## Abstract

Population growth metrics such as *R*_{0} are usually asymmetric functions of temperature, with cold-skewed curves arising when the positive effects of a temperature increase outweigh the negative effects, and warm-skewed curves arising in the opposite case. Classically, cold-skewed curves are interpreted as more beneficial to a species under climate warming, because coldskewness implies increased population growth over a larger proportion of the species’ fundamental thermal niche than warm-skewness. However, inference based on the shape of the fitness curve alone, and without considering the synergistic effects of net reproduction, density, and dispersal may yield an incomplete understanding of climate change impacts. We formulate a moving-habitat integrodifference equation model to evaluate how fitness curve skewness affects species’ range size and abundance during climate warming. In contrast to classic interpretations, we find that climate warming adversely affects populations with cold-skewed fitness curves, positively affects populations with warm-skewed curves and has relatively little or mixed effects on populations with symmetric curves. Our results highlight the necessity of considering the synergistic effects of fitness curve skewness, density, and dispersal in climate change impact analyses, and that the common approach of mapping changes only in *R*_{0} may be misleading.

## Introduction

Numerous species are undergoing range shifts in response to climate change, typically polewards in latitude or upwards in altitude^{1-3}. Underlying these shifts are complex spatial dynamics that may include (i) regional extirpations in areas where conditions are becoming unsuitable, (ii) persistence, but with altered population dynamics in regions where conditions have changed but remain suitable, and (iii) dispersal, followed by potential establishment and growth, into regions where conditions have become newly suitable^{4,5}. Depending on the respective strengths of these processes, the net effect may be an overall range contraction, a range shift, or a range expansion, as well as an overall decline or increase in abundance. Examples of such changes exist from almost all major taxa^{1-3} including for some pathogens and pests, as well as for some species that provide ecosystem services^{6-9}.

One key element for determining the impacts of a warmer climate on a species’ range and abundance is the temperature sensitivity of its population growth, which in turn is a consequence of the temperature sensitivities of the underlying life history components^{10}. Mortality, for example, tends to increase exponentially with temperature within a species’ thermal tolerance range, while fecundity in contrast exhibits a hump-shaped relationship - increasing first to some optimal temperature before decreasing again to zero at high temperatures (see ^{10} and references therein). In many insects and vertebrate ectotherms ^{10-12}, these patterns combine to yield a unimodal, cold-skewed temperature-population growth relation that increases gradually to some optimum before dropping steeply to zero near the upper boundary of the species’ tolerance range (Figure 1b). Environmentally transmitted nematode parasites, by contrast, can exhibit a warm-skewed population growth curve, because the mortality of free-living stages is directly affected by temperature, but reproduction - when occurring within an endotherm definitive host - is temperature-independent^{13} (Figures lc and 2b; see also ref^{14}.). Intermediate cases also exist (i.e. more symmetric population growth curves in bacteria^{15}; Figure 1d), and skewness may further vary between^{16} and within taxa^{17}. In general, fitness curves are expected to be cold-skewed when the positive effects of a temperature increase (e.g. increased reproduction) outweigh the negative effects (i.e. increased mortality), and are expected to be warm-skewed in the opposite case^{18,19}.

The shape of a species’ temperature-dependent population growth curve has been highlighted as a key determinant of whether or where a species is likely to persist in a warming climate. For example, researchers often estimate a species’ intrinsic rate of increase, *r*, or net reproductive number, *R*_{0}, as a function of temperature, and then evaluate whether population growth would increase or decrease in different locations under future climates ^{11,20}-^{22}, sometimes also comparing the total land area where increases are likely against the total area of likely decreases as a measure of the anticipated overall net climate change impact^{21}. The logical consequence of this perspective is that species with a cold-skewed population growth curve should have an inherent advantage over species with a warm-skewed curve in a warming climate, because cold-skew maximizes the area over which population growth will increase in response to warming^{7,18} (Figure Ib-d). This interpretation does not, however, explicitly consider the ability of a species to disperse into and colonize new habitats, nor does it account for potential population dynamics differences along the temperature gradients of a species’ range. Here, we show that the explicit consideration of dispersal and population dynamics processes suggests the opposite interpretation, that is, that a warm-skewed population growth curve is more beneficial to a species under climate warming than a cold-skewed curve.

