Leroux et al.
Species range shift under climate change
Mechanistic models for the spatial spread of species under climate change
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Shawn J. Leroux1,2, Maxim Larrivée1,3, Véronique Boucher-Lalonde1,4, Amy Hurford5, Juan
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Zuloaga1,6, Jeremy T. Kerr1,7, Frithjof Lutscher8
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30 Marie Curie, Ottawa, ON, Canada, K1N 6N5.
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Newfoundland, 232 Elizabeth ave, St John’s, NL, Canada, A1B 3X9. Tel. (709) 864-3042, Fax
Canadian Facility for Ecoinformatics Research, Department of Biology, University of Ottawa,
corresponding author current address: Department of Biology, Memorial University of
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(709) 864-3018, Email:
[email protected], 3current address:
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[email protected], Montreal Insectarium, 4581 rue Sherbrooke Est, Montreal,
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QC, Canada, H1X 2B2,
[email protected],
[email protected], MPrime Centre for
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Disease Modelling, YIHR 5021 TEL Building, York University, 4700 Keele Street, Toronto ON,
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M3J 1P3; current address: Department of Biology, Memorial University of Newfoundland, 232
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Elizabeth ave, St John’s, NL, Canada, A1B 3X9,
[email protected],
[email protected],
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King Edward Ave, Ottawa, ON, Canada, K1N 6N5.
[email protected], Department of Mathematics and Statistics, University of Ottawa, 585
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Running title: Species range shift under climate change
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Type of article: Article
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Abstract word count: 282
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Manuscript word count: 8314
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Number of references: 84
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Number of figures: 6
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Species range shift under climate change
Number of tables: 4
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Species range shift under climate change
ABSTRACT Global climate change is a major threat to biodiversity. The most common methods for
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predicting the response of biodiversity to changing climate do not explicitly incorporate
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fundamental evolutionary and ecological processes that determine species’ responses to changing
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climate such as reproduction, dispersal, and adaptation. We provide an overview of an emerging
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mechanistic spatial theory of species’ range shifts under climate change. This theoretical
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framework explicitly defines the ecological processes that contribute to species range shifts via
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biologically meaningful dispersal, reproductive, and climate envelope parameters. We present
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methods for estimating the parameters of the model with widely available species occurrence and
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abundance data and apply these methods to empirical data for 12 North American butterfly
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species to illustrate the potential use of the theory for global change biology. The model predicts
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species persistence in light of current climate change and habitat loss. On average, we estimate
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the climate envelopes of our study species are shifting north at a rate of 3.25 km/yr (± 1.36
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km/yr) and that our study species produce 3.46 viable offspring per individual per year (± 1.39).
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Based on our parameter estimates, we are able to predict the relative risk of our 12 study species
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lagging behind changing climate. This theoretical framework improves predictions of global
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change outcomes by facilitating the development and testing of hypotheses, providing
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mechanistic predictions of current and future range dynamics and encouraging the adaptive
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integration of theory and data. The theory is ripe for future developments such as the
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incorporation of biotic interactions and evolution of adaptations to novel climatic conditions and
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has the potential to be a catalyst for the development of more effective conservation strategies to
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mitigate losses of biodiversity from global climate change.
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Species range shift under climate change
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Keywords: butterflies, climate change, climate envelope, climate velocity, dispersal, global
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change, intrinsic growth rate, invasive species, mathematical model, mechanistic model, range
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shift, reaction-diffusion
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INTRODUCTION
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Anthropogenic global changes including habitat loss and fragmentation, pollution, exotic
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species invasions, and climate change threaten biodiversity and associated ecosystem services
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(Vitousek 1997, Foley et al. 2005, Kerr et al. 2007). Predicting the response of biodiversity to
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climate change, in particular, has become a burgeoning field of study (Bellard et al. 2012)
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because climate change is emerging as a major threat to biodiversity in the next few decades
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(Thomas et al. 2004, Leadley et al. 2010). The distribution and persistence of many species is
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constrained by climate (Bryant et al. 1997, Hill et al. 2001), and recent species range expansions
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and shifts show patterns consistent with contemporary climate warming (e.g., Parmesan et al.
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1999, Parmesan and Yohe 2003, Root et al. 2003, Chen et al. 2011, DeVictor et al. 2012). In the
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face of changing climate, species may persist by moving or dispersing to track preferred
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conditions (Hickling et al. 2006, Parmesan 2006), demonstrating in situ plastic or acclimatory
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responses to changing climate (Nussey et al. 2005, Durant et al. 2007), or evolving adaptations to
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novel climatic conditions (Visser 2008, Gardner et al. 2009). For example, European bird and
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butterfly communities are moving northward (Devictor et al. 2012) and Dutch great tits (Parus
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major) show plasticity in the timing of reproduction over a 32 year period which is consistent
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with climate change (Nussey et al. 2005). A mechanistic framework for disentangling the role of
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these three main strategies for species responses to climate change will be an invaluable
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predictive tool for global change biology.
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Species range shift under climate change
A number of different approaches have been developed for predicting the response of
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species to global change. Many of these approaches, however, do not explicitly incorporate
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fundamental evolutionary and ecological processes that may determine the ability of a species to
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respond to changing climate such as rates of reproduction, dispersal, and adaptation (Keith et al.
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2008, Kearney and Porter 2009, Buckley et al. 2010, Chevin et al. 2010, Zhou and Kot 2011).
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For example, correlative species' distribution models (e.g., Maxent (Phillips et al. 2006),
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BIOMOD (Thuiller 2003)) relate species’ occurrence records to environmental conditions to
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infer abiotic correlates of a species’ realized niche. Mechanistic distribution models (reviewed in
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Kearney and Porter 2009, Buckley et al. 2010) or habitat suitability models coupled with
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stochastic population models (Keith et al. 2008, Araújo and Peterson 2012) are alternatives to
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correlative models as they relate species processes (e.g., activity levels, survivorship, fecundity,
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etc) to environmental conditions. But, these models require more detailed data than correlative
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models (Keith et al. 2008, Thuiller et al. 2008, Buckley et al. 2010) and it remains unclear
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whether current mechanistic distribution models perform better than correlative models in
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predicting the current and future distribution of species (Kearney and Porter 2009, Morin and
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Thuiller 2009, Buckley et al. 2010). We present a theoretical framework for improving our
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predictions of global change outcomes. This framework explicitly defines the ecological
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processes that contribute to species range shifts via biologically meaningful dispersal,
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reproductive, and climate envelope parameters.
