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doi: 10.1098/rstb.2012.0085, 368 2013 Phil. Trans. R. Soc. B

Matthew Miles Osmond and Claire de Mazancourt How competition affects evolutionary rescue

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ResearchCite this article: Osmond MM, deMazancourt C. 2012 How competition affects

evolutionary rescue. Phil Trans R Soc B 368:

20120085.

http://dx.doi.org/10.1098/rstb.2012.0085

One contribution of 15 to a Theme Issue

‘Evolutionary rescue in changing

environments’.

Subject Areas:ecology, evolution, theoretical biology

Keywords:adaptation, adaptive dynamics, competition,

environmental change, mathematical model,

persistence

Author for correspondence:Matthew Miles Osmond

e-mail: [email protected]

& 2012 The Author(s) Published by the Royal Society. All rights reserved.

†Present address: Station d’Ecologie

Experimentale, Centre National de la Recherche

Scientifique, 09200 Moulis, France.

Electronic supplementary material is available

at http://dx.doi.org/10.1098/rstb.2012.0085 or

via http://rstb.royalsocietypublishing.org.

How competition affectsevolutionary rescue

Matthew Miles Osmond and Claire de Mazancourt†

Redpath Museum, McGill University, 859 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0C4

Populations facing novel environments can persist by adapting. In nature,the ability to adapt and persist will depend on interactions between coexist-ing individuals. Here we use an adaptive dynamic model to assess how thepotential for evolutionary rescue is affected by intra- and interspecific com-petition. Intraspecific competition (negative density-dependence) lowersabundance, which decreases the supply rate of beneficial mutations, hinder-ing evolutionary rescue. On the other hand, interspecific competition can aidevolutionary rescue when it speeds adaptation by increasing the strength ofselection. Our results clarify this point and give an additional requirement:competition must increase selection pressure enough to overcome the nega-tive effect of reduced abundance. We therefore expect evolutionary rescue tobe most likely in communities which facilitate rapid niche displacement. Ourmodel, which aligns to previous quantitative and population genetic modelsin the absence of competition, provides a first analysis of when competitorsshould help or hinder evolutionary rescue.

1. IntroductionIndividuals are often adapted to their current environment [1]. When theenvironment changes individuals may become maladapted, fitness may drop,and population abundances may decline [2]. If the changes in the environmentare severe enough, populations may go extinct. But populations can alsoevolve in response to the stress and thereby return to healthy abundances [3,4].Why some populations are capable of rescuing themselves from extinctionthrough evolution, while others go extinct, is a central question to both basicevolutionary theory and conservation [5].

Ecological and evolutionary responses to changing environments are con-tingent on the community in which the change occurs [6–10]. A population’sability to adapt and persist in changing environments will therefore alsohinge on the surrounding community [11] (see also [12]). By including the eco-logical community in a formal theory of adaptation to changing environments,we may better predict the response of natural communities to contemporarystresses, such as invasive species [13,14] and global climate change [15,16].

Competition reduces population abundance [17–20]. Since less abundantpopulations are more likely to go extinct when exposed to new environ-ments [21,22], competition may therefore lower the potential for evolutionaryrescue. But competition can also increase selective pressure [23], speed nicheexpansion [24–26] and increase rates of evolution [27], possibly allowing popu-lations to adapt to new conditions faster. These potentially contrasting effectsmay account for the unanticipated population dynamics and patterns ofpersistence in competitive communities [6] (but see [10]).

Currently, most theories on adaptation to abrupt environmental changeconsider only isolated populations [3,28–33], and many of these studiesassume unbounded population growth, thus ignoring intraspecific competitionas well. The studies that do consider intraspecific competition, in the form ofnegative density-dependence, give inconsistent conclusions, stating thatdensity-dependence has no effect [29] or decreases [30,34] persistence. Of thehandful of studies that examine the effect of interspecific competition on adap-tation to environmental change, nearly all predict slower adaptation and moreextinctions (reviewed in [35]). One notable exception suggests that interspecific

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orig

inal

env

iron

men

tne

w e

nvir

onm

ent

z0* zn

*zNc trait value (z)

K

Kn

Nc

0

Figure 1. Our initially adapted population is monomorphic for the optimalphenotype in the original environment ẑ ¼ z�0 (grey). When theenvironment changes, the carrying capacity function shifts (black). The newcarrying capacity of our population Kn ¼ k(z*0, z*n) is the height of theintersection of the original trait value z*0 and the new carrying capacityfunction. The population evolves towards the new optimal phenotype z*n. Thepopulation is at risk of extinction while its abundance is less than Nc, or

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competition can aid persistence in a continuously changingenvironment, by adding a selection pressure that effectively‘pushes’ the more adapted populations in the direction ofthe moving environment [36].

Here we use the mathematical framework of adaptivedynamics to describe the evolutionary and demographic dy-namics of a population experiencing competition and anabrupt change in the environment. Adaptive dynamics allowsus to incorporate both intra- and interspecific competitionin an evolutionary model while maintaining analyticaltractability. We assess the potential for evolution to rescuepopulations by measuring the ‘time at risk’, i.e. the time a popu-lation spends below a critical abundance [3]. First, we derive anexpression for the ‘time at risk’ in a population undergoing anabrupt change in isolation. We then compare our results to pre-vious studies and test the robustness of our results by relaxing anumber of simplifying assumptions using computer simu-lations. Finally, we examine how a population’s ability toadapt and persist to an abrupt environmental change isimpacted by the presence of competing species.
equivalently, while z , zNc . 85
2. Model and results(a) One-population modelWe first examine how, in the absence of competitors,an asexual population with density- and frequency-dependent population growth responds to an abruptchange in the environment.

We assume that each individual in the population has atrait value z, and that a phenotype’s growth rate is deter-mined by both its own trait value as well as the trait valueof all other individuals within the population. Populationdynamics are described by the logistic equation (eqn 2 in [37])

dnidt¼ niR 1�

Ðaðzi; zjÞnj dzj

kðzi; z�Þ

� �; ð2:1Þ

where ni is the number of individuals with trait value zi, R isthe per capita intrinsic growth rate, a(zi,zj) is the per capitacompetitive effect of individuals with trait zj on individualswith trait zi, and k(zi,z*) is the carrying capacity of individualswith trait zi in an environment where the trait value givingmaximum carrying capacity is z*. We describe carryingcapacity k as a Gaussian distribution (eqn 1 in [37])

kðzi; z�Þ ¼ K e�ðzi�z�Þ2=2s2k ; ð2:2Þ

where K is the maximum carrying capacity and sk .0 is the‘environmental tolerance’, which describes how strongly car-rying capacity varies with zi. For a given deviation from z*,smaller variances sk

2 mean larger declines in carryingcapacity k. We therefore refer to sk

22 as the strength of stabil-izing selection. Data on yeast responses to salt [5,38] fitGaussian carrying capacity functions, as described inequation (2.2) (see the electronic supplementary material).