To explore how fitness curve skewness may affect a species’ range size and abundance during climate change-induced range shifts, we formulate a moving-habitat^{23} integrodifference equation (IDE) model that explicitly tracks population growth and dispersal along a thermally non-uniform shifting niche, and evaluates the transient and long-term spatiotemporal dynamics that emerge from these interacting processes. IDEs are particularly well-suited for describing the typical ‘travelling wave’ patterns of range expansions^{24}, and have been applied to invasive species, trees, and wildlife^{25}. More recently, IDE models have also been formulated for situations where the niche itself is shifting due to climate change, showing how the speed of climate change, the size of a species’ niche, as well as its population growth rate and dispersal ability interact to determine if a species will keep pace with its moving niche^{26-26}. Such moving-habitat IDE models^{26-28}, and closely-related partial differential equation models ^{29,30}, have, however, only considered vastly simplified representations of the temperature-dependence of population growth, typically assuming a constant, spatially uniform growth rate within the niche, and a constant, spatially uniform decline rate elsewhere (but see ^{31}). Our model (i) accounts for temperature dependencies in reproduction and survival using relationships suggested by the Metabolic Theory of Ecology (MTE) ^{18,32}, (ii) uses these relationships to formulate temperature-dependent population growth curves of varying degrees of skewness (Figure 2), (iii) uses numerical simulations to subject populations to climate change-induced habitat shifts, and (iv) evaluates how skewness affects range size, abundance, and the lags between the invasion front and the niche boundaries during climate change (Figure 1). Generally, we find that warming adversely affects populations with cold-skewed fitness curves, positively affects populations with warm-skewed fitness curves, has relatively little or mixed effects on populations with symmetric fitness curves, and that these results are largely robust against different choices of population growth and dispersal mechanisms.

## Methods

IDE models describe the spatiotemporal dynamics of a population, treating space and time as continuous and discrete variables, respectively. The population density, *n*_{t+1}(*x*), at location ** x** in year

*t*+

*1*is given by

where *f*(*n*_{t}(*y*),*T*(*y,t*)) describes the density- and temperature-dependent population growth at location *y* in year *t,* and the dispersal kernel *k*(*x-y*) describes the probability of an individual dispersing from location *y* to location ** x** after the population growth phase. Following population growth and dispersal, the integral totals the numbers of individuals arriving at location

**from all possible origins (**

*x**y*in Ω), thus giving the new population density,

*n*

_{t+1}(

*x*). For simplicity, we model Ω as a one-dimensional domain [

*-L,L*], corresponding to either a latitudinal temperature gradient from the equator to a pole, or to an elevation gradient from low to high. For convenience, but without loss of generality, we discuss our model for a latitudinal temperature gradient in the northern hemisphere, hereafter referring to

**-**

*x*=*L*as the “south” and

*x*=

*L*as the “north” and assuming a linear temperature decrease from

*y*=

*-L*to

*y*=

*L*(Figure 1a).

For generality, we evaluate our model for compensatory (Beverton-Holt) and overcompensatory (Ricker) functions of population growth. Furthermore, we consider both exponentially bounded (Laplace) and fat-tailed (Cauchy) dispersal kernels to evaluate how climate warming effects on species range shifts may depend on the organisms’ ability to disperse long distances (a larger proportion of the population disperses long distances with the Cauchy kernel than with the Laplace kernel^{24}). In the main text, we focus our discussion of methods and results on the Beverton-Holt/Laplace scenario (Figures 3-4), and then evaluate the robustness of our conclusions to other combinations of growth and dispersal in the Supplementary Online Material (SOM Sections 3-5) summarizing these comparisons in Table 1.

The Laplace kernel for dispersal is given by

and assumes that the probability of dispersal from location *y* to location *x* only depends on the distance |*x-y*| with a mean dispersal distance *D* (see ref.^{33} for a mechanistic derivation).

The Beverton-Holt model for population growth can be viewed as the discrete time analogue to the continuous time logistic equation, and is given by

where *K* represents carrying capacity and *R*_{0}(*T*) is the temperature-dependent net reproductive number, which we refer to as the ‘fitness curve’. Both *R*_{0} and *K* could be impacted by environmental conditions in multiple ways, but we focus only on temperature dependencies in the former to allow disentangling of the role of fitness curve shape on the range change dynamics without confounding these analyses via changes in habitat. We assume a fixed generation time of 1 year and describe the temperature dependence of population growth via the underlying temperature dependencies of fecundity and mortality,

where *ρ* is fecundity at low population density, *σ* is the annual survival probability at low population density, and *T*(*y,t*) is temperature at location *y* at time *t.* For notational simplicity, we write *T*(*y,t*)=*T* henceforth.