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Mathematical biologists have developed spatial theory that has been widely used to
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predict the spread of species invasions (reviewed in Shigasada and Kawasaki 1997, Hastings et
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al. 2005). For example, reaction-diffusion and integro-difference models have been applied to
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predict the spatial spread of a range of taxa including house finches (Veit and Lewis 1996), grey
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Species range shift under climate change
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squirrels (Okubo et al. 1989), muskrat (Andow et al. 1990), wolves (Hurford et al. 2006), and
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cabbage white butterflies (Andow et al. 1990). Recognizing that the spatial spread of invasive
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species is a similar mathematical problem as the spatial spread of species in response to changing
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climate, Potapov and Lewis (2004) developed a general mathematical theory of species range
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shifts under changing climate. Their analytical model relates the velocity of a species’ specific
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climate envelope to basic species processes of reproduction and dispersal. Dispersal is a
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fundamental process that can facilitate (or restrict) a species’ range by enabling (or preventing) a
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species to reach suitable sites (Stevens et al. 2010, Boulangeat et al. 2012). Having high vagility,
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however, is not sufficient to guarantee species persistence as persistence also is dependent on
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species-specific growth rates and the speed at which the suitable climate zone is moving. Since
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the initial derivation by Potapov and Lewis (2004), there has been some theoretical development
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(see Roques et al. 2008, Berestycki et al. 2009, Zhou and Kot 2011) but the theory has yet to be
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confronted with empirical data, and methods for empirically estimating parameters of the models
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have not been developed. Our goal is to bridge the gap between the simple analytical predictions
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of Potapov and Lewis (2004) and empirical observations of species spread under climate change.
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We set out to make this theory accessible to ecologists as we believe this framework will help to
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organize the current research agenda, inform data needs and best use practices, and disentangle
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the multiple ways that biodiversity may respond to changing climate.
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Here we provide a brief primer on the use of reaction-diffusion equations in spatial
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ecology and on recent theoretical developments to include climate change into these models.
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Then, we present methods for estimating parameters of a simple reaction-diffusion model with
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changing climate and apply these methods to 12 North American butterfly species. We compare
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our mechanistic, species-specific mobility predictions to realized mobility estimates for our study
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species in order to determine the relative risk of these species lagging behind climate change. We
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end by discussing the advantages and future directions in the development and application of this
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theory to improve predictions of global change outcomes.
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AN OVERVIEW OF SPATIAL THEORY OF SPECIES SPREAD UNDER CLIMATE
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CHANGE
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Reaction-diffusion equations have been used extensively in spatial ecology since the
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seminal papers of Skellam (1951) and Kierstead and Slobodkin (1953). When applied to climate
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change, reaction-diffusion equations allow us to derive conditions for a species to keep pace with
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changing climate (Pease et al. 1989, Potapov and Lewis 2004, Berestycki et al. 2009, Chevin et
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al. 2010). These conditions depend on the speed at which a species can move and the minimum
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patch size necessary for it to persist. We present both of these properties below.
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A reaction-diffusion equation describes the change in the density of a population (u(t,x))
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through time (t) and space (x). In the simplest case, individuals move randomly in one-
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dimensional space with diffusion rate D and reproduce at a constant per capita rate r. The
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corresponding equation reads (for variables, parameters, and their units, see Table 1)
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2 u D 2 u ru t x
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(1)
In invasion ecology, this equation can be used to predict the speed of spatial spread of a
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locally introduced species in an unbounded homogeneous landscape as c * 2 Dr (reviewed in
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Shigesada and Kawasaki 1997, Okubo and Levin 2001, Hastings et al. 2005).
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In conservation biology, one can predict the minimal size required for a certain habitat to
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support a given species. One assumes that equation (1) holds on a bounded domain of length L,
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and, as a worst case, that the surroundings are completely hostile. This setup gives a critical
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patch size of Lc D /r (Skellam 1951, Kierstead and Slobodkin 1953). Dispersal can induce 7
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Species range shift under climate change
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loss from a given patch. If the patch is small and the surroundings are hostile, then this dispersal-
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induced loss can cause population decline and eventual extinction (Perry 2005, Kenkre and
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Kumar 2008).
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Recently, theoreticians have begun to consider the effect of global change on the critical
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patch size in this reaction-diffusion framework (Pease et al. 1989, Potapov and Lewis 2004,
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Berestycki et al. 2009, Chevin et al. 2010). Potapov and Lewis (2004) implemented the effects of
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a latitudinal shift of temperature isoclines by considering the x-axis as a north-south section
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through the landscape. They assumed that the species’ growth rate is positive in some patch [x1,
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x2] of length L and negative outside. They furthermore assumed that the boundaries of the
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favorable patch move northward with constant speed q, i.e., xi(t) = xi,0 + qt (Fig. 1) so that, the
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size of the patch remains constant over time. Parameter q represents the rate of movement of a
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species’ climate envelope. In the special case that the environment outside the favorable patch is
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completely hostile, the critical patch size with moving temperature isoclines is (Potapov and
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Lewis 2004)
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1 q Lc (q) D /r 1 2 Dr
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provided
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q c* 2 Dr .
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(2)
(3)
For q = 0, we recover the critical patch size from above. If the climate envelope moves
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more quickly, as q approaches c* from below, the critical patch size increases nonlinearly to
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infinity. When the speed of the temperature isoclines (q) is faster than the spread rate of the
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population in a homogeneous landscape (c*), the population will not persist in any patch of finite
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size. Formally, a population will not keep up with changing climate if q c* 2 Dr .
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Species range shift under climate change
When the conditions outside of the favorable patch are not completely hostile (i.e., the
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population has a finite death rate) then the explicit expression for the critical patch size is more
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cumbersome (see Potapov and Lewis 2004). However, the key model prediction that the
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population cannot keep up with changing climate on any patch of finite size when q > c* still
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holds.