We do not give a specific form for intraspecific compe-tition a, but instead give requirements that are satisfied bya wide range of functions. First, we assume that individualswith the same trait value compete most strongly, that is(d/dz)a(z,z) ¼ 0 and (d2/dz2)a(z,z) , 0. This is biologicallyreasonable and could describe, for instance, the effect ofbeak size on finches competing for seeds, where individualswith similar-sized beaks compete strongly for similar-sized

seeds [39]. And we arbitrarily set a(z,z) ¼ 1, meaning thatindividuals with the same trait value take up one ‘unit’ ofcarrying capacity.

Trait value z is assumed to be determined by a largenumber of loci, each with equal and small effect, makingthe range of possible phenotypes continuous and unbounded(i.e. z [ R). To proceed analytically, we first assume thatmutations are rare. The population remains monomorphic,with all individuals having ‘resident’ trait value ẑ. The evo-lutionary trajectory is determined by the per capita growthrate of rare mutants in the neighbourhood of ẑ (adaptivedynamics; [40]). When mutations are sufficiently rare,evolution occurs slowly enough for us to consider the popu-lation at demographic equilibrium on an evolutionarytime-scale. This stands in contrast with previous modelswhich jointly model demography and evolution [3,34].The time-scale separation between demography and evolu-tion allows us to incorporate intra- and interspecificcompetition while maintaining analytical tractability. Welater use computer simulations to examine how our analyticalresults perform when demography and evolution occur onsimilar time-scales.

In appendix A, we show that when (d2/dz2 )a(z,z) . sk22

the ‘optimal trait value’ z* is both convergence stable (i.e. bysmall steps the resident trait converges to z*) and evolution-arily stable (i.e. once ẑ ¼ z� no other strategies can invade;z* is an ESS, sensu Maynard Smith & Price [41]). Weassume (d2/dz2)a(z,z) . sk

22 for the remainder of the paper,which means frequency-dependence is weak enough [42].Our results apply for any function a, as long as z* is bothconvergence and evolutionarily stable.

Let our population begin in a constant environment withoptimal trait value z� ¼ z�0. In time, all individuals becomeperfectly adapted ẑ ¼ z�0. The population will reach equilib-rium abundance ~n ¼ K, and its growth rate will becomezero (figure 1). Let us call this original abundance K0.

Suppose then that the environment suddenly changes sothat the new optimal trait value is z�n = z

�0. Our monomor-

phic population, with trait value ẑ ¼ z�0, then immediatelyhas equilibrial abundance kðz�0; z�nÞ , K0 (figure 1). The

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trai

t val

ue (

z)ab

unda

nce

(n)

(a)

(b)

time (t)50 000trobstr0

KK

Nc

z0*

zn*

zNc

Figure 2. Adaptation following an abrupt change in the environment. (a)Population trait value ẑ evolves towards the new optimal z*n (equation (2.7)).The time it takes to evolve a trait value zNc, which gives a critical abundanceNc, is the expected ‘time at risk’ tr (equation (2.10)). (b) Populationabundance ~n increases as the population adapts to the new environment(equation (2.8)). Solid lines are analytical predictions (equations (2.7)and (2.8)). Greyscale is trait value weighted by abundance in a computersimulation, with dark common and white rare. The thick dashed line is totalabundance at each time step in simulation. The observed time at risk isdenoted trobs.


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environmental change serves to decrease the carryingcapacity of the population. The population will initiallysurvive the abrupt change if kðz�0; z�nÞ � 1 or, equivalently

jz�0 � z�nj � skffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lnðKÞ

q; 4z�: ð2:3Þ

Note that setting ~n � 1 as the extinction threshold scalespopulation abundance in units of minimal viable populationsize [37,43]. Because z*n is the new evolutionarily and conver-gence stable strategy, if the population survives the change itwill evolve towards the new optimal trait value, ẑ! z�n.According to the canonical equation of adaptive dynamics [44],the monomorphic trait value ẑ will change at rate

dẑdt¼

ms2m2

~nðẑ; z�nÞgðẑ; z�nÞ; ð2:4Þ

where m is the per capita per generation mutation rate, sm2 is the

mutational variance (mutations symmetrically distributedwith mean of parental value) and gðẑ; z�nÞ is the local fitnessgradient (appendix A):

gðẑ; z�nÞ ¼@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �Rðẑ� z

�nÞ

s2k; ð2:5Þ

where nm and zm are a rare mutant’s abundance and traitvalue, respectively, and ẑ is the resident trait value [40]. Thelocal fitness gradient describes the slope of the fitness functionin the neighbourhood of the parental trait value. Steeper slopessignify greater fitness differences between individuals withsimilar but unequal trait values [45]. Notice that R/sk

2 is thestrength of stabilizing selection per unit time.

The rate of change in trait value is then

dẑdt¼ �

ms2mRðẑ� z�nÞ2s2k

K e�ðẑ�z�nÞ

2=2s2k : ð2:6Þ

We cannot solve equation (2.6) explicitly for ẑðtÞ, butusing a first-order Taylor expansion, we derive an approxi-mate solution, describing evolution and demographyfollowing the abrupt change (appendix B):

ẑðtÞ � z�n þ ðz�0 � z�nÞ eð�ms2mK0R=2s

2k Þt ð2:7Þ

and

~nðtÞ � K exp �ððz�0 � z�nÞ eð�ms

2mK0R=2s

2k ÞtÞ2

2s2k

" #: ð2:8Þ

Taking the Taylor expansion about z*0 2 z*n ¼ 0 results inthe assumption that the environmental change jz�0 � z�nj issmall relative to environmental tolerance ak (i.e. a weak ‘initialstress’). Our first-order approximation of the Gaussian kis therefore taken at the maximum z ¼ 0, which is a linewith slope zero and height K0. This means we assumemutational input mk is constant at mK0, effectively decouplingthe demographic and evolutionary dynamics of the recoveringpopulation. Our first-order approximation is the highest-orderfor which we can obtain an analytical solution.

Now, let Nc be the abundance below which demographicor environmental stochasticity are likely to cause rapid extinc-tion [3,46]. We use this heuristic Nc, in the place of stochasticmodels, for simplicity. We are interested in the amount oftime a population spends below this threshold, i.e. howlong the population is at risk of extinction.