We describe the temperature sensitivities of both fecundity, *ρ* (*T*), and survival, *σ*(*T*), using relationships suggested by the Metabolic Theory of Ecology (MTE)^{18,32}. Fecundity is typically cold-skewed, increasing exponentially with temperature within a species’ fundamental thermal niche, and dropping steeply to zero near the upper and lower temperature boundaries of that range (Figure 2a). Assuming that this temperature dependency is largely driven by the temperature sensitivity of underlying metabolic processes, we use the Sharpe-Schoolfield model^{18} to represent *ρ* (*T*) and write

where *ρ*_{0} is the fecundity at a reference temperature *T*_{0} (units: K), κ = 8.62xl0^{-5} eV K^{-1} is Boltzmann’s constant, *E*_{ρ} is the average activation energy (units: eV) of the rate-limiting enzyme driving reproduction (determining the temperature sensitivity of fecundity at intermediate temperatures of the organism’s niche), and island are the inactivation energies (units: eV) determining how abruptly fecundity drops to zero at the thermal tolerance boundaries.

Survival is typically also a unimodal function of temperature but, unlike fecundity, is usually warm-skewed, peaking near the lower boundary of the organism’s fundamental niche and declining exponentially as temperatures increases (Figure 2b). As with fecundity, we use an MTE-based formulation^{18} to capture these patterns, representing the temperature-dependent rate of mortality by

and the proportion of individuals that survive one year by

As with fecundity, the parameters *E*_{μ}, and determine the temperature sensitivity of mortality within and outside the lower and upper temperature thresholds, and , and *T*_{0}, is a reference temperature at which mortality equals *μ*_{0}.

To evaluate how the shape of the temperature-dependent fitness curve *R*_{0}(*T*) (equation 4) influences the range change dynamics, we establish cold-skewed, warm-skewed, and symmetric *R*_{0}(*T*) curves by assuming temperature sensitivities in fecundity only, mortality only, or both (Figure 2, cf. also ref. ^{18}) and by setting the inactivation energies such that they further accentuate the direction of any skew in *R*_{0}(*T*) whilst ensuring that the fundamental niche is contained within the spatial domain (see SOM Section 1 for details). The magnitude of the skew is further manipulated in each of these cases by considering a biologically plausible range of activation energies^{34} (0.2 ≤ *E*_{ρ}, *E*_{μ} ≤ 1.1). We standardized each fitness curve so that the total reproductive potential of an organism across the entire landscape is always a constant. This calibration ensures that the shape of *R*_{0}(*T*) is the main source of variation in all comparisons, and that qualitative differences in climate change impacts can therefore be attributed to differences in this fitness curve shape (SOM Section 2).

### Model simulations and parameter values

We used computer simulations to explore the range change dynamics given by equations 1-7 when climate warming occurs. We allowed the population density to equilibrate before the onset of warming (at t=0), after which we increase the temperature at all locations by *w*=0.1°C yr^{-1} up until an increase of 10°C is achieved, and evaluate how warming alters a population’s range size and abundance, as well as the lag between the northern boundary of the population range and of its thermal niche (see Figure 1e for definitions). Range size and abundance changes are reported as percentage changes relative to the range and abundance at the onset of warming (*t*=0), whereas lag is reported as the difference between future lags and the lag at *t*=0 to facilitate comparisons between fitness curves with different shapes (see SOM Section 2.1 for details). The MATLAB code used to generate all simulations is provided on Figshare^{35} (doi: 10.6084/m9.figshare.6955370) and further details of the simulation implementation are available in the SOM (Sections 1 and 2).

## Results

The numerical solutions of equations 1-7 with simulated climate warming reveal that the species’ distribution lags behind its shifting fundamental niche, a dynamic that is discussed in detail in Svenning and Sandel (2013)^{5}. We observe regions of ‘colonization credit’^{4}, where suitable habitat (*R*_{0}(*T*)>1) has yet to be fully colonized, and of ‘extinction debt’^{4,35}, where individuals continue to persist temporarily in unsuitable habitat (*R*_{0}(*T*)<1; Figure 3a-c; see Figure le for definitions of colonization credit and extinction debt).

During climate warming, population density retains its general shape, but the skew of *R*_{0}(*T*) affects the length of habitat where the population is at carrying capacity (Figure 3a-f). Populations with a warm-skewed *R*_{0}(*T*) have small lags (Figures 3k and 4k) and small colonization credits (Figure 3h), which means that most of the fundamental niche has been fully colonized, and suggests increased range sizes (Figures 3k. and 4e) and increased abundances (Figures 3k and 4h) under climate warming. The opposite holds for cold-skewed *R*_{0}(*T*), where a larger lag (Figure 3j and 4j) results in large colonization credits (Figure 3g), meaning that a large region of the fundamental niche has yet to be fully colonized, and leading to decreased range sizes (Figures 3j and 4d) and decreased abundances (Figures 3j and 4g).