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There is a parallel body of literature on mathematical models for the spread of
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populations with discrete, non-overlapping generations, so-called integro-difference equations
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(e.g., Kot and Schaffer 1986, Kot et al. 1996). Integro-difference equations also can
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accommodate more detailed distribution of dispersal distances; a key trait when dealing with
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species with frequent long-distance dispersal events. Zhou and Kot (2011) investigated the
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effects of shifting climate zones on population persistence in these models and arrived at
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qualitatively similar results.
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In summary, this theory predicts that if q 2 Dr a population cannot keep up with
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changing climate and will eventually go extinct. If q 2 Dr then the population can persist
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provided its favorable habitat is large enough.
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To apply this theory, we must estimate three parameters (i.e., r, D, q). Estimates for
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dispersal, and D in particular, are notoriously difficult to come by (Grosholtz 1996) because D
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estimates usually require detailed multisite mark-recapture studies (see Stevens et al. 2010 for a
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review of methods for quantifying butterfly dispersal). However, we can use existing data to
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obtain estimates for the population growth rate, r, and the climate envelope movement rate, q,
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and then find a threshold value for D, Dc. A species will be able to keep pace with climate if D >
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Dc. After re-arranging eqn 3, we find Dc
q2 . This elegant theoretical prediction allows 4r
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empiricists to readily test the influence of different processes (i.e., reproduction (r), movement
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(D), climate change (q)) on species persistence in light of climate change depending on the data
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that is available to them. Next, we illustrate how this theory can be used to predict the relative
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ability of 12 North American butterfly species to keep pace with changing climate.
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AN APPLICATION OF THE THEORY
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Empirical evidence shows many species expanding or shifting their ranges in the
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direction of changing climate, however, many of these species are not actually keeping pace with
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the velocity of climate change (Loarie et al. 2009, Devictor et al. 2012). The dynamics of
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species’ ranges during a period of climate change are determined by a suite of ecological and
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evolutionary processes such as rates of reproduction and dispersal (Gaston 2009, Atkins and
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Travis 2010, Angert et al. 2011) but much of the empirical evidence of expanding ranges does
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not quantify the role of reproduction or dispersal. Predicting which species are at a higher risk of
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lagging behind climate change is critical for identifying future risks, supporting development of
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proactive strategies to reduce climate change impacts on biodiversity and prioritizing policy
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initiatives (Bellard et al. 2012). The body of theory we have summarized in the previous section
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predicts that D Dc
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q2 words, a species must have a movement rate above a certain threshold (i.e. ) in order to keep 4r
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pace with changing climate. Here, we derive methods for estimating q and r parameters for a
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suite of North American butterfly species. Then we use these estimates to calculate the threshold
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D values for these species, which enables us to determine the ability of each species to track
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climate. Our analysis was conducted on all ecozones of the Canadian mainland east of the
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Canadian Rockies and south of the Northern Arctic (Fig. 2). We excluded ecozones with high
q2 is necessary for a population to keep up with climate change. In other 4r
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Species range shift under climate change
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elevations from our study area because these areas are highly heterogeneous on small scales and
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therefore do not match out model assumptions.
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Estimating the climate envelope movement rate, q
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We obtained occurrence data for 12 butterfly species, 3 species of Lycaenidae, 5 species
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of Nymphalidae, 1 species of Papilionidae, and 3 species of Pieridae butterflies from the
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Canadian Biodiversity Information Facility. This database contains ~ 300,000 precisely
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georeferenced dated records for 297 Canadian butterfly species (Layberry et al. 1998) from
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specimens stored at one of many museums across Canada. See Kharouba et al. (2009) for more
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details on these data. These 12 species were the only ones for which we could obtain both
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occurrence and abundance data. We had a mean of 106 geographically unique occurrence
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records for the time period 1960-1970 per study species (sd = 59). Phenological and range-shift
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responses for species in North America are predominantly subsequent to 1970 (Parmesan 2006),
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as is directional climate change, which is very likely to be attributable to human activities
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(Hansen et al. 1999), so the period between 1960-1970 was used as a historical baseline in which
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to construct species climate envelope models.
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We used Maxent (Phillips et al. 2006) to model a potential climate envelope for each
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species based on 1960-1970 occurrence records. Maxent predicts where a species may be found
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across geographical space derived from its occurrence records relative to environmental
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predictors. We used minimum winter temperature, mean summer temperature, annual
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precipitation and seasonality of precipitation as our environmental predictors. These variables
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reflect previously documented environmental limits for butterflies in this region (Kharouba et al.
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2009). Climate observations were constructed using ANUSPLIN, a regression splines
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interpolation, across all available weather station data for North America (McKenney et al.
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Species range shift under climate change
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2006). Data are available at 10 arc minute resolution annually from 1961 to 2006. These data
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were developed at the Canadian Forest Service and are used in climate reporting by the
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Government of Canada. We projected species climatic envelopes through time based on 5 year
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climate normals (i.e., 1971-1975, 1976-1980, etc.). A mean projected suitability envelope was
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produced based on 10 iterations of the model to derive the final output.
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For each model, probability of occurrence was converted into a binary map of areas
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predicted to be part of the species’ range (i.e., suitable) and outside the species’ range (i.e.,
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unsuitable). The threshold suitability value was calculated by taking the average of the lowest 10
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predicted suitability values of the true presences used to test the 10 model iterations for the
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baseline 1960-1970 model (see methods in Liu et al. 2005, Kharouba et al. 2009). This
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thresholded output provides a species-specific estimate of the potential climate envelope for each
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5 year period. This method assumes that the species is in equilibrium with climate and that data
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collected between 1960-1970 are representative of the species climate niche prior to significant
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climate change.
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The extent of our data does not cover the full climate envelope of each species but rather
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the northern portion of its range. Consequently, we estimate q as the expansion of the northern
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climate envelope edge in Canada. For each species, we extracted the northern climate envelope
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edge of contiguous range patches (i.e., we excluded range “islands” distant from the main
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predicted range) for each 5 year climate envelope period (Fig. 3). We calculated the mean
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distance from the full length (i.e., east-west) of the 1960-1970 pre-climate change baseline
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climate envelope edge to the full length of the climate envelope edge of each 5 year period (i.e.,
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distance from 1960-1970 to 1971-1975, from 1960-1970 to 1976-1980, etc., Fig. 3). Once the
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northern edge of a climate envelope hit a coastal boundary (e.g., Hudson Bay), we excluded all
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Species range shift under climate change
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further points along this boundary from our q calculations. We excluded these points because the
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climate envelope of terrestrial species is bounded by such physical boundaries. Their inclusion
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would therefore systematically underestimate the true climatic shift, q that affects species. We
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estimated q as the slope of a linear regression of cumulative distance between climate envelope
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edges (km) vs time.