The population will never be at risk of extinction if itsequilibrial abundance ~n remains above the critical abundance

Nc. In this model, equilibrial abundance strictly increases inevolutionary time in a constant environment. Abundance istherefore at a minimum immediately following the abruptshift in the environment. The population will avoid allchance of extinction if Nc , k(z*0,z*n) or, rearranging,

jz�0 � z�nj , sk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln

KNc

� �s; 4z��: ð2:9Þ

Here, we are most interested in the case where the popu-lation initially survives the abrupt change but abundancedrops below the critical abundance: 4z�� , jz�0 � z�nj � 4z�,as this is when evolution is required to rescue populationsfrom extinction.

From equation (2.2), we can find the trait value zNcrequired for a carrying capacity of Nc. Plugging zNc intoequation (2.7) and solving for t gives the time it will take apopulation to evolve to this safe trait value zNc, which wewill call the ‘time at risk’ tr (figure 2)

tr ¼s2k

ms2mK0Rln

ðz�0 � z�nÞ2

2s2k lnðK=NcÞ

" #: ð2:10Þ


tr

tr

tr

0

Nc/K1

00

sk**sk* sk0

Dz*D z**z0* – zn*

(a)

(b)

(c)

Figure 3. (a) Time at risk tr (equation (2.10)) increases monotonically withthe magnitude of environmental change jz�0 � z�nj. Magnitudes of changesmaller than Dz** are not large enough to put the population at risk ofextinction (equation (2.9)) and magnitudes of change larger than Dz* causeimmediate extinction (equation (2.3)). (b) Time at risk tr increases as thecritical abundance Nc approaches maximum abundance K. As the criticalabundance approaches the maximum abundance, Nc/K! 1, the ratio has astronger effect on the time at risk. (c) Time at risk tr is a unimodal function of‘environmental tolerance’ sk, where extinction is most likely at intermediatevalues. We must have sk . s*k for the population to survive the initialchange in the environment and sk , s**k for the population abundance todrop below Nc (s*k and s**k are derived by rearranging equations (2.3)and (2.9), respectively).


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So the time at risk tr increases with the strength of theinitial stress jz�0 � z�njs�1k and the ratio of critical abundanceto maximum carrying capacity Nc/K and decreases withthe mutational input mK0, mutational variance sm

2 and thestrength of stabilizing selection per unit time R/sk

2. Time atrisk tr is a unimodal function of environmental tolerance sk,with longest times at intermediate tolerances (figure 3).Time at risk is reduced at small and large environmental

tolerances because small tolerances cause strong selection(and hence fast evolution) and large tolerances allow greaterabundances for a given degree of maladaptation.

(b) Comparison of one-population model toprevious work

Here, we compare our one-population model to previousdiscrete-time quantitative genetic models [3,34]. We firstshow how our adaptive dynamics approach gives a qualita-tively similar description of trait dynamics over time andthen compare our predictions of time at risk.

In a model without frequency- or density-dependence,Gomulkiewicz & Holt [3] describe the evolutionary trajectoryof the population mean trait value as a geometrical approachto the optimum (eqn 5 in [3]):

dt ¼ d0wþ ð1� h2ÞP

wþ P

� �t; ð2:11Þ

where dt is the distance of the population mean trait value fromthe trait value giving maximum growth rate at time t, w is thevariance of the growth rate function, h2 is the trait heritabilityand P is the constant phenotypic variance [3]. We derive aqualitatively similar trajectory (equation (2.7)), in continuoustime, from adaptive dynamics. Adaptive dynamics providesgreater ecological context by including intrinsic growth rateand maximum carrying capacity as parameters in the evol-utionary trajectory. The trajectories are identical when

wþ ð1� h2ÞPwþ P ¼ exp

�ms2mK0R2s2k

" #: ð2:12Þ

Gomulkiewicz & Holt [3] refer to equation (2.12) asthe evolutionary ‘inertia’ of a trait. Inertia is bounded betweenzero and unity in both models. When inertia is unity there is noevolution. In Gomulkiewicz & Holt [3], evolution halts whentrait heritability h2 or phenotypic variance P is zero. In ourmodel, inertia is determined by mutational input mK0, andevolution halts when there are no mutations. For a given wand h2=0, inertia is minimized and evolution proceeds at amaximum rate in Gomulkiewicz & Holt [3] as phenotypic var-iance goes to infinity P!1. In our model, for a given strengthof stabilizing selection per unit time R/sk

2, inertia approacheszero and the rate of evolution is maximized as mutationalinput goes to infinity mK0!1.

Note that to maintain analytical tractability both modelsassume the material which selection acts upon (phenotypicvariance P or mutational input mK0) is constant. Bothmodels will therefore be more accurate when the environ-mental change is relatively small. Large changes in theenvironment are likely to cause strong selection and largevariation in abundance, which could greatly alter phenotypicvariance and mutational input [30]. Since phenotypic vari-ance and mutational input are expected to decline understrong stabilizing selection and reduced abundance [47],respectively, the analytical results of both models will tendto underestimate a population’s time at risk.

Our evolutionary trajectory aligns even more closely withthat of Chevin and Lande (eqn 10 in [34]; also see eqn 18ain [48]), who incorporated both density-dependence andphenotypic plasticity. The two trajectories are identicalwhen there is constant plasticity w ¼ 0, additive geneticvariance is equivalent to the supply rate of beneficial


expected time at risk (tr)

obse

rved

tim

e at

ris

k

5000

10 000

15 000

20 000

0 2000 4000 6000 8000 10 000 12 000 14 000

Figure 4. Accuracy of analytical prediction, in the one-population case. Eachpoint represents the mean + s.e. for 10 replicated simulation runs. Solidline is 1 : 1 line; points falling on line represent perfect predictions of timeat risk tr. Squares, mK log(K ) � 0.1; circles, mK log(K ) � 1; triangles, mKlog(K ) . 1; black, jz�0 � z�njs�1k ¼ 1:2; grey, jz�0 � z�njs�1k ¼ 2:1.Parameters: m ¼ f1027, 1026, 1025, 1024g, K ¼ f104, 105, 106g,sm ¼ f0.01, 0.05g, R ¼ 1, sk ¼ 1, sa ¼ 1.5 and Nc is 1000 greaterthan the minimum abundance of each run.


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mutations multiplied by mutational size sa2 ¼ msm2 K0/2, and

the two measures of stabilizing selection strength per unittime are the same g* ¼ R/sk2.