These differences in the range change dynamics of warm-skewed, symmetric, and cold-skewed *R*_{0}(*T*) can be understood by considering the population’s potential for range expansions at the leading range edge. Populations with warm-skewed *R*_{0}(*T*) are sensitive to beneficial temperature increases in the north, where the slope of the *R*_{0}(*T*) curve is steep. In addition, warm-skewed populations have higher densities at the leading edge of their fundamental niche (Figure 3b) due to smaller lags (Figure 4k), and thus a large potential for colonizing newly available northern habitats via dispersal, resulting in an increased range size and abundance during warming (Figure 4e,h). For populations with a cold-skewed *R*_{0}(*T*), these same mechanisms act in the opposite direction, implying decreased range sizes and abundance. Extinctions at the trailing edge of the population density also determine the net impact of climate warming, but the extinction debt is similar for *R*_{0}(*T*) of all skewnesses (Figures 3g-i and S2). Populations with a symmetric *R*_{0}(*T*) are mostly unaffected by climate warming, but demonstrate slight decreases in both range size and abundance (Figure 4f,i). These slight decreases occur where the population lags behind its fundamental niche, resulting in a negative net effect of colonization versus extinction, despite the symmetry of *R*_{0}(*T*).

In contrast to the suggestion by Molnár *et al.* (2013)^{18} that cold-skewed populations experience increased population growth over the majority of their habitat (orange bars in Figure 1b) and warm-skewed populations experience increased growth over only a small portion of their habitat (orange bars in Figure 1c), we find that generally climate warming adversely affects the range size, abundance and lag of populations with a cold-skewed *R*_{0}(*T*), positively affects populations with a warm-skewed *R*_{0}(*T*), and has relatively little or mixed effects on populations with a symmetric *R*_{0}(*T*) (Figure 4; Table 1). For cold- and warm-skewed *R*_{0}(*T*), the degree of the skew further magnifies these effects. This result is a consequence of the interaction between population growth, density, and dispersal, which is masked when one considers population growth in isolation as a predictor of climate warming affects.

The influence of *R*_{0}(*T*)’s skewness on a population’s response to climate warming is largely insensitive to mean dispersal distance, the choice of population growth function, and the choice for dispersal kernel (SOM Sections 3-5; summarized in Table 1). Using an overcompensatory population growth function (Ricker) instead of the compensatory Beverton-Holt function generally yields final ranges, abundances and lags that are similar to those outlined for the Beverton-Holt scenario (Table 1; SOM Section 3). However, for large values of *R*_{0}(*T*), Ricker growth may give rise to oscillatory or chaotic travelling waves that do not occur with Beverton-Holt growth^{26}, and it is noteworthy that these oscillations can be dampened by climate warming due to the temperature dependence of *R*_{0}(*T*) (SOM Section 4). Using a fattailed (Cauchy) dispersal kernel instead of the Laplace kernel yields some minor differences, but our general results remain qualitatively unaffected (Table 1; SOM Section 5).

## Discussion

The shape of temperature-dependent population growth measures such as *R*_{0}(*T*) contains critical information on a species’ sensitivity to climate change. The common approach of only considering how population growth metrics such as *R*_{0}(*T*) or *r*(*T*) would change in different locations under climate change (e.g. *R*_{0}(*T*)*-* or r(7)-based impact maps for insects^{11}, wildlife pathogens^{22}, and human pathogens^{20,21}), however, can yield an incomplete and potentially misleading picture. Cold-skewed *R*_{0}(*T*) have previously been interpreted as more beneficial to species’ persistence under warming because cold-skewness implies fitness increases over a comparatively larger portion of habitat^{7,18} (Figure 1b-d). Explicit consideration of the population growth-dispersal dynamics in our moving-habitat IDE framework, however, revealed that species with a cold-skewed *R*_{0}(*T*) may in fact be more likely to experience adverse impacts during warming than species with a warm-skewed *R*_{0}(*T*). With a warm-skewed *R*_{0}(*T*) steep fitness increases and relatively large densities at the northern range limit allow for rapid abundance increases in areas that have not yet been fully colonized, while more gradual fitness declines combined with smaller densities simultaneously limit the extinction risk at the trailing niche edge. The opposite is true for species with cold-skewed *R*_{0}(*T*). It is these synergistic effects of *R*_{0}(*T*) skewness, spatially varying population densities, and dispersal that underly our findings that a warm-skewed *R*_{0}(*T*) results in range size and abundance increases and small lags between the moving leading edge of the fundamental niche and the population’s range edge, whereas a cold-skewed *R*_{0}(*T*) results in range contractions, decreased abundances, and large lags.