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The hotspots of predicted species range overlap in 1960-1970 for our sample of species
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occurs in southern Ontario and Manitoba, south-eastern Ontario and south-western Quebec but
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some species have a predicted 1960-1970 climate envelope as far north as Inuvik, Northwest
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Territories (Fig. 2). For all species, there was high variability in the cumulative distance between
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northern range edges through time with R2 ranging from 0.02 to 0.65 (Table 2). Our estimated q
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ranged from 1.14 km/yr (Papilio canadensis, 95% C.I. 0, 6.12 km/yr) to 5.51 km/yr (Callophrys
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niphon, 95% C.I. 0.75, 10.27 km/yr) with a mean q value of 3.25 km/yr (sd = 1.36 km/yr, Table
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2, Fig. 4). The lower 95% C.I. estimate for 10/12 species was zero indicating the case of no
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northern shift in these species’ climate envelope. These results predict that, on average, the
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climate envelopes of our study species are shifting at a rate of 3.25 km/yr (± 1.36 km/yr).
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Estimating the per capita growth rate, r
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We obtained abundance time series data for our 12 butterfly species from Ross Layberry,
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Canadian butterfly expert and lead author of The Butterflies of Canada (Layberry et al. 1998).
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Since 1989, Layberry intensively sampled a 300 ha patch of mixed-wood, open habitat in Eastern
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Ontario, Canada several times during the butterfly flight season and recorded species identity and
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abundances. We used the maximum abundance estimates per season for every species with at
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least 9 consecutive years of abundance data (mean = 13 years) for estimating the population
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growth rate parameter, r, of our model. All data were collected between 1989 and 2009.
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Species range shift under climate change
We used the Ricker model to estimate the population growth rate, r, of the 12 butterfly
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species (Ricker 1954). The Ricker model is a widely used phenomenological model of
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population dynamics that incorporates density dependence as the mechanism preventing
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unbounded growth (Clark et al. 2010). We used a density dependent model because there is
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evidence for density dependent population dynamics in a range of taxonomic groups including
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insects (Brook and Bradshaw 2006). The Ricker model can be written formally as: N r1 t 0, K
2
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N t 1 N t e
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where Nt is population abundance at the current time t. The per capita growth rate at low
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abundance is er, and the population carrying capacity is K. Following Brook and Bradshaw
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(2006) and Clark et al. (2010), we model process error, ε, as normally distributed with zero mean
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and variance, σ2.
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(4)
We estimated r, K, and σ with Maximum Likelihood implemented with the bbmle
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(Bolker 2008) package in R v.2.14.1 (R Development Core Team 2011). We re-arranged eqn 4
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for fitting as follows:
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N N ln t 1 r1 t 0, 2 N t K
290 291 292
(5)
Ninety-five percent confidence intervals for r were calculated directly from the likelihood profiles using the confint function in R. All 12 species’ Maximum Likelihood Ricker model r estimates were identifiable (Table
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3). These r estimates ranged from 0.69 (Polygonia comma, 95% C.I. 0.23, 1.14) to 1.72
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(Glaucopsyche lygdamus, 95% C.I. 0.9, 2.54) with a mean r value of 1.24 (sd = 0.33, Table 3).
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These results suggest that, on average, our study species produce 3.46 viable offspring per
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individual per year (± 1.39).
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Species range shift under climate change
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
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q2 minimum value of D required for a species to track climate change (i.e., Dc ). We calculate 4r
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Dc for our mean estimates of r and q values. An upper confidence interval for Dc was calculated
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with our mean estimate of q and lower 95 % C.I. estimate of r and a lower confidence interval
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for Dc was calculated with our mean estimate of q and upper 95 % C.I. estimate of r (see Tables
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2, 3 for parameter estimates). We ranked species according to their mean calculated Dc values to
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determine their relative movement rate required to keep pace with climate change.
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Our estimates of Dc only provide us with a prediction for how vagile a species must be to
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track climate change, they do not tell us anything about the actual mobility of a species. To
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determine the relative ability of each species to track climate change we must compare the
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predicted Dc to an actual measure of species mobility. An estimate of actual mobility for our
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study species was obtained from Burke et al. (2011). Burke et al. (2011) asked 51 North
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American lepidopterists to score Canadian butterfly species on their mobility from 0 (sedentary)
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to 10 (extremely mobile). They summarized the mean scores for the group of experts into a
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relative mobility index for 297 butterfly species in Canada. Expert opinions may reflect
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migration propensity of butterflies instead of realized dispersal (Stevens et al. 2010) but these
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data represent the best available mobility data for our study species. We ranked our species
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according to their mobility index score and calculated the difference between the mobility index
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score rank and the critical Dc estimate rank. We present this simple rank difference method as a
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first pass at comparing predicted vs observed butterfly dispersal abilities. Future comparisons
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should use empirical dispersal data where available.
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Species range shift under climate change
Mean Dc ranged from 0.21 km2/yr (Papilio canadensis, C.I. 0.13, 0.48 km2/yr) to 8.83
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km2/yr (Callophrys niphon, C.I. 4.77, 63.25 km2/yr) with a grand mean Dc value across all
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species of 2.71 km2/yr (sd = 2.36 km2/yr, Table 4, Fig. 5a). Species in the family Pieridae
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require relatively high Dc to keep pace with climate change. Burke et al. (2011) mobility index
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scores ranged from 3.71 (Celastrina lucia) to 9.50 (Danaus plexippus) for our study species
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(Table 4) with a mean mobility index score of 6.25 (sd = 1.67, Table 4).
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The difference in the mobility index relative rank and the Dc value relative rank ranged
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from -10 to 10 (median = -1, Fig. 5b). Five species (Callophrys niphon, Glaucopsyche lygdamus,
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Polygonia comma, Pieris oleracea, Enodia anthedon) differed in their relative rank by -3 or less,
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3 species (Celastrina lucia, Phyciodes cocyta, Pieris rapae) differed in their relative ranking by -
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1 to 1 and four species (Colias philodice, Limenitis arthemis, Danaus plexippus, Papilio
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canadensis) ranked relatively higher on the mobility index than critical Dc scale (Fig. 5b).