Although our evolutionary trajectory aligns closelywith those of Gomulkiewicz & Holt [3] and Chevin &Lande [34], we uncover an analytical approximation for thetime at risk tr by assuming a time-scale separation betweendemographics and evolution. Gomulkiewicz & Holt [3] andChevin & Lande [34] do not assume such a time-scale separ-ation, leading to more complex population dynamics and theneed to calculate tr numerically. This makes a quantitativecomparison with our time at risk approximation impossible.However, Gomulkiewicz & Holt [3] agree that the time at risktr should increase with initial maladaptation (i.e. magnitudeof environmental change) jz�0 � z�nj and that at high degreesof maladaptation the relationship with time at risk shouldbe close to linear (figure 3; fig. 5A in [3]). In addition, inboth Gomulkiewicz & Holt [3] and Chevin & Lande [34]strengthening selection 1/v!1 increases the rate of adap-tation while decreasing abundance (through a decline inmean fitness). Time at risk should therefore be minimizedat an intermediate selection strength, as in our model(figure 3c), although they do not explore this explicitly.Gomulkiewicz & Holt [3] also argue that the time at risk trshould decrease with the abundance before environmentalchange, since the population declines geometrically begin-ning at this abundance. In our model, time at risk alsodecreases with abundance before environmental change K0,but for a different reason. Recall that because of our first-order approximation we assume a small initial stress andhence a small change in abundance. This allows us toassume that mutations are supplied at a constant rate mK0,where m is the per capita mutation rate and K0 is the abun-dance before environmental change. A greater abundancebefore environmental change K0 therefore causes faster evo-lution resulting in less time at risk.

(c) SimulationsAdaptive dynamics assumes mutations are rare enough suchthat, on the time-scale of evolution, the population remainsmonomorphic (i.e. a mutation fixes or is lost before the nextarises [49]) and at demographic equilibrium (i.e. demographyis faster than evolution), and that mutations are small enoughto allow local stability analyses to determine evolutionarystability [40,45]. Our approximation of time at risk tr(equation (2.10)) also rests on the assumption that the initialstress jz�0 � z�njs�1k is weak. We therefore performed computersimulations to examine how well our analytical result (time atrisk tr) holds when we relax these assumptions. To do this,we varied (i) mutation rate m and maximum carryingcapacity K, (ii) mutational variance sm

2 , and (iii) the strengthof the initial stress jz�0 � z�njs�1k . Computer simulations allowmultiple phenotypes to coexist and introduces stochasticityin mutation rate and size.

Simulations describe the numerical integration ofequation (2.1), using a fourth-order Runge–Kutta algorithmwith adaptive step size, and stochastic mutations. Mutationsoccur in a phenotype with probability mnDt, where m isthe per capita per time mutation rate, n is the abundance ofthe phenotype and Dt is the realized time step. For eachmutation occurring in a phenotype with trait value z, oneindividual is given a new trait value, randomly chosen

from a normal distribution with mean z and standard devi-ation sm. Trait values are rounded to the third decimal toprevent the accumulation of overly similar phenotypes. Phe-notypes with abundance below unity were declared extinct.Simulations began with the population at maximum carryingcapacity K and all individuals optimally adapted with traitvalue z ¼ z0* . At the time-step 500, the optimal trait valueinstantaneously shifted to zn* = z0* . Simulations were termin-ated at time-step 50 000. Code available upon request;implemented in R [50].

Parameter values for m, K and jz�0 � z�njs�1k were chosen inthe range of those observed for yeast exposed to increasedsalt concentration [5]. We estimated sk from fig. S1 in Bell &Gonzalez [5] (see the electronic supplementary material).

In all simulations, the population evolved towards zn* ,

and, if successful in reaching zn* , remained there. Likewise,

population size always approached carrying capacity, asexpected (figure 2).

The transient dynamics, however, showed varyingdegrees of congruence with our prediction (equations (2.7)and (2.8); figure 4). In simulations, the amount of standingphenotypic variance increases with mutation rate m multi-plied by population size. Our time-scale assumption, whichimplies zero phenotypic variance, is thought to becomeunrealistic as mKlog(K ) approaches unity [51]. The thresholdof mKlog(K ) is obtained because mK is the mutational inputand log(K ) is the typical time of fixation for a successfulmutant when the population is well adapted [51]. Over ourparameter range (m ¼ f1027, 1026, 1025, 1024g, K ¼ f104,105, 106g) mKlog(K) seemed to be an excellent predictor ofaccuracy; our predictions were much more accurate whenmKlog(K) , 1. When mKlog(K) . 1, we greatly underesti-mated the time at risk (triangles in figure 4).

Mutational variance sm2 seemed to have little effect on the

accuracy of our predictions, at least over the range of par-ameter space explored here (sm ¼ f0.01, 0.05g; figure 4).However, our analytical prediction did perform consistentlybetter when the initial stress jz�0 � z�njs�1k was small, for allparameter combinations (compare black jz�0 � z�njs�1k ¼ 1.2and grey jz�0 � z�njs�1k ¼ 2.1 points in figure 4).



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(d) CompetitionWe now introduce interspecific competition. Let the popu-lation dynamics of the focal population be described by thelogistic growth equation

dnidt¼ niR 1�

Ðaðzi; zjÞnj dzj þ Cðzi; tÞ

kðzi; z�Þ

� �; ð2:13Þ

where C(zi,t) � 0 is the effect of interspecific competition onindividuals in the focal population with trait value zi attime t. We do not model the coevolution of the competitorsexplicitly; we instead keep interspecific competition C(zi,t)as general as possible, allowing it to depend on focal traitvalue zi and vary in time t with any other biotic or abioticfactor (including the trait values and abundance of the focaland competing populations). For evolutionary rescue ofthe focal population, the only relevant dependency iswith zi. Our formulation allows competition C to encompassall possible types of coevolution feedback. In fact, Ccould even be interpreted as an abiotic selection pressure.However, for brevity, we limit our discussion to C as theeffect of a competitor. Previous studies have explicitly mod-elled the coevolution of competing species in a constantenvironment [37,52,53], at the expense of analytical results.All other variables in equation (2.13) are defined as in theone-population case.

We again assume that mutations are rare, so that our focalpopulation remains monomorphic with trait value ẑ andequilibrial abundance ~n. In the presence of competition, equi-librium abundance of the focal population is

~nðẑ; z�; tÞ ¼ kðẑ; z�Þ � Cðẑ; tÞ: ð2:14Þ

Comparison with the one-population case, where ~n ¼ k,shows how competition reduces abundance.

