed Dc (the critical, or threshold, diffusion rate) for 12 North American butterfly species.
Solid points represent mean Dc estimates; the upper and lower bars represent upper and lower 95% CI estimates of r, respectively
(see Tables 2 and 3). (b) Difference in the relative rank of 12 North American butterfly species based on a mobility index assigned
by naturalists (i.e., realized mobility; see Table 4 and Burke et al. [2011]) and the relative rank of these same species based on our
mean Dc estimates (i.e., predicted mobility). Larger negative differences may indicate species that are more at risk of not keeping
pace with climate change, whereas larger positive differences may indicate species that are more likely to keep pace with climate.
biologists have access to extensive data sets that could be
used to test model predictions and to determine the
validity of model assumptions. Here we outline a few
simplifying assumptions that could be relaxed in order
to improve predictions of species range shifts under
changing climate.
The model formulation that we present assumes that a
uniformly suitable patch of constant size moves in an
otherwise hostile environment. This formulation is most
useful for investigating latitudinal climate change over
relatively flat terrain. The assumption of a uniformly
suitable patch is limiting, as small-scale spatial heterogeneity is apparent in many natural communities due to
consumer–resource distributions, habitat quality differences, and elevational gradients (Pickett and Cadenasso
1995). Furthermore, matrix habitat outside the patch
need not be uniformly hostile. Shigesada et al. (1986)
introduced habitat heterogeneity into a reaction–diffusion model of an invasive species by allowing periodic
variation in dispersal and reproductive rates, and
FIG. 6. Potapov and Lewis (2004) show that
to persist a species must keep pace with climate
change and the length of available habitat must
be sufficient. Eqs. 2 and 3 define the relative risks
of extinction due to climate change (dark gray)
and insufficient habitat (light gray) as they
depend on the species’ reproduction (r) and
climate envelope shift rate (q) estimates. The
figure is parameterized for Phyciodes cocyta and
suggests that if Phyciodes cocyta can track
climate change, it will probably persist because
the habitat requirements for D . 2.30 are
modest.
June 2013
SPECIES RANGE SHIFT UNDER CLIMATE CHANGE
Lutscher and Seo (2011) investigated the persistence of
invasive species in a seasonal river environment.
Interestingly, the rate of spread of invasive species is
determined by the harmonic mean of the diffusion
constant and the arithmetic mean of the growth rates in
different environments (Shigesada et al. 1986, Shigesada
and Kawasaki 1997). In a temporally varying environment, the spread rate is given by the arithmetic means of
D and r (Lutscher and Seo 2011). We modeled
population dynamics as a continuous process, whereas
population dynamics of insects may be better represented with distinct growth and dispersal stages. Discrete
integro-difference models probably would capture the
population dynamics and dispersal of butterflies better
(Kot et al. 1996, Zhou and Kot 2011), but these models
require more and higher-resolution data to parameterize
(but see Clark et al. 2001), and they may arrive at
qualitatively similar results (Zhou and Kot 2011). Our
model parameterization focuses on the northern edge of
the range and assumes that the southern edge is
retracting at the same rate as the northern edge expands.
Although there is some evidence of range retraction in
the southern parts of ranges (Parmesan et al. 1999, Kerr
2001), it is largely unknown whether the rate of southern
retraction is as fast as the rate of northern expansion.
Future developments of the model may consider
modeling a flexible habitat patch in which northern
and southern edges can move at different speeds, and
future empirical tests of the theory should look at the
entire range of a species. Finally, our simple model
assumes that dispersal and growth rate remain unchanged as climate changes. There is some evidence for
plasticity in movement (e.g., Cormont et al. 2011) and
growth rate (e.g., Boggs and Inouye 2012) of individuals
under climate change, and this is a key direction for
future research.
FUTURE DIRECTIONS OF A SPATIAL THEORY OF SPECIES
SPREAD UNDER CLIMATE CHANGE
Future developments of the theory of species spread
under climate change could consider a number of
processes not currently incorporated in the initial model
formulation of Potapov and Lewis (2004). In particular,
the current formulation focuses on a species’ ability to
move as its main response to climate change, while
ignoring the two other main pathways for species
responses to climate change, phenotypic plasticity or
evolution of adaptations to novel climatic conditions
(Atkins and Travis 2010, Angert et al. 2011, Bellard et
al. 2012). Recently, a number of developments have
been made in modeling adaptations to changing
environments. For example, Chevin et al. (2010) and
Duputié et al. (2012) offer two modeling approaches for
including phenotypic and genetic adaptation of key
traits in changing environments. Further developments
may integrate these recent efforts with work done on the
spread of invasive species. For example, Garcı́a-Ramos
and Rodrı́guez (2002) model the influence of local
825
adaptation on invasion in a spatially heterogeneous
environment, and Perkins (2012) models the influence of
evolutionary lability in an invasive predator and native
prey on the speed of the invasion front. A theoretical
framework that incorporates trade-offs and interactions
between the three main strategies for species to respond
to climate change will be an invaluable predictive tool
for global change biology (Pease et al. 1989, Chevin et
al. 2010, Lavergne et al. 2010). A second class of
processes that should be considered in future developments of this theory is species interactions (e.g.,
competition, predation, mutualism). Potapov and Lewis
(2004) did investigate the dynamics of two competing
species under changing climate, but there is mounting
evidence that consumer–resource interactions and other
biotic interactions can influence the outcome of species
responses to climate change (e.g., Araújo and Luoto
2007, Suttle et al. 2007; reviewed in Gilman et al. 2010,
Lavergne et al. 2010). For example, the long-term
response of a northern California grassland food web
to simulated climate change (i.e., increased precipitation) can be explained by the lagged effects of altered
competitive and trophic interactions (Suttle et al. 2007).