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Andow et al. (1990) estimated a diffusion coefficient for Pieris rapae between 4.8 and
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129 km2/yr based on mark-recapture data collected by Jones et al. (1980). Our Dc estimate for
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Pieris rapae ranges between 2.74 and 6.90 km2/yr which falls in the lower range of the realized
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mobility estimate from Andow et al. (1990).
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IMPROVING PREDICTIONS OF GLOBAL CHANGE OUTCOMES
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Potapov and Lewis (2004) developed a framework for a general mathematical theory of
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species range shifts under changing climate based on reaction-diffusion models for invasive
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species. The main prediction of this theory relates the velocity of climate (q) to species’
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q2 a population is at risk of not keeping track reproduction (r) and diffusion (D); if D Dc 4r
340
with changing climate and will eventually go extinct. We provide a road map for the application
341
of this theory by presenting methods for estimating the parameters of this model and applying 16
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Species range shift under climate change
342
these methods to parameterize the model for 12 North American butterfly species. The
343
application of this theory allowed us to identify the relative risk of 12 butterfly species not
344
keeping pace with climate change.
345
Global change biologists have assembled extensive large-scale data on species
346
distribution (e.g., Global Biodiversity Information Facility) and abundance (e.g., Global
347
Population Dynamics Database) as well as global climate (e.g., WorldClim) and land cover data
348
(e.g., Global Landcover 2000). As we have shown here, these data correspond to parameters that
349
have been defined in theoretical ecology and can be put to good use testing theoretical
350
predictions on the spatial spread of species under changing climate.
351
The advantages of adopting a theoretical framework in global change biology are many.
352
First, a formal mathematical theory will facilitate testing existing hypotheses and generating
353
novel ones on the conditions that allow biodiversity to persist in the face of environmental
354
change. For example, we might derive competing models of species dynamics in light of the
355
processes of habitat loss and climate change and confront the models with empirical data to
356
determine the relative role of habitat loss and changing climate on species persistence (Warren et
357
al. 2001, Thullier et al. 2008). In fact, Lewis and Popatov (2004) organize and relate species
358
extinction risk due to habitat loss and climate change through their common dependence on
359
species’ reproduction, dispersal and climate. The results of Potapov and Lewis (2004) emphasize
360
that tracking climate change alone does not guarantee that a species’ will thrive, as persistence
361
depends also on the size of available habitat (eqn 2, Fig. 6). With sufficient data, our theoretical
362
framework allows one to quantify the relative risk of species extinction due to either insufficient
363
habitat and/or inability to keep pace with climate change (see Fig. 6 for an example for
17
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Species range shift under climate change
364
Phyciodes cocyta). In essence, the theory can become an organizing and predictive framework in
365
the quickly emerging field of global change biology.
366
Second, mechanistic mathematical models incorporate key ecological and evolutionary
367
processes (e.g., dispersal) that determine the ability of a species to respond to environmental
368
change a priori. Consequently, these models should better predict range dynamics under future
369
environmental projections than would a purely correlative model (Keith et al. 2008, Buckley et
370
al. 2010, Chevin et al. 2010, Araújo and Peterson 2012). Approaches that neglect dispersal (e.g.,
371
correlative species distribution models) may overestimate species persistence under changing
372
climate (Zhou and Kot 2011). Consequently, explicitly stating how the processes of reproduction
373
and dispersal combine to determine species persistence may be critical for accurate prediction.
374
Third, a formal mathematical framework provides analytical solutions and thresholds that
375
can be used to predict past, present, and future species responses to changing climate. Analytical
376
solutions identify key variables to empirically measure, which encourages feedback between the
377
model formulation and data. Continually confronting our models with empirical data in an
378
adaptive process is a necessary reality check to identify competing hypotheses which are most
379
consistent with empirical data, highlight data needs and future theoretical developments and
380
ultimately lead to better predictions of range shifts and extinction risks under environmental
381
change (Sexton et al. 2009).
382
LIMITATIONS OF THE THEORY
383
Current models of species spread under climate change require a number of simplifying
384
assumptions. The theoretical predictions of these models should be confronted with empirical
385
data from a range of ecosystems and taxa in order to determine to what extent species spread
386
rates under climate change can be captured by the simple mechanisms currently incorporated in
18
Leroux et al.
Species range shift under climate change
387
the models. As stated above, global change biologists have access to extensive data sets that
388
could be used to test model predictions and to determine the validity of model assumptions. Here
389
we outline a few simplifying assumptions that could be relaxed in order to improve predictions
390
of species range shifts under changing climate.
391
The model formulation we present, assumes that a uniformly suitable patch of constant
392
size moves in an otherwise hostile environment. This formulation is most useful for investigating
393
latitudinal climate change over relatively flat terrain. The assumption of a uniformly suitable
394
patch is limiting as small-scale spatial heterogeneity is apparent in many natural communities
395
due to consumer-resource distributions, habitat quality differences, and elevational gradients
396
(Pickett and Cadenasso 1995). Furthermore, matrix habitat outside the patch need not be
397
uniformly hostile. Shigesada et al. (1986) introduced habitat heterogeneity into a reaction-
398
diffusion model of an invasive species by allowing periodic variation in dispersal and
399
reproductive rates, and Lutscher and Seo (2011) investigated the persistence of invasive species
400
in a seasonal river environment. Interestingly, the rate of spread of invasive species is determined
401
by the harmonic mean of the diffusion constant and the arithmetic mean of the growth rates in
402
different environments (Shigesada et al. 1986, Shigesada and Kawasaki 1997). In a temporally
403
varying environment, the spread rate is given by the arithmetic means of D and r (Lutscher and
404
Seo 2011). We modeled population dynamics as a continuous process, whereas population
405
dynamics of insects may be better represented with distinct growth and dispersal stages. Discrete
406
integro-difference models likely capture the population dynamics and dispersal of butterflies
407
better (Kot et al. 1996, Zhou and Kot 2011) but these models require more and greater resolution
408
data to parameterize (but see Clark et al. 2001) and they may arrive at qualitatively similar
409
results (Zhou and Kot 2011). Our model parameterization focuses on the northern edge of the
19
Leroux et al.
