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Ecological Applications, 23(4), 2013, pp. 815–828� 2013 by the Ecological Society of America

Mechanistic models for the spatial spread of speciesunder climate change

SHAWN J. LEROUX,1,4 MAXIM LARRIVEE,1,5 VERONIQUE BOUCHER-LALONDE,1 AMY HURFORD,2,6 JUAN ZULOAGA,1

JEREMY T. KERR,1 AND FRITHJOF LUTSCHER3

1Canadian Facility for Ecoinformatics Research, Department of Biology, University of Ottawa, 30 Marie Curie,Ottawa, Ontario K1N 6N5 Canada

2MPrime Centre for Disease Modelling, YIHR 5021 TEL Building, York University, 4700 Keele Street,Toronto, Ontario M3J 1P3 Canada

3Department 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 commonmethods for predicting the response of biodiversity to changing climate do not explicitlyincorporate fundamental evolutionary and ecological processes that determine speciesresponses to changing climate, such as reproduction, dispersal, and adaptation. We providean overview of an emerging mechanistic spatial theory of species range shifts under climatechange. This theoretical framework explicitly defines the ecological processes that contributeto species range shifts via biologically meaningful dispersal, reproductive, and climateenvelope parameters. We present methods for estimating the parameters of the model withwidely available species occurrence and abundance data and then apply these methods toempirical data for 12 North American butterfly species to illustrate the potential use of thetheory for global change biology. The model predicts species persistence in light of currentclimate change and habitat loss. On average, we estimate that the climate envelopes of ourstudy species are shifting north at a rate of 3.25 6 1.36 km/yr (mean 6 SD) and that our studyspecies produce 3.46 6 1.39 (mean 6 SD) viable offspring per individual per year. Based onour parameter estimates, we are able to predict the relative risk of our 12 study species forlagging behind changing climate. This theoretical framework improves predictions of globalchange outcomes by facilitating the development and testing of hypotheses, providingmechanistic predictions of current and future range dynamics, and encouraging the adaptiveintegration of theory and data. The theory is ripe for future developments such as theincorporation 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 conservationstrategies 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

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

Manuscript received 15 August 2012; revised 28 November2012; accepted 2 January 2013. Corresponding Editor: J.Franklin.

4 Present address: Department of Biology, MemorialUniversity of Newfoundland, 232 Elizabeth Ave, St John’s,Newfoundland A1B 3X9 Canada. E-mail: [email protected]

5 Present address: Montreal Insectarium, 4581 Rue Sher-brooke Est, Montreal, Quebec H1X 2B2 Canada.

6 Present address: Department of Biology, MemorialUniversity of Newfoundland, 232 Elizabeth Ave., St John’s,Newfoundland A1B 3X9 Canada.

815

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 devel-

oped 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 (Phil-

lips 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, Araujo 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 theo-

ry 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 persis-

tence, because persistence also is dependent on species-

specific 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 mechanis-

tic, 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

UNDER CLIMATE CHANGE

Reaction–diffusion equations have been used exten-

sively 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:

]

]tu ¼ D

]2

]x2uþ ru: ð1Þ

In invasion ecology, this equation can be used to predict

the speed of spatial spread of a locally introduced

SHAWN J. LEROUX ET AL.816 Ecological ApplicationsVol. 23, No. 4

species in an unbounded homogeneous landscape as c*¼2ffiffiffiffiffiffiDrp

(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. This setup gives a

critical patch size of Lc ¼ pffiffiffiffiffiffiffiffiD=r

p(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, Pota-

pov 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

LcðqÞ ¼ pffiffiffiffiffiffiffiffiD=r

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q

2ffiffiffiffiffiffiDrp

r� ��1

ð2Þ

provided

q , c� ¼ 2ffiffiffiffiffiffiDrp

: ð3Þ

For q¼ 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

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 will not keep up with changing

climate if q . c* ¼ 2ffiffiffiffiffiffiDrp

.

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 persis-

tence in these models and arrived at qualitatively similar

results.

In summary, this theory predicts that if q . 2ffiffiffiffiffiffiDrp

, a

population cannot keep up with changing climate and

will eventually go extinct. If q , 2ffiffiffiffiffiffiDrp

, 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 (Grosh-

olz 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

TABLE 1. Reaction–diffusion model variables/parameters, definitions, and units for the spatial spread of species under changingclimate.