Now, let the competing populations coexist in a constantenvironment with z* ¼ z*0. The equilibrial abundance ñ of thefocal population is not necessarily maximized at z*0, but at a‘competitive optimal’ z*c,0 (appendix C). Assuming z*c,0 is a fit-ness maximum (appendix C), the focal population willeventually evolve to the competitive optimal ẑ ¼ z�c;0. Wethen let the competitive optimal change abruptly, to newtrait value z*c,n = z*c,0. This change could arise from a shift incompetition C or in the optimal trait value z* ¼ z*n. The abun-dance of the focal population is now k(z*c,0,z*n) 2 C(z*c,0,t). Theamount of competition a population feels immediately fol-lowing the environmental change C(zc,0

* , t) will depend onthe type of environmental change as well as the response ofthe competitors. Competition may be close to negligible ifresources remain plentiful but the abundance of competitorsare greatly reduced (e.g. when a pollutant causes severemortality in the competitor). However, competition may beexceptionally strong if the change in environment is a shiftin available resources, so that the supply of resources islimiting (e.g. seed size changes on an island supporting mul-tiple species of finch [54]). Persistence requires k(zc,0

* ,zn* ) 2

C(zc,0* ,t) � 1, and therefore persistence following environmental

change is more likely when competition C(zc,0* ,t) is weak.

In appendix C, we derive the local fitness gradientof the focal population. In the new environment, with

z* ¼ z*n, it can be written as

gðẑ; z�n; tÞ ¼@

@zm

1nm

dnmdt

� ��zm¼ẑ

¼ R ð@=dẑÞðkðẑ; z�nÞ � Cðẑ; tÞÞ

kðẑ; z�nÞ

� �: ð2:15Þ

The population evolves in a direction that increases abundancek 2 C until ð@=@ẑÞðk � CÞ ¼ 0, which occurs when the popu-lation reaches the competitive optimal in the new environmentẑ ¼ z�c;n (figure 5). We assume that z*c,n is a fitness maximum,such that the population remains monomorphic (appendix C).

From equation (2.15) we see that, relative to the one-popu-lation case (equation (2.5)), competition can alter the strengthand direction of selection, depending on how competitionchanges with trait value (figure 5). Competition increasesthe strength of selection when jð@=@ẑÞðk � CÞj . jð@=@ẑÞkj.This is will always occur when competition selects in thesame direction as carrying capacity (i.e. @k=@ẑ and @C=@ẑare of different signs). Competition decreases selectionwhen jð@=@ẑÞðk � CÞj , jð@=@ẑÞkj, which will occur whencompetition weakly selects in the opposite direction to carry-ing capacity (i.e. @k=@ẑ and @C=@ẑ are of the same sign andj@C=@ẑj is small). When competition selects in the oppositedirection as carrying capacity and has a stronger selectiveeffect j@C=@ẑj . j@k=@ẑj, it will reverse the direction of selec-tion and the population will evolve away from z�n.Competition has no effect on selection when it is independentof trait value @C=@ẑ ¼ 0.

Combining equations (2.14) and (2.15), we compute therate of adaptation, as described by the canonical equation [44]:

dẑdt¼�ms2m

2½kðẑ; z�nÞ

� Cðẑ; tÞ�R ð@=dẑÞðkðẑ; z�nÞ � Cðẑ; tÞÞ

kðẑ; z�nÞ

� �: ð2:16Þ

The rate the focal population adapts dẑ=dt depends on howcompetition affects abundance relative to selection. Owingto the added complexity of competition we are unable tosolve equation (2.16) for trait value as a function of timeẑðtÞ and are therefore unable to compute a time at risk tr, aswe did in the one-population case. However, we can showwhen competition will help or hinder adaptation, and there-fore when competition has the potential to increase ordecrease the likelihood of evolutionary rescue. Rearrangingequation (2.16) and comparing with the one-populationcase (equation (2.6)) show that competition will increase therate of adaptation when (appendix D)

@

@ẑðkðẑ; z�nÞ � Cðẑ; tÞÞ

�� . kðẑ; z�nÞkðẑ; z�nÞ � Cðẑ; tÞ

@kðẑ; z�nÞ@ẑ

��; ð2:17Þ

and decrease the rate of adaptation when the inequality isreversed. Competition will tend to speed adaptation whencompetition C is weak and gets much weaker as the focalpopulation evolves towards zc,n

* (dotted-dashed curve infigure 6). Note that although competition may increase therate of adaptation, and therefore cause a greater rate of increasein abundance, abundance will still be depressed by compe-tition. Competition’s effect on evolutionary rescue (thetime at risk tr) will therefore depend on both its effect onadaptation and the abundance k 2 C relative to critical abun-dance Nc (figure 6c). As maximal abundance [k 2 C]ẑ ¼ z*c,napproaches the critical value Nc evolutionary rescue becomes


zn*

= z*c,n z

*n

(c)

K

z z

(d )

z*n = z*

c,n

(b)

k(z)

K

z*n = z*

c,n

(a)

C(z)

k(z)

C(z)

z*c,n

Figure 5. Selection pressures from carrying capacity and competition. The population evolves in the direction which increases abundance according to equation (2.15).Population size is carrying capacity minus competition k 2 C (solid curve minus dashed curve). Populations can persist in communities only when they have positivepopulation size (region of persistence; solid line higher than the dashed line). The selection pressure in the new environment is proportional to the selection for carryingcapacity (slope of solid curve) minus the selection for competition (slope of dashed curve). The population will therefore evolve towards the trait value for which the slopesof the two curves are equal ẑ ! z�c;n. The effective selection pressure will depend on the shape of the two curves and the position of the population in trait space. (a)Competition increases selection pressure. Competition decreases as carrying capacity increases, meaning both carrying capacity and competition select in the samedirection. (b) Competition reduces selection pressure. Competition increases as carrying capacity increases, meaning carrying capacity and competition exert opposingselection pressures. Note that if the competition curve was steeper than carrying capacity competition could reverse the direction of evolution. (c) Competition affects allphenotypes equally, and therefore has no effect on selection pressure. (d ) Competition increases or decreases selection pressure. When ẑ , z�c;n competition and carryingcapacity exert opposing selection pressures. When ẑ . z�n competition and carrying capacity select in the same direction, towards z

�n .


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less likely, and regardless of the rate of adaptation, when[k 2 C]ẑ ¼ z*c,n � Nc evolutionary rescue is impossible.

3. DiscussionIn nature, population abundance cannot increase indefin-itely [55]. One of the main ‘checks of increase’ [56] iscompetition for resources [17,19,57–59]. Because populationswith lower abundances are more likely to go extinct [46], anyfactor which limits abundance is likely to hinder persistence,especially when the environment changes [22]. However,when we consider that populations can persist in new environ-ments by adapting [3,5], competition has a second effect, inaddition to lowering population size, which could potentiallyhelp populations persist in novel environments. Since the ratea population adapts depends on the strength of selection itexperiences [44,60], competition which increases the strengthof selection may speed up adaptation [61] possibly increasingthe chances of persistence in the face of change.