Species range shift under climate change
410
range and assumes that the southern edge is retracting at the same rate as the northern edge
411
expands. While there is some evidence of range retraction in the southern parts of ranges
412
(Parmesan et al. 1999, Kerr 2001), it is largely unknown whether the rate of southern retraction
413
is as fast as the rate of northern expansion. Future developments of the model may consider
414
modeling a flexible habitat patch where northern and southern edges can move at different
415
speeds and future empirical tests of the theory should look at the entire range of a species.
416
Finally, our simple model assumes that dispersal and growth rate remain unchanged as climate
417
changes. There is some evidence for plasticity in movement (e.g., Cormont et al. 2011) and
418
growth rate (e.g., Boggs and Inouye 2012) of individuals under climate change and this is a key
419
direction for future research.
420
FUTURE DIRECTIONS OF A SPATIAL THEORY OF SPECIES SPREAD UNDER
421
CLIMATE CHANGE
422
Future developments of the theory of species spread under climate change could consider
423
a number of processes not currently incorporated in the initial model formulation of Potapov and
424
Lewis (2004). In particular, the current formulation focuses on a species’ ability to move as its
425
main response to climate change while ignoring the two other main pathways for species
426
responses to climate change, phenotypic plasticity or evolution of adaptations to novel climatic
427
conditions (Atkins and Travis 2010, Angert et al. 2011, Bellard et al. 2012). Recently, a number
428
of developments have been made in modelling adaptations to changing environments. For
429
example, Chevin et al. (2010) and Duputié et al. (2012) offer two modeling approaches for
430
including phenotypic and genetic adaptation of key traits in changing environments. Further
431
developments may integrate these recent efforts with work done on the spread of invasive
432
species, for example, García-Ramos and Rodríguez (2002) model the influence of local
20
Leroux et al.
Species range shift under climate change
433
adaptation on invasion in a spatially heterogeneous environment, and Perkins (2012) models the
434
influence of evolutionary lability in an invasive predator and/or native prey on the speed of the
435
invasion front. A theoretical framework which incorporates trade-offs and interactions between
436
the three main strategies for species to respond to climate change will be an invaluable predictive
437
tool for global change biology (Pease et al. 1989, Chevin et al. 2010, Lavergne et al. 2010). A
438
second class of processes that should be considered in future developments of this theory is
439
species interactions (e.g., competition, predation, mutualism, etc.). Potapov and Lewis (2004) did
440
investigate the dynamics of two competing species under changing climate but there is mounting
441
evidence that consumer-resource interactions and other biotic interactions can influence the
442
outcome of species responses to climate change (e.g., Araujo and Luoto 2007, Suttle et al. 2007
443
and reviewed in Gilman et al. 2010, Lavergne et al. 2010). For example, the long-term response
444
of a northern California grassland food web to simulated climate change (i.e., increased
445
precipitation) can be explained by the lagged effects of altered competitive and trophic
446
interactions (Suttle et al. 2007). What is more, consumers will not be able to track climate if their
447
resources are lagging behind, therefore, it is critical to investigate match/mismatches in the
448
phenology of consumers and resources generated by climate change (Parmesan 2006, Durant et
449
al. 2007). Biotic interactions are often ignored in species distribution modeling (but see
450
Boulangeat et al. 2012) but biotic interactions can easily be included in reaction-diffusion or
451
integro-difference equation models (see Okubo et al. 1989, Potapov and Lewis 2004, Roques et
452
al. 2008 for examples). Further inclusion of biotic interactions into our theoretical framework
453
will be challenging but rewarding as it will facilitate predictions of whole community responses
454
to climate change. With theoretical progress occurring on multiple fronts as outlined above, we
21
Leroux et al.
Species range shift under climate change
455
are making good strides towards achieving a synthetic theory for predicting species responses
456
under changing climate.
457
A future direction for empirical tests of this theory lies in the collection of better
458
movement data for a range of taxa. Even for widely studied species like birds and butterflies, we
459
have a rudimentary understanding of their movement patterns at different spatial scales (Groholtz
460
1996, Okubo and Levin 2001). The framework we present uses a simple representation of
461
movement, diffusion. Diffusion rates have previously been approximated with mean
462
displacement data from mark-recapture studies (e.g., Andow et al. 1990, Veit and Lewis 1996)
463
and expert opinions are commonly used to quantify mobility of large groups of species (e.g.,
464
Burke et al. 2011). While diffusion does not prohibit long distance dispersal, it may not capture
465
the frequency of long dispersal events in some species (e.g. Byasa impediens, Li et al. in press).
466
Consequently, if model predictions do not fit empirical observations of range shift, more
467
complex dispersal kernels (e.g. power law or Cauchy distribution) or population dynamics data
468
may be needed to adequately capture range dynamics (Marco et al. 2011). In a recent meta-
469
analysis of dispersal in butterflies, Stevens et al. (2010) found that dispersal estimates made from
470
multisite mark-recapture experiments, genetic studies, experimental assessments, expert opinions
471
and transect surveys generally converged. Comparative studies of this nature are sorely needed
472
for other taxa and will prove invaluable for testing theoretical predictions of reaction-diffusion
473
models.
474
CONCLUSION
475
Global climate change is a major threat to biodiversity (Fischlin et al. 2007, Leadley et al.
476
2010). In response to changing climate, species can move or disperse to keep pace with their
477
preferred climatic conditions, or acclimatize or evolve adaptations to novel climatic conditions
22
Leroux et al.
Species range shift under climate change
478
(Angert et al. 2011, Bellard et al. 2012). Species that cannot shift their range or adapt fast enough
479
will be at risk of extinction (Thomas et al. 2004, Visser 2008). The velocity of climate change
480
and species’ traits will determine which strategy species can adopt and the ultimate fate of the
481
species.