Our model is deliberately simple to facilitate a sound understanding of the key processes determining range changes (dispersal, fecundity, and mortality). It is intended for separating out the influence of fitness curve skewness arising from imbalances in the temperature sensitivity of life history components (here, fecundity and mortality) from multiple other factors that may also influence range change dynamics. Our analyses highlight the synergistic effects of fitness curve skewness, density, and dispersal as a previously unrecognized mechanism for why some species may benefit from warming while others experience range contractions and population declines, but these mechanisms should be viewed as complementary to previously recognized influences, such as the role of interspecific interactions^{38} or dispersal limitation^{39} in confining species to only part of their fundamental niche.

The muskox lungworm *Umingmakstrongyls pallikuukensis,* for example, has a warm-skewed *R*_{0}(*T*)^{13} and has been closely tracking the northern boundary of its shifting fundamental niche during a warming-induced range expansion^{6} as would be expected (Figures 3-4). No changes were reported in the southern boundary of the lungworm’s distribution^{6}, which is consistent with our results that warm-skewed *R*_{0}(*T*) still have large extinction debts (Figure 3h), and may further be explained by the worm benefitting from the behavioral thermoregulation of its intermediate slug host and the shelter it provides from lethal high-temperature extremes^{13} Also in agreement with our results are the range contractions observed in tree species in the eastern United States^{40} and southern Africa^{41}, where limited dispersal resulted in large dispersal lags at the poleward limits, perhaps further exacerbated by apparently cold-skewed fitness curves (cf. Figure 3 in Talluto et al ^{4}). Lepidopterans^{8,9} and Odonata^{42}, by contrast, have largely expanded their ranges polewards in response to warming - often while maintaining stable or more slowly moving equatorward boundaries - despite generally cold-skewed fitness curves (see Deutsch et al. 2008^{11}, their Dataset SI). It has, however, been suggested that these range change patterns could be a result of temperature limiting species’ distribution at the poleward boundary, and factors other than temperature (e.g. competition) currently limiting distribution at the equatorward boundary^{8,9,42}. In other words, these species may not be fully occupying the warmest end of their fundamental thermal niche, and would thus temporarily be shielded from the worst consequences of warming despite having a cold-skewed fitness curve.

The dynamics of climate warming-induced range changes are complex and our analyses are not meant to imply that skewness will uniquely determine whether warming benefits or harms a species. Dedicated meta-analyses are needed to disentangle the various factors affecting range changes (i.e., dispersal limitation^{39}, sensitivities to other abiotic variables^{43}, biotic interactions^{44}, and landscape features^{45,46}), but are currently limited by (i) unknown fitness curve shapes for most species, (ii) the difficulty of inferring fitness curve shape from local densities when carrying capacity is temperature-independent as assumed here (cf. Figure 3a-f, showing ‘rectangular travelling waves with soft edges’ regardless of fitness curve skew), and (iii) a paucity of studies^{47} documenting distribution and abundance changes over a species’ entire range and on long-enough time scales to capture the dynamics suggested here. Nevertheless, given the robustness of our results to different forms of density dependence and dispersal (Table 1), we expect that imbalances in the temperature sensitivities of different life history traits and the resultant fitness curve skewness may explain some of the observed variation in distribution and abundance change responses to climate warming.

Our models can be easily extended in multiple ways by relaxing simplifying assumptions or adding additional population regulating mechanisms. The list of potential factors influencing range change dynamics is long, and includes potential temperature dependencies in other life history components (i.e. increased temperature would imply: decreased carrying capacity, since faster metabolism accelerates the depletion of a fixed resource supply^{12}; decreased development times, due to faster metabolism^{32}; and shortened dispersal distances in marine plankton^{48}, due to decreased larval development time^{32}), local adaptation (e.g. different temperature sensitivities of individuals at the leading and trailing edge^{17}), environmental stochasticity (e.g. influencing individuals at the cold and warm ends of their range differently due to Jensen’s inequality^{49-49}), limiting abiotic factors other than temperature^{43} (e.g. moisture), age-structure^{27} (e.g. for species where dispersal only occurs in certain life stages such as trees or many insects), Allee effects limiting range shift speeds at the leading edge^{24,25}, as well as landscape heterogeneities^{45,46}, or species interactions^{29,38,44}. Each of these might dampen or amplify the patterns suggested by our analyses.