Intraspecific competition often has relatively little impact onselective pressures [58,62] (but see [63]), and therefore the effectit has on evolutionary rescue will often be determined primarilyby the effect it has on abundance. Previous computer simu-lations have suggested that negative density-dependence willhave little effect on population persistence because survival

depends on the dynamics of populations which are wellbelow carrying capacity [29]. More recent analytical work hascome to a different conclusion, showing that, relative to thedensity-independent case, density-dependence can increasethe rate at which abundance declines as well as decrease therate abundance recovers, therefore increasing the time a popu-lation spends at risk of extinction [34]. The conflicting results aredue to the different types of density-dependence used in thetwo studies. In Boulding & Hay [29], density-dependence islinear (i.e. per capita growth rate declines linearly with abun-dance) while in Chevin & Lande [34] density-dependence isstronger than linear at low abundances (the per capita growthrate declines logarithmically with abundance). Since it isthe effect of density-dependence at low abundances that is criti-cal for population persistence, this explains why Chevin &Lande [34] claim density-dependence increases the chances ofextinction. A similar trend is expected in biological invasions,where populations experiencing strong density-dependence atlow abundances are predicted to invade slowly [64].

Here we assume evolution is slow, and hence, on the time-scale of evolution, populations are always at carrying capacity.Carrying capacity therefore indicates how well a population isadapted; populations below carrying capacity will increase inabundance without evolving, and hence may not require evol-utionary rescue if their carrying capacity is large enough. In ourmodel, it is the maximum carrying capacity that affects the


z

K(a)

(b)

(c)

0

z

t0

z0

zn

t0

n~

K

Nc,high

Nc,low

k(z)

C(z)

zn*z

0*

Figure 6. Competition can help or hinder evolutionary rescue. (a) Carryingcapacity k (solid curve) as a function of trait value ẑ and two competition Cscenarios: complete niche overlap (dashed curve) or partial niche overlap (dotted-dashed curve). (b) With complete niche overlap (dashed curve) competitionincreases as the population adapts, and the population therefore adapts slowerthan it would without competition (solid curve). With partial niche overlap (dot-dashed curve) competition decreases as the population adapts, and the populationadapts faster. (c) The time a population spends at risk of extinction (the timeabundance ~n is below critical abundance Nc) depends on competition’s effect onabundance and evolution as well as on the value of the critical abundance. Forinstance, when the critical abundance is low Nc,low both competition scenariosincrease the time at risk relative to when there is no competition (solid curve)because they depress the focal population’s abundance. However, when the criticalabundance is high Nc,high partial niche overlap (dotted-dashed curve) decreases thetime at risk relative to the no competition case (solid curve) because it sufficientlyincreases the rate of adaptation.


SocB368:20120085

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potential, and need, for evolutionary rescue. Since abundanceasymptotically approaches maximum carrying capacity inevolutionary time (figure 2), maximum carrying capacity willhave a larger effect on the time at risk as it approaches thecritical abundance (figure 3).

Notice that maximum carrying capacity plays both a demo-graphic and evolutionary role; for a given environmentalchange, larger values keep populations at larger abundances(K in equation (2.8)) and, following the change, increase therate of evolution (K0 in equation (2.7)). Here we assume greaterabundances lead to faster evolution because they cause greatermutational inputs. In previous models [3,34], where the rate ofevolution is determined by additive genetic variation instead ofmutational input, the relationship between population size andthe rate of evolution can be weaker (reviewed in [65]).Although non-additive genetic effects, such as epistasis anddominance, and temporal fluctuations in abundance (leadingto lower effective population sizes) can weaken the relationshipbetween population size and the rate of evolution [66], they donot qualitatively alter our results, but merely lead to a slowerrate of evolution than predicted.

Given the differences between quantitative geneticsand adaptive dynamics [51], our results are surprisinglyconsistent with previous quantitative genetic models of evo-lutionary rescue [3,34]. We derive a similar evolutionarytrajectory and agree with Gomulkiewicz & Holt [3] on howtime at risk should increase with initial maladaptation anddecrease with abundance before environmental change.

There is, however, one major difference between ourapproach and previous models of evolutionary rescue. Allprevious models assume the environmental change affectsintrinsic growth rate, and that it is the intrinsic growth ratethat must evolve fast enough to allow persistence. In ourmodel, intrinsic growth rate R has no effect on abundancesince populations are assumed to remain at demographicequilibrium, which is independent of R. In particular, theenvironmental change might affect R with no effect on abun-dance (so long as R . 0). Intrinsic growth rate is thereforeirrelevant for evolutionary rescue in our model. Here rescuedepends on the effect of the environmental change on carry-ing capacity k, and the evolution of k. Past models describeevolutionary rescue under r-selection while we describe evo-lutionary rescue under K-selection [67,68]. Hence, our modelis more applicable to situations where density-dependenceremains strong following the environmental change, duringsubsequent adaptation. Density-dependence will remainstrong when the demand for resources continues to equalthe supply. Obviously, density-dependence will remainstrong when an environmental change acts only to reducethe supply of resources. This describes how a populationof Darwin’s finches has responded to drought [54]. Thedrought lowered the supply of seeds the finches ate, causinga rapid decline in finch abundance. Competition for smallseeds intensified following drought and the finch populationremained at carrying capacity, a carrying capacity which hadbeen reduced by decreased food supply. Density-dependencecan also be maintained when an environmental change leavesthe supply of resources unaffected but increases the per capitademands. For instance, if stress tolerance requires increasedenergetic demands, a population exposed to a stress may con-tinue to experience strong density-dependence despite adecline in abundance and unaffected resources. This maydescribe the situation observed in recent experiments of



SocB368:20120085

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evolutionary rescue in yeast populations exposed to salt,where glucose concentration was unaffected [5,38].

Simulations indicate that our analytical approximationsare sensitive to mutational input and the fixation times ofnew beneficial mutations. When mutations are too frequentor fixation times are too long, we consistently underestimatethe time at risk (figure 4). The underestimate probably arisesfrom the adaptive dynamic assumption that fixation occursinstantaneously and the population remains monomorphic.In simulations that permit greater polymorphism, less fitphenotypes compete with those closer to the adaptiveoptimum, imposing a demographic load on the population.The continued existence of less fit phenotypes slows theincrease of carrying capacity, causing populations toremain at risk of extinction for longer than expected. Thisis similar to what, in microbial evolution, is referred toas ‘clonal interference’ [69]. However, many populationsshould conform to our low mutation input assumption. Forinstance, the mutations rate of Saccharomyces cerevisiae salt tol-erance is approximately m ¼ 1027 mutations per genome pergeneration [5]. Since our analytical approximations are accur-ate when mKlog(K ) , 1, our method can handle yeastpopulations of about one million cells or fewer.