482
At expanding climate fronts, colonization rates are determined by rates of reproduction,
483
dispersal, and adaptation (Gaston 2009, Chevin et al. 2010, Angert et al. 2011) but current
484
methods for predicting the response of biodiversity to changing climate fronts (e.g., correlative
485
and mechanistic species distribution models) do not explicitly quantified these dynamic
486
processes. Consequently, these methods may be better suited for investigating large-scale
487
changes in species’ distributions than for predicting the persistence of species under global
488
change (Chevin et al. 2010). We present a predictive theoretical framework which explicitly
489
accounts for the key processes of reproduction and dispersal in biodiversity responses to climate
490
change, develop methods for estimating the parameters of this model and provide an empirical
491
estimation of the main prediction of this theory for 12 North American butterfly species. Similar
492
theory has been successfully developed and applied in invasion ecology (reviewed in Hastings et
493
al. 2005) and we believe global change biology will benefit by adopting such a theoretical
494
framework at this important juncture of the field. Paired with correlative distribution models and
495
other mechanistic models, this new theory can help develop more effective conservation
496
strategies to mitigate losses of biodiversity from global climate change.
497
ACKNOWLEDGMENTS
498
SJL was supported by a PDF from the Natural Sciences and Engineering Research
499
Council of Canada (NSERC). ML was supported by a PDF from the Fonds Québecois de
500
Recherche du Québec – Nature et Technologies. JTK and FL were supported by a Discovery
23
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Species range shift under climate change
501
Grant from NSERC and VBL was supported by a doctoral scholarship from NSERC. JTK also
502
was supported by infrastructure from the Canadian Foundation for Innovation and Ontario
503
Ministry of Research and Innovation. We thank David Currie, members of the Kerr and Currie
504
labs and anonymous reviewers for comments on this work.
505
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challenges. Perspectives in Plant Ecology, Evolution and Systematics 9:137-152. Thuiller, W. 2003. BIOMOD: optimizing predictions of species distributions and projecting potential future shift under global change. Global Change Biology 9:1353-1362. Veit, R. R., and M. A. Lewis. 1996. Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion in eastern North America. American Naturalist 148:255-274. Visser, M. E. 2008. Keeping up with a warming world, assessing the rate of adaptation to climate change. Proceedings of the Royal Society of London B 275:649-659. Vitousek, P. M., H. A. Mooney, J. Lubchenco, and J. M. Melillo. 1997. Human domination of Earth's ecosystems. Science 277:494-499. Warren, M. S., et al. 2001. Rapid responses of British butterflies to opposing forces of climate and habitat change. Nature 414:65-68. Zhou, Y., and M. Kot. 2011. Discrete-time growth-dispersal models with shifting species ranges. Theoretical Ecology 4:13-25.
689
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Leroux et al.
Species range shift under climate change
690
Table 1 Reaction-diffusion model variables/parameters, definitions and units for the spatial
691
spread of species under changing climate. Variable/
Definition
Units
u
Population density
Number of individuals/km2
x
Space
km
t
Time
years
D
Diffusion rate
km2/year
r
Per capita growth rate
Number of individuals/year
q
Climate envelope movement rate
km/year
L
Bounded habitat domain
km
Lc
Critical patch size; D / r
km
Lc(q)
Critical patch size with moving temperature
km
parameter
q isoclines; D / r 1 2 Dr c*
1
Speed of spatial spread of a locally introduced
km/year
species; 2 Dr Dc
Threshold value of D for a species to keep pace with km2/year changing climate;
q2 4r
692
33
Leroux et al.
Species range shift under climate change
693
Table 2 Results of linear regression models of cumulative spread (km) vs time. q is the slope of
694
this linear regression. Negative values for lower 95% C.I. were replaced with zero because zero
695
represents the case of no northern shift in the climate envelope. Cumulative spread is distance
696
between northern range edges 1960-1970 (baseline) to successive five year periods (i.e., baseline
697
to 1971-75, baseline to 1976-80, etc.). See Fig. 3 for illustration of methods for calculating q and
698
Fig. 4 for representation of data and regression line fits. q
Lower 95% C.I.
Upper 95% C.I.
R2
Callophrys niphon
5.51
0.75
10.27
0.64
Celastrina lucia
1.24
0
10.47
0.02
Glaucopsyche lygdamus
4.22
0.62
7.82
0.65
Danaus plexippus
2.97
0
7.22
0.39
Enodia anthedon
3.57
0
7.64
0.50
Limenitis arthemis
1.79
0
5.51
0.24
Phyciodes cocyta
3.12
0
6.41
0.54
Polygonia comma
3.70
0
9.07
0.39
1.14
0
6.12
0.07
Colias philodice
3.62
0
10.91
0.25
Pieris oleracea
3.13
0
10.76
0.18
Pieris rapae
5.04
0
11.31
0.46
Species Lycaenidae
Nymphalidae
Papilionidae Papilio canadensis Pieridae
699 34
Leroux et al.
Species range shift under climate change
700
Table 3 Number of years of continuous abundance data (n) and Maximum Likelihood estimates
701
for population growth rate parameter r (+/- 95% confidence intervals) for 12 butterfly species. Species
n
r
Lower 95% C.I.
Upper 95% C.I.
Callophrys niphon
13
0.86
0.12
1.59
Celastrina lucia
20
0.95
0.26
1.63
Glaucopsyche lygdamus
9
1.72
0.90
2.54
Danaus plexippus
9
1.35
0.13
2.59
Enodia anthedon
12
1.40
0.15
2.66
Limenitis arthemis
10
1.15
0.27
2.04
Phyciodes cocyta
21
1.06
0.34
1.78
Polygonia comma
12
0.69
0.23
1.14
13
1.56
0.67
2.45
Colias philodice
21
1.48
0.71
2.25
Pieris oleracea
17
0.99
0.36
1.62
Pieris rapae
12
1.62
0.92
2.32
Lycaenidae
Nymphalidae
Papilionidae Papilio canadensis Pieridae
702 703
35
Leroux et al.
Species range shift under climate change
704
Table 4 Dc estimates and mean mobility index scores (Burke et al. 2011) for 12 North American
705
butterfly species. Mean Dc estimates are based on our mean estimates of q and r, whereas the
706
upper and lower Dc estimates are for our mean C.I. estimate of q and 95% lower and upper C.I.