Variable or parameter Definition Units

u population density no. individuals/km2

x space kmt time yearsD diffusion rate km2/yearr per capita growth rate no. individuals/yearq climate envelope movement rate km/yearL bounded habitat domain kmLc critical patch size; p

ffiffiffiffiffiffiffiffiD=r

pkm

Lc(q) critical patch size with moving temperature isoclines; pffiffiffiffiffiffiffiffiD=r

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q

2ffiffiffiffiffiffiDrp

r !�1

km

c* speed of spatial spread of a locally introduced species; 2ffiffiffiffiffiffiDrp

km/year

Dc threshold value of D for a species to keep pace with changing climate; q2/4r km2/year

June 2013 817SPECIES RANGE SHIFT UNDER CLIMATE CHANGE

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 estimat-

ing 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 Nympha-

lidae, 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 (SD¼ 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 season-

ality of precipitation as our environmental predictors.

These variables reflect previously documented environ-

mental 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

FIG. 1. The Potapov and Lewis (2004) modelfor the spatial spread of species under climatechange 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 climatechange. The size of the suitable climate enveloperemains constant over time.


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 species-

specific 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 R2 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

(SD¼ 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 butter-

fly expert and lead author of The Butterflies of Canada

(Layberry et al. 1998). Since 1989, Layberry has

FIG. 2. Study area in Canada and butterfly species richness (number of species) based on occurrence records and baselineMaxent model predictions for the period 1960–1970. The study area includes all ecozones of the Canadian mainland east of theCanadian Rockies and south of the Northern Arctic.


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

Ntþ1 ¼ Nterð1�Nt=KÞþeð0;r2Þ 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.

FIG. 3. Methods for estimating the rate of northern shift of the species-specific climate envelope, q. (a) We use occurrencerecords for each butterfly species from 1960–1970 to build a baseline map of the species distribution. Data shown here and in otherpanels 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 distancebetween the climate envelope in 1960–1970 and the range in 2001–2005, we convert the northern edge of 1960–1970 to points andcalculate the mean distance (di ) between each point from 1960–1970 to the nearest point on the northern edge of 2001–2005.


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:

lnNtþ1

Nt

� �¼ r 1� Nt

K

� �þ eð0;r2Þ: ð5Þ

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 (SD¼ 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

TABLE 2. Results of linear regression models of cumulativespread (km) vs. time with q as the slope of this linearregression, for 12 North American butterfly species.

Species, by family qLower95% CL

Upper95% CL R2

Lycaenidae

Callophrys niphon 5.51 0.75 10.27 0.64Celastrina lucia 1.24 0 10.47 0.02Glaucopsyche lygdamus 4.22 0.62 7.82 0.65

Nymphalidae

Danaus plexippus 2.97 0 7.22 0.39Enodia anthedon 3.57 0 7.64 0.50Limenitis arthemis 1.79 0 5.51 0.24Phyciodes cocyta 3.12 0 6.41 0.54Polygonia comma 3.70 0 9.07 0.39

Papilionidae

Papilio canadensis 1.14 0 6.12 0.07

Pieridae

Colias philodice 3.62 0 10.91 0.25Pieris oleracea 3.13 0 10.76 0.18Pieris rapae 5.04 0 11.31 0.46

Notes: Negative values for lower 95% confidence limits (CL)were replaced with zero because zero represents the case of nonorthern shift in the climate envelope. Cumulative spread is thedistance 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 anillustration of methods for calculating q, and Fig. 4 forrepresentation of data and regression line fits.

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) fitfor 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.


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 atcomparing 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 (median¼�1; Fig. 5b). Five species (Callophrys niphon, Glauco-

psyche lygdamus, Polygonia comma, Pieris oleracea,

Enodia anthedon) differed in their relative rank by �3or 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 ¼q2/4r, 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

TABLE 3. Number of years of continuous abundance data (n)and maximum-likelihood estimates for population-growth-rate parameter r (with 95% confidence limits) for 12 NorthAmerican butterfly species.