Although our approximations are most sensitive to highmutational inputs and slow fixation times, our assumptionthat mutational input is constant throughout adaptation (simi-lar to assuming constant phenotypic variance [3,48]) becomesless realistic as the initial stress becomes larger (figure 4).Assuming constant mutational input is necessary for ananalytical solution, but causes us to consistently underestimatethe time at risk. In reality, environmental changes will causereductions in abundance which will decrease the supply rateof new mutations (or phenotypic variance [48]), effectively‘pulling the rug out from under evolutionary rescue’ [30].Both ours and the traditional quantitative genetic [48]analytical approximations are less accurate under strong selec-tion [29]. Because high mutation rates, long fixation times andlarge initial stresses all cause our approximation to underesti-mate the time at risk, our analytical results can be considereda best-case scenario for population persistence.

Competition between individuals of distinct species islikely to cause dramatic changes in selective pressures [62,70].If competition is strong enough to drive rapid adaptation,competitors can potentially help a population adapt and per-sist following an environmental change. In a continuouslychanging environment, computer simulations of two compet-ing populations have shown that competition can aid thepersistence of the better-adapted population by increasingselective pressure, effectively ‘pushing’ the phenotype ofthe better-adapted population towards the moving opti-mal [36]. Our results clarify this point—competition can aidpopulation persistence when it increases the selectivepressure to evolve to the new environment—and give anadditional requirement: competition must increase selectionpressure enough to overcome the negative effect of reducedabundance. The effect of competition on evolutionaryrescue can be explained in terms of the overlap between thecompetitor’s niche and the niche to which the focal popu-lation is attempting to adapt. When the focal population isforced to adapt to a niche already occupied by a competitor(strong niche overlap), competition will hinder adaptationbecause competition selects in the opposite direction as thenew environment (dashed curve in figure 6). On the other

hand, when the competitor has a niche which only partiallyoverlaps the niche to which the focal population is attemptingto adapt, it can speed adaptation by depressing the fitness ofindividuals in the focal population which are farther fromthe new niche (dotted-dashed curve in figure 6). We can illus-trate this concept by returning to the example of Darwin’sfinches. Drought reduced the supply of small seeds, shiftingthe niche available to the medium ground finch (Geospizafortis) to larger seeds. In general, this caused fortis populationsto evolve to larger size [54]. However, in the presence of thelarge ground finch G. magnirostris, who eat large seeds(strong niche overlap), larger fortis were outcompeted by mag-nirostris, preventing fortis from evolving to larger size [71,72].Meanwhile, in the presence of the small ground finch G. fuligi-nosa, who eat small seeds (partial niche overlap), smaller fortiswere outcompeted by fuliginosa, causing fortis to evolve to alarger size faster than they did in the absence of competi-tors [61]. Populations of fortis approached the new adaptivepeak faster when in competition with fuliginosa because fuligi-nosa increased selection pressure towards the peak. Whatremains to be seen, and what is pivotal for evolutionaryrescue, is whether the increased adaptation of fortis in the pres-ence of fuliginosa overcame the reduction in fortis abundancecaused by competition with fuliginosa.

On the other hand, competition may be the very reasonevolutionary rescue is required for persistence in the firstplace. Invasive species, for example, can greatly reduce theabundance of pre-existing competitors, putting many popu-lations at risk of extinction (reviewed in [14]). Our resultssuggest that some invading populations, which are them-selves the cause of extinction risk, hinder evolutionaryrescue in their competitors, while other invaders maypermit rapid adaptation. The model presented here maytherefore help predict if an invasive species is likely tocause niche displacement or extinction (reviewed in [13]).Since few examples of extinction are associated with competi-tive interactions between native and invasive species [13],invading competitors may often allow rapid adaptation.

Although we have shown that competition can helpevolutionary rescue under specific circumstances, we havesimultaneously shown that in other circumstances competitionwill surely hinder persistence. Interspecific competition is alsoexpected to reduce rates of adaptation in the context of species’range limits [72] and gradual environmental changes in meta-communities [73]. When competition hinders adaptation, weexpect evolutionary rescue to be more common in commu-nities with reduced niche overlap [74] or greater characterdisplacement [75], since in these communities there shouldbe less interspecific competition.

Coevolution can alter the demographic costs and selectionpressures imposed by competition, therefore impactingpopulation persistence [70]. In our case, altering the strengthand selection pressure of competition means a shift in theheight and slope of the competition curve (figure 5), respect-ively, as the focal population evolves. A number of previousstudies have investigated the effect of coevolution betweencompetitors (although not in the context of evolutionaryrescue; [37,52,53]). Here, instead of asking how a specificform of coevolution influences persistence, we ask a moregeneral question: what types of coevolution help (or hinder)evolutionary rescue? For example, if coevolution is expectedto cause strong character displacement [53], not only willthe less-adapted population ‘push’ the better-adapted



SocB368:20120085

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population to even greater levels of adaptation, but thebetter-adapted population will also ‘push’ the less-adaptedpopulation away from it, possibly reducing the positiveeffect of competition on evolutionary rescue.

Although our analytical approach sometimes requiresstricter assumptions than simulation studies (e.g. constantmutational input), it avoids the finite choice of parametervalues demanded in simulation studies, and thereby providesmore general results. For instance, our expression for time atrisk (equation (2.10)) shows a unimodal relationship withenvironmental tolerance (figure 5), indicating that extinctionis most likely at intermediate tolerances. Extinction is mostprobable at intermediate environmental tolerances becausesmall tolerances cause strong selection pressures and hence—if the population can survive the initial stress—fast evolution,while large tolerances allow high degrees of maladaptationwithout a demographic cost. To our knowledge, this is thefirst time this relationship has been clearly demonstrated.

In a recent experiment of adaptation to a novel environmentunder competition, Collins [9] subjected pairs of competingphotosynthetic microbe strains to increased carbon dioxidelevels. Despite the loss of one of the competing strains partway through the experiment, the presence of a competitor atthe beginning of the experiment always reduced the final abun-dance of the survivor. Collins [9] partitioned the effects ofphysiology, evolution to increased carbon dioxide levels, andcompetitive ability on final abundance. She found that whencompetition had an effect it was always opposing evolution tocarbon dioxide. In other words, when competition affectedadaptation, it was because the superior competitor went extinctwhile the strain most capable of adapting to the new environ-ment evolved slower than it would have in monoculture. Atrade-off between competitive ability and the ability to adaptto abiotic change lowered the abundance of both strains, imped-ing evolutionary rescue of all. In our model, this amounts toa positive correlation between carrying capacity and compe-tition during the initial stages of adaptation. When thispositive correlation exists, competition will nearly alwaysimpede evolutionary rescue.