We used temperature-dependent relationships only for fecundity and survival but did not explore the effects of temperature dependence on generation time. This assumption most closely corresponds to organisms with a fixed, one-year generation, for example driven by obligatory diapause to overwinter a cold season (e.g. Lepidoptera in temperate forests). However, even in these cases, generation time may decrease if development increases sufficiently fast with increasing temperatures to allow for multiple generations within a year. Such changes would not affect *R*_{0}(*T*) (because it measures population growth per generation)^{10,52}, but would shift the relationship between temperature and the intrinsic rate of population growth increase, *r*(*T*), towards more cold-skewed, likely with the corresponding consequences for abundance and range changes that were outlined above. Our combined MTE-IDE framework allows easy incorporation of such additional factors.

The MTE describes temperature effects from the individual to the ecosystem level^{32} and has provided many insights on how a warmer climate may alter population and community dynamics^{18,34,46,53}, but analyses have generally been limited to non-spatial effects to date. Likewise, there exists a rich literature that uses integrodifference equation models to explore how various intrinsic and extrinsic mechanisms^{24,25,27,33,45,46} affect range change dynamics, but few studies have explicitly explored the role of temperature in these dynamics. Combining these two bodies of theory has enormous potential for unravelling the complexities of temperature-dependent spatial population dynamics.

## Supplementary Material

### 1 Additional simulation details and parameter values

The main text, provides a general overview describing the simulation of our moving-habitat integrodifference equation model with spatially non-uniform net reproduction (equations 1-7). Additional specific details are provided in this section and the MATLAB code used to generate all simulations is archived at Figshare [1] (doi: 10.6084/m9.figshare.6955370).

#### 1.1 Simulation details

Temperature is modelled as a linear gradient along the spatial domain [*–L,L*]that peaks at *y* = *—L,* and warming is simulated by increasing temperatures at all locations at a constant rate:
where *T*_{max,0} and *T*_{min,0}are the respective temperatures at the southern (*y* = *—L*) and northern (*y* = *L*) limits of the spatial domain at time 0, and *w* is the rate of temperature increase. For simplicity, we ignored seasonal temperature fluctuations in all simulations, assuming a constant *T* within each year (Figure la).

We used computer simulations to explore the range change dynamics given by equations 1-7 in the main text· To do this, we set the initial abundance to 1 on a small interval about *x* = 0, and calculate population densities across the entire spatial domain for 400 years with no change in climate (*w* = 0) to let the population density equilibrate before the onset of warming. Subsequently we increase the temperature at all locations by *w* = 0.1°C yr^{-l} up until an increase of 10°C is achieved after 100 years, and evaluate how warming alters a population’s range and abundance, as well as the lag between the northern boundary of the population range and of its thermal niche. We recognize that a warming of 10°C is far more than expected under current climate change predictions, but note that this value was simply chosen to allow easy visualization of how climate change impacts range, abundance, and lag. All results are qualitatively insensitive to the total amount of warming imposed.

To evaluate how the shape of the temperature-dependent, fitness curve *R*_{0}(*T*) (equation 4) influences the range change dynamics, we establish cold-skewed, warm-skewed, and symmetric *R*_{0}(*T*) curves by assuming temperature sensitivities in fecundity only, mortality only, or both (cf. also ref. [2]). The strongest cold-skew in *R*_{0}(*T*) arises in populations with highly temperature-dependent, fecundity (large *E*_{ρ}), but temperature-independent, mortality (*E*_{μ} = 0). A strongly temperature-dependent, mortality (large *E*_{μ}) combined with temperature-independent, fecundity (*E*_{ρ} = 0), by contrast, leads to a, strongly warm-skewed *R*_{0}(*T*), and equal temperature sensitivities (*E*_{ρ} = *E*_{μ}) yield an approximately symmetric *R*_{0}(*T*) (Figure 2). We present simulations for these three extreme eases and note that they generate a wide variety of *R*_{0}(*T*) shapes that bound other, more common, combinations of *E*_{ρ} and *E*_{μ} (Figure 4; i.e. *E*_{ρ} ≠ *E*_{μ} but both >0).