Species, by family n rLower95% CL

Upper95% CL

Lycaenidae

Callophrys niphon 13 0.86 0.12 1.59Celastrina lucia 20 0.95 0.26 1.63Glaucopsyche lygdamus 9 1.72 0.90 2.54

Nymphalidae

Danaus plexippus 9 1.35 0.13 2.59Enodia anthedon 12 1.40 0.15 2.66Limenitis arthemis 10 1.15 0.27 2.04Phyciodes cocyta 21 1.06 0.34 1.78Polygonia comma 12 0.69 0.23 1.14

Papilionidae

Papilio canadensis 13 1.56 0.67 2.45

Pieridae

Colias philodice 21 1.48 0.71 2.25Pieris oleracea 17 0.99 0.36 1.62Pieris rapae 12 1.62 0.92 2.32

7 http://www.gbif.org8 http://www3.imperial.ac.uk/cpb/databases/gpdd9 http://www.worldclim.org/10 http://bioval.jrc.ec.europa.eu/products/glc2000/glc2000.

php


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 incorpo-

rate key ecological and evolutionary processes (e.g.,

dispersal) that determine the ability of a species to

respond to environmental change a priori. Consequent-

ly, 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, Araujo 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 predic-

tion.

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 incorpo-

rated in the models. As previously stated, global change

TABLE 4. Dc estimates and mean mobility index scores (Burke et al. 2011) for 12 North Americanbutterfly species.

Species, by family

Our critical Dc estimate Burke mobility index

Dc Lower Dc Upper Dc Rank Mean Rank

Lycaenidae

Callophrys niphon 8.83 4.77 63.25 12 4.20 2Celastrina lucia 0.40 0.24 1.48 2 3.71 1Glaucopsyche lygdamus 2.59 1.75 4.95 9 5.37 5

Nymphalidae

Danaus plexippus 1.63 0.85 16.96 4 9.50 12Enodia anthedon 2.28 1.20 21.24 6 5.12 3Limenitis arthemis 0.70 0.39 2.97 3 6.97 8Phyciodes cocyta 2.30 1.37 7.16 7 5.43 6Polygonia comma 4.96 3.00 14.88 11 6.64 7

Papilionidae

Papilio canadensis 0.21 0.13 0.48 1 7.79 11

Pieridae

Colias philodice 2.21 1.46 4.61 5 7.33 9Pieris oleracea 2.47 1.51 6.80 8 5.36 4Pieris rapae 3.92 2.74 6.90 10 7.56 10

Notes: Mean Dc estimates are based on our mean estimates of q and r, whereas the upper andlower 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 12butterfly species are derived from Burke et al. (2011). We report the relative ranking of each speciesaccording to our estimates of Dc and the mobility index scores of Burke et al. (2011). Species thatrank relatively higher on the Dc scale than on the mobility index may be most at risk of not keepingpace with changing 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 hetero-

geneity is apparent in many natural communities due to

consumer–resource distributions, habitat quality differ-

ences, 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–diffu-

sion model of an invasive species by allowing periodic

variation in dispersal and reproductive rates, and

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 assignedby naturalists (i.e., realized mobility; see Table 4 and Burke et al. [2011]) and the relative rank of these same species based on ourmean Dc estimates (i.e., predicted mobility). Larger negative differences may indicate species that are more at risk of not keepingpace with climate change, whereas larger positive differences may indicate species that are more likely to keep pace with climate.

FIG. 6. Potapov and Lewis (2004) show thatto persist a species must keep pace with climatechange and the length of available habitat mustbe sufficient. Eqs. 2 and 3 define the relative risksof extinction due to climate change (dark gray)and insufficient habitat (light gray) as theydepend on the species’ reproduction (r) andclimate envelope shift rate (q) estimates. Thefigure is parameterized for Phyciodes cocyta andsuggests that if Phyciodes cocyta can trackclimate change, it will probably persist becausethe habitat requirements for D . 2.30 aremodest.


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 environ-

ment, 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 represent-

ed 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 un-

changed 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

Duputie 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, Garcıa-Ramos

and Rodrıguez (2002) model the influence of local

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 develop-

ments 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., Araujo 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 precipita-

tion) 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 distribu-

tion 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 whole-

community responses to climate change. With theoret-

ical 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 frame-

work 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


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 experi-

ments, 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 biodiver-

sity (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 mecha-

nistic models, this new theory can help in the develop-

ment 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 NaturalSciences and Engineering Research Council of Canada(NSERC). M. Larrivee was supported by a PDF from theFonds Quebecois de Recherche du Quebec–Nature etTechnologies. J. T. Kerr and F. Lutscher were supported by aDiscovery Grant from NSERC, and V. Boucher-Lalonde was

supported by a doctoral scholarship from NSERC. J. T. Kerralso was supported by infrastructure from the CanadianFoundation for Innovation and Ontario Ministry of Researchand Innovation. We thank R. Layberry for graciously sharinghis butterfly abundance data, and D. Currie, members of theKerr and Currie labs, and anonymous reviewers for commentson this work.

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