What is more, consumers will not be able to track
climate if their resources are lagging behind. Therefore,
it is critical to investigate matches/mismatches in the
phenology of consumers and resources generated by
climate change (Parmesan 2006, Durant et al. 2007).
Biotic interactions are often ignored in species distribution modeling (but see Boulangeat et al. 2012), yet biotic
interactions easily can be included in reaction–diffusion
or integro-difference equation models (for examples, see
Okubo et al. 1989, Potapov and Lewis 2004, Roques et
al. 2008). Further inclusion of biotic interactions into
our theoretical framework will be challenging but
rewarding, as it will facilitate predictions of wholecommunity responses to climate change. With theoretical progress occurring on multiple fronts as we have
outlined, we are making good strides toward achieving a
synthetic theory for predicting species responses under
changing climate.
A future direction for empirical tests of this theory lies
in the collection of better movement data for a range of
taxa. Even for widely studied species like birds and
butterflies, we have a rudimentary understanding of
their movement patterns at different spatial scales
(Grosholz 1996, Okubo and Levin 2001). The framework that we present uses a simple representation of
movement: diffusion. Diffusion rates previously have
been approximated with mean displacement data from
mark–recapture studies (e.g., Andow et al. 1990, Veit
and Lewis 1996), and expert opinions are commonly
used to quantify mobility of large groups of species (e.g.,
Burke et al. 2011). Although diffusion does not prohibit
long-distance dispersal, it may not capture the frequency
of long-dispersal events in some species (e.g., Byasa
impediens; Li et al. 2013). Consequently, if model
predictions do not fit empirical observations of range
826
Ecological Applications
Vol. 23, No. 4
SHAWN J. LEROUX ET AL.
shift, more complex dispersal kernels (e.g., power law or
Cauchy distribution) or population dynamics data may
be needed to adequately capture range dynamics (Marco
et al. 2011). In a recent meta-analysis of dispersal in
butterflies, Stevens et al. (2010) found that dispersal
estimates made from multisite mark–recapture experiments, genetic studies, experimental assessments, expert
opinions, and transect surveys generally converged.
Comparative studies of this nature are sorely needed
for other taxa and will prove invaluable for testing
theoretical predictions of reaction–diffusion models.
CONCLUSION
Global climate change is a major threat to biodiversity (Fischlin et al. 2007, Leadley et al. 2010). In
response to changing climate, species can move or
disperse to keep pace with their preferred climatic
conditions, or acclimatize or evolve adaptations to
novel climatic conditions (Angert et al. 2011, Bellard
et al. 2012). Species that cannot shift their range or
adapt fast enough will be at risk of extinction (Thomas
et al. 2004, Visser 2008). The velocity of climate change
and species traits will determine which strategy species
can adopt and the ultimate fate of the species.
At expanding climate fronts, colonization rates are
determined by rates of reproduction, dispersal, and
adaptation (Gaston 2009, Chevin et al. 2010, Angert et
al. 2011), but current methods for predicting the
response of biodiversity to changing climate fronts
(e.g., correlative and mechanistic species distribution
models) do not explicitly quantify these dynamic
processes. Consequently, these methods may be better
suited for investigating large-scale changes in species
distributions than for predicting the persistence of
species under global change (Chevin et al. 2010). We
present a predictive theoretical framework that explicitly
accounts for the key processes of reproduction and
dispersal in biodiversity responses to climate change. We
develop methods for estimating the parameters of this
model and provide an empirical estimation of the main
prediction of this theory for 12 North American
butterfly species. Similar theory has been successfully
developed and applied in invasion ecology (reviewed in
Hastings et al. 2005), and we believe that global change
biology will benefit by adopting such a theoretical
framework at this important juncture of the field. Paired
with correlative distribution models and other mechanistic models, this new theory can help in the development of more effective conservation strategies to
mitigate losses of biodiversity from global climate
change.
ACKNOWLEDGMENTS
S. J. Leroux was supported by a PDF from the Natural
Sciences and Engineering Research Council of Canada
(NSERC). M. Larrivée was supported by a PDF from the
Fonds Québecois de Recherche du Québec–Nature et
Technologies. J. T. Kerr and F. Lutscher were supported by a
Discovery Grant from NSERC, and V. Boucher-Lalonde was
supported by a doctoral scholarship from NSERC. J. T. Kerr
also was supported by infrastructure from the Canadian
Foundation for Innovation and Ontario Ministry of Research
and Innovation. We thank R. Layberry for graciously sharing
his butterfly abundance data, and D. Currie, members of the
Kerr and Currie labs, and anonymous reviewers for comments
on this work.
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