#### 1.2 Parameter values

The three cases of cold-skewed, warm-skewed, and symmetric *R*_{0}(*T*) are thus implemented by considering temperature dependence in (i) fecundity only (*E*_{ρ} > 0, *E* _{μ}=0), (ii) survival only (*E*_{ρ} = 0, *E* _{μ}> 0), and (iii) both in fecundity and survival (*E*_{ρ} = *E*_{μ} *>* 0). The magnitude of the skew is further manipulated in each of these cases by considering a biologically plausible range of activation energies [3] (0.2 ≤ *E*_{ρ}, *E*_{μ} ≤ 1.1), and by setting the inactivation energies such that they further accentuate the direction of any skew in *R*_{0}(*T*) whilst ensuring that, the fundamental niche is contained within the spatial domain (case (i): case (ii): case (iii): Figure 4). The remaining model parameters were fixed arbitrarily, but chosen such that dispersal is large enough that the population can keep pace with its moving niche (*D* = *0.5, K* = 1, *μ*_{0}=-ln(0.2), *ρ*_{0} = 50, , *T*_{0} = 12.5°C, *T*_{min,0} = −15°*C, T*_{max,0} = 30*°C;* note that all temperatures are reported in °C for simplicity but converted to Kelvin for use in equations 5-8). Finally, we standardized each fitness curve so that the total reproductive potential of an organism across the entire landscape is always a constant (chosen as where *c*_{1} is the calibration parameter). This calibration ensures that the shape of *R*_{0}(*T*) is the main source of variation in all comparisons, and that qualitative differences in climate change impacts can therefore be attributed to differences in this fitness curve shape (see Section 2). While the values of *ρ*_{0}, *μ*_{0}, *T*_{L},*T*_{H} and were chosen arbitrarily, our results are not sensitive to these choices because it is the shape of the fitness curve, rather than the specific details of how the curves are produced, that determines the impact of climate warming.

### 2. Standardization of the results

#### 2.1 How we standardize comparisons between fitness curves with different skewness

Throughout this study, we quantify the impacts of climate warming by comparing range size, abundance, and lag relative to their initial values. The range change (%) is calculated by taking the range size at time *t* divided by the initial range size (when climate warming begins at *t* = 0) and multiplied by 100, and the abundance change (*%*) is calculated in the same way. The lag is calculated by subtracting the lag at time *t* = 0 from the lag at time *t.* Final range size, final abundance, and final lag are calculated as described above, but with *t* = 100 (the time when climate warming ends). In addition, we normalized all fitness curves so that the reproductive potential of a population (the integral of *R*_{0}(*T*) across all of space) is equal to the same value regardless of fitness curve skew. This normalization was necessary to ensure that results reflect differences in fitness curve skewness and are not due to associated changes in reproductive potential.

#### 2.2 Differences in the initial range size and initial abundance cannot explain the observed impact of climate change

The normalization of the reproductive potential for all fitness curves to does not ensure that the initial range sizes, initial abundances, or initial lags are also equal for different *R*_{0}(*T*) (Table S.l) because these quantities are also influenced by density-dependent processes that are not captured by *R*_{0}(*T*), which describes only the net reproduction at low densities (see equation 3 in the main text). In Figure S.l, we demonstrate that differences in initial range size and initial abundance between different *R*_{0}(*T*) curves cannot explain the systematic climate change impact differences, which are instead driven by the differences in fitness curve skewness. To interpret Figure S.l, note that fitness curves with cold-skew (*), warm-skew (Δ), and symmetric (•) shapes may all have initial range sizes close to 41 (left column), but the initial range size of 41 corresponds to a wide range of final range sizes (between −6 and +2% change), whereas cold-skewed curves (*), of any initial range size, correspond to a narrower range of final range sizes (between +2 and +4% change; Figure S.la). This pattern is generally consistent for different initial abundances (right column), and if different climate warming impacts are considered (rows), and so we conclude that differences in initial range size or initial abundance are not responsible for the different climate warming impacts that we observe.

### 3. Effect of mean dispersal distance

In this section, we analyze how mean dispersal distance affects a population’s colonization credits, extinction debts, range size, abundance, and lag during range changes.

#### 3.1 Colonization credit and extinction debt

Colonization credit occurs in locations where the population is below its carrying capacity despite *R*_{0}(*T*) > 1, and is quantified as the difference between the carrying capacity and the population density numerically integrated across locations meeting these criteria. Colonization credit quantifies the additional abundance the environment could support, if it were not for dispersal limitation. In contrast, extinction debt occurs in areas where the population remains temporarily present despite *R*_{0}(*T*) < 1, and is defined as the numerical integral of the population density across all these locations. Extinction debt is thus the abundance of this population that can only persist temporarily (see Figure 1 in the main text).

Figure S.2 shows that both colonization credits and extinction debts decrease as the mean dispersal distance. The decrease in colonization credit as the mean dispersal distance increases (Figure S.2, green solid lines) is since a larger proportion of offspring disperse longer distances from their natal sites, and are thus better able to colonize new habitat and keep pace with their moving fundamental niche. There are two factors that explain why extinction debt decreases with increasing mean dispersal distance (Figure S.2, pink dashed lines). Firstly, the population is better able to track the location of the fundamental niche with larger dispersal distances, meaning that fewer individuals begin their dispersal from a position behind the trailing edge of the niche, resulting in smaller extinction debts. Secondly, any group of individuals that does start behind the trailing edge, with larger dispersal distances, will see a larger proportion of dispersers catching up with the niche, again lessening the extinction debt. Despite both colonization credit and extinction debt decreasing with mean dispersal distance, populations with cold-skewed fitness curves remain more adversely affected by climate warming (Figure S.22a.) in comparison to their warm-skewed counterparts (Figure S.2b) for all choices of mean dispersal distance (Figure S3). Figure S.2 shows that if colonization credit is greater than extinction debt, then this holds for all values of the mean dispersal distance (and visa versa), which in turn implies that abundance changes for fitness curves of a given skewness are robust to different mean dispersal distances.