707
estimates of r, respectively (see Tables 2 and 3 for parameter estimates). Mean mobility index
708
scores for our 12 butterfly species are derived from Burke et al. (2011). We report the relative
709
ranking of each species according to our estimates of Dc and Burke et al. (2011)’s mobility index
710
scores. Species that rank relatively higher on the Dc scale than on the mobility index may be
711
most at risk of not keeping pace with changing climate. Our critical Dc estimate Species
Burke mobility index
Dc
Lower Dc
Upper Dc
Rank
Mean
Rank
Callophrys niphon
8.83
4.77
63.25
12
4.20
2
Celastrina lucia
0.40
0.24
1.48
2
3.71
1
Glaucopsyche lygdamus
2.59
1.75
4.95
9
5.37
5
Danaus plexippus
1.63
0.85
16.96
4
9.50
12
Enodia anthedon
2.28
1.20
21.24
6
5.12
3
Limenitis arthemis
0.70
0.39
2.97
3
6.97
8
Phyciodes cocyta
2.30
1.37
7.16
7
5.43
6
Polygonia comma
4.96
3.00
14.88
11
6.64
7
0.21
0.13
0.48
1
7.79
11
Lycaenidae
Nymphalidae
Papilionidae Papilio canadensis Pieridae
36
Leroux et al.
Species range shift under climate change
Colias philodice
2.21
1.46
4.61
5
7.33
9
Pieris oleracea
2.47
1.51
6.80
8
5.36
4
Pieris rapae
3.92
2.74
6.90
10
7.56
10
712 713
37
Leroux et al.
Species range shift under climate change
714
FIGURE LEGENDS
715
Figure 1 Potapov and Lewis’ (2004) model for the spatial spread of species under climate
716
change models a suitable climate envelope [xi,t, x2,t] of length, L moving with a constant speed, q,
717
which is determined by the velocity of climate change. The size of the suitable climate envelope
718
remains constant over time.
719
Figure 2 Study area in Canada and butterfly species richness based on occurrence records and
720
baseline Maxent model predictions for the period 1960-1970. The study area includes all
721
ecozones of the Canadian mainland east of the Canadian Rockies and south of the Northern
722
Arctic.
723
Figure 3 Methods for estimating the rate of northern shift of the species’ specific climate
724
envelope, q. We use occurrence records for each butterfly species from 1960-1970 to build a
725
baseline map of the species distribution (a). Then we project species distributions through time
726
(5 year time period 2001-2005 shown here) based on a changing climate envelope (b). We
727
extract the northern edge for each time period (c, d). To calculate the distance between the
728
climate envelope in 1960-70 to the range in 2001-05, we convert the northern edge of 1960-70 to
729
points and calculate the mean distance (di) between each point from 1960-70 to the nearest point
730
on the northern edge of 2001-05 (e). Data shown here are for Danaus plexippus.
731
Figure 4 Mean distance (km) between northern range edges 1960-70 (baseline) to 1971-75,
732
1960-70 to 1976-80, 1960-70 to 1981-85, etc (this is cumulative spread) with regression line
733
(cumulative spread vs time (5 year periods)) fit for 12 species of butterflies organized in four
734
families. Values in parenthesis are the slopes of the regression lines for each species.
735
Figure 5 a) Dc estimates for 12 North American butterfly species. Solid points represent mean
736
Dc estimates, whereas the upper and lower bars represent Dc estimates for mean q and 95% lower
38
Leroux et al.
Species range shift under climate change
737
and upper C.I. estimates of r, respectively (see Tables 2 and 3). b) Difference in the relative rank
738
of 12 North American butterfly species based on a mobility index assigned by naturalists (i.e.,
739
realized mobility, Burke et al. 2011, Table 4) and the relative rank of these same species based
740
on our mean Dc estimates (i.e., predicted mobility). Larger negative differences may indicate
741
species that are more at risk of not keeping pace with climate change and larger positive
742
differences may indicate species that are more likely to keep pace with climate.
743
Figure 6 Potapov and Lewis (2004) show that to persist a species must keep pace with climate
744
change and the length of available habitat must be sufficient. Eqns (2) and (3) define the relative
745
risks of extinction due to climate change (dark grey) and insufficient habitat (light grey) as they
746
depend on the species’ reproduction (r) and climate envelope shift rate (q) estimates. The figure
747
is parameterized for Phyciodes cocyta and suggests that if Phyciodes cocyta can track climate
748
change it will likely persist because the habitat requirements for D > 2.30 are modest.
749 750 751 752 753
39
Leroux et al.
Species range shift under climate change
Northern edge (x2,t+1) L = space (km)
Time = t + 1 Southern edge (x1,t+1) q = space/time (km/yr) Northern edge (x2,t) Time = t 754 755
L = space (km)
Southern edge (x1,t) Figure 1
756
40
Leroux et al.
Species range shift under climate change
Legend Study Area Provincial Boundaries Waterbodies
Species Richness High : 12 Medium : 6 Low : 0
757 758
Figure 2
759
41
Leroux et al.
760 761
Species range shift under climate change
Figure 3
42
Leroux et al.
Species range shift under climate change
200
D. plexippus (2.973) E. anthedon (3.571) L. arthemis (1.790) P. cocyta (3.120) P. comma (3.698)
100 0 -100
-200 100
1970
1980
1990 2000 Papilionidae
2010
1970
1990 Pieridae
2000
2010
2000
2010
0 50
50
1980
C. philodice (3.624) P. oleracea (3.125) P. rapae (5.040)
150
P. canadensis (1.142)
-100
-100
-50
0
Cumulative spread (km)
0
100
C. niphon (5.512) C. lucia (1.238) G. lygdamus (4.220)
1970
1980
1990
2000
2010
1970
1980
1990
Time (year)
762 763
Nymphalidae
300
Lycaenidae
Figure 4
764
43
Leroux et al.
Species range shift under climate change
5
a)
b) C. niphon
4
G. lygdamus
3
P. comma P. oleracea
ln(Dc) 1 2
E. anthedon C. lucia P. cocyta
-1
0
P. rapae C. philodice L. arthemis
765 766
P.
ca na de ns is C .l uc L. ia ar t D hem .p le is xi p C . p pus hi l E. odi an ce th ed on P. co cy P. ta ol er G ac .l yg ea da m us P. ra pa P. e co m m C .n a ip ho n
-2
D. plexippus P. canadensis
-10 -5
0
5
10
Realized - Predicted Relative Vagility
Figure 5
767
44
Leroux et al.
Species range shift under climate change
768 769
Figure 6
45