#### 3.2 Final range size, final abundance, and final lag

Populations are better able to keep pace with climate change as mean dispersal distance (*D*) increases, resulting in smaller lags, and larger final range sizes and final abundances (Figure S.3). Our qualitative results from the main text remain unchanged for different choices of mean dispersal distance (Figure S.3): climate warming results in range size and abundance decreases for organisms with cold-skewed fitness curves and in range size and abundance increases for organisms with warm-skewed fitness curves (Figure S.3). Mean dispersal distances smaller than 0.5 were not considered in our simulations as the population with the cold-skewed fitness curve fails to keep pace with climate change in these cases.

### 4. Simulations using the Ricker growth function

In the main text, we evaluated range change dynamics for the case of compensatory density dependence using a Beverton-Holt, population growth model (equation 3). Here, we assess whether results differ for overcompensatory density dependence using the Ricker model for population growth. The Ricker growth function is given by,

where *r*(*T*(*t,y*)) = In(*R*_{0}(*T*(*y,t*))) and *K* is the carrying capacity. As in the Beverton-Holt scenario, this carrying capacity is independent of *R*_{0}, as can be seen from rewriting equation S.2 as,

Combining the Ricker growth function with the Laplace dispersal kernel yields near identical values for our summary statistics as in the Beverton-Holt/Laplaee scenario (Figures S.4 and S.5). The only exception is for large values of *R*_{0}(*T*) as discussed in the next section.

#### 4.1 Population cycles

Climate change may dampen or eliminate population cycles. For large values of *R*_{0}(*T*), prior to climate change the Ricker population growth function gives rise to a two-cycle in population density (Figure S.6a), but when climate warming occurs the population cycles dampen (Figure S.6b,d). After the end of climate warming, the population density returns to an identical two-cycle pattern, but displaced northwards due to the warming-induced northern shift in the fundamental niche.

### 5. Simulations using the Cauchy dispersal kernel

In the main text, we evaluated range change dynamics for the case of an exponentially bounded (Laplace) dispersal kernel. Here, we assess whether results differ for a fat-tailed dispersal kernel (Cauchy) because population spread rates are known to be greatly influenced by the tail of the kernel [5]. The Cauchy kernel is commonly used to model ‘fat-tailed’ dispersal and is given by,

where *β* is a shape parameter. Compared to the Laplace distribution, which exhibits exponential decay in the tail of the distribution, the Cauchy distribution is ‘fat-tailed’ due to the power law decay of the tail. The associated higher number of long-distance dispersal events explains both why range sizes are larger under a Cauchy kernel and why abundances are smaller (more individuals are lost due to long-distance dispersal into hostile habitat; Figure S.5). Nevertheless, with the Cauchy dispersal kernel our general result remains: climate warming will adversely affect populations with cold-skewed fitness curves (substantially decreasing abundance), benefit populations with warm-skewed fitness curves, and will have mixed effects on populations with symmetric fitness curves.

Simulations that use the Cauchy dispersal kernel were run with *β* = 0.005, where *2β* is the width of the distribution at half its maximum (for reference, the corresponding width for the Laplace distribution is *2D* In 2, where *D* is the mean dispersal distance). Figure S.7 is analogous to Figure 4 of the main text except that the Cauchy dispersal kernel is used. Ricker population growth combined with a Cauchy dispersal kernel yields near identical results to the combination of the Beverton-Holt growth function with the Cauchy dispersal kernel (Figure S.5, triangle symbols).

## Acknowledgements

This research was catalyzed by a Banff International Research Station workshop (13w5095) and further facilitated by an Atlantic Association for Research in the Mathematical Sciences Collaborative Research Grant. AH (RGPIN 2014-05413) and PKM (RGPIN 2016-06301) were supported by National Sciences and Engineering Research Council of Canada Discovery grants, and PKM was further supported by the Canada Foundation for Innovation John R. Evans Leader Funds, and MRIS Ontario Research Funds. CC was supported by the Biotechnology and Biological Sciences Research Council (BB/P004202/1). Part of this work was carried out while CC was a funded visiting scholar at the Fields Institute for Research in Mathematical Sciences.