To our knowledge, this is the first analytical workto investigate the effect of interspecific competition onevolutionary rescue. In doing so, we have highlightedthe general ecological and evolutionary settings wherecompetition should help or hinder persistence toenvironmental change.

We thank Helene Weigang, Ophélie Ronce, Peter Jackson, RobertD. Holt and an anonymous reviewer for helpful comments onthe manuscript. M.M.O. was funded by a Alexander Graham BellCanada Graduate Scholarship from the National Sciences andEngineering Research Council of Canada, the Quebec Centre for Bio-diversity Science, and the Dr Neal Simon Memorial Scholarship.C.d.M. acknowledges a Discovery Grant from the Natural Sciencesand Engineering Research Council of Canada.

Appendix AHere we find the singular strategy in the one-populationcase and evaluate its stability. Detailed methods can befound in Geritz et al. [40]. From equation (2.1), the local fit-ness gradient is

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R @

@zm

aðzm; ẑÞnrkðzm; z�Þ

� �zm¼ẑ

; ðA 1Þ

where zm is the trait value of a rare mutant with abundancenm and ẑ is the trait value of the resident with abundance nr.Dropping the arguments of the functions and denoting@=@zm with prime gives

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R nr

a0k � ak0k2

� �� zm¼ẑ

: ðA 2Þ

Assuming (d/dz)a(z,z) ¼ 0 and a(z,z) ¼ 1, evaluating atzm ¼ ẑ gives

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ Rnr

k0

k2: ðA 3Þ

Specifying k as a Gaussian function (equation (2.2)) withmean z* and variance sk

2,

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R ðẑ� z

�Þs2k

: ðA 4Þ

The local fitness gradient is zero when ẑ ¼ z� (i.e. z* is thesingular strategy). If z* maximizes the local fitness gradient, itis a fitness maximum and therefore evolutionarily stable(ESS). If z* minimizes the local fitness gradient, it is a fitnessminima and evolutionary branching may occur [40]. Thesingular strategy is a fitness maximum when

@2

@z2m

1nm

dnmdt

� ��zm¼ẑ¼z�

, 0 ðA 5Þ

or, equivalently

�Rnr@

@zm

a0k � ak0k2

� �� zm¼ẑ¼z�

, 0: ðA 6Þ

Evaluating at zm ¼ ẑ ¼ z� gives

� R a00 � k00

K

� �, 0; ðA 7Þ

and z* is therefore evolutionarily stable when

a00 .k00

K: ðA 8Þ

Specifying k as equation (2.2), z* is evolutionarily stable when

a00 . � 1s2k: ðA 9Þ

The population will converge on the singular strategy z*only if

@2

@z2m

1nm

dnmdt

� �� zm¼ẑ¼z�

,@2

@ ẑ21

nm

dnmdt

� �� zm¼ẑ¼z�

ðA 10Þ

and

� R a00 � k00

K

� �, 0; ðA 11Þ

and so, if the singular point is evolutionarily stable itis also convergence stable. Throughout the paper, weassume equation (A11) holds to simplify our analysis ofevolutionary rescue.

Appendix BHere we derive approximations for the ecological andevolutionary dynamics in the one-population case (equa-tions (2.7) and (2.8)). We first move all terms of equation (2.6)with ẑ to the left-hand side and bring dt to the right. Then



SocB368:20120085

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taking the integral,ðeðẑ�z

�nÞ

2=2s2k

ðẑ� z�nÞdẑ ¼

ð�ms2mKR2s2k

dt: ðB 1Þ

Since there is no analytical solution for the indefiniteintegral on the left-hand side, we use the Taylor expansionabout x¼ 0, ex2=a=x ¼Pðx2n�1=n!anÞ, with x ¼ ẑ� z�n and a ¼2sk

2. Taking the Taylor series about ẑ� z�n ¼ 0 leads us toassume a small change in abundance and hence constant muta-tional input mK. We therefore replace K with K0 to indicatethat mutational input depends on the original abundance.We now haveðX1

n¼0

ðẑ� z�nÞ2n�1

n!ð2s2kÞn dẑ ¼

�ms2mK0R2s2k

t; ðB 2Þ

ð1

ẑ� z�nþ ẑ� z

�n

2s2kþ ðẑ� z

�nÞ

3

8s4kþ � � �

!dẑ ¼

�ms2mK0R2s2k

t ðB 3Þ

and

lnðẑ� z�nÞ þðẑ� z�nÞ

2

4s2kþ � � � þ C ¼

�ms2mK0R2s2k

t: ðB 4Þ

Approximating to the first-order

lnðẑ� z�nÞ þ C ��ms2mK0R

2s2kt; ðB 5Þ

and solving for ẑ gives

ẑ � z�n þ eð�ms2mK0R=2s

2k Þt�C: ðB 6Þ

At t ¼ 0, we have ẑ ¼ z�0, so C ¼ �lnðz�0 � z�nÞ and we getequation (2.7):

ẑðtÞ � z�n þ ðz�0 � z�nÞ eð�ms2mK0R=2s

2k Þt: ðB 7Þ

Subbing equation (B7) into equation (2.2) gives an approximatedescription of population abundance across evolutionary time(equation (2.8)).

Appendix CHere we find the singular strategies for a populationexperiencing interspecific competition and evaluate theirstability. From equation (2.13), the local fitness gradient is

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R @

@zm

aðzm; ẑÞnr þ Cðzm; tÞkðzm; z�Þ

� �� zm¼ẑ

:

ðC 1Þ

where zm and nm are the trait value and abundance of arare mutant, respectively, in a population with residenttrait value ẑ and abundance nr. We drop the argumentsof the functions and denote @=@zm with prime. Expand-ing gives

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R nr

a0k � ak0k2

þ C0k � Ck0

k2

� �zm¼ẑ

:

ðC 2Þ

And from equation (2.14):

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R ðk � CÞa

0k � ak0k2

þ C0k � Ck0

k2

� �zm¼ẑ

:

ðC 3Þ

Evaluating at zm ¼ ẑ:

@

@zm

1nm

dnmdt

� ��zm¼ẑ¼R a

0k2�akk0 �a0CkþaCk0 þC0k�Ck0k2

� �:

ðC4Þ

Assuming intraspecific competition a is maximal whenindividuals share the same trait value, (@=@zi)a(zi,zi) ¼ 0,and a(zi,zi) ¼ 1:

gðẑ; z�Þ ¼ @@zm

1nm

dnmdt

� ��zm¼ẑ¼ �R k

0 � C0k

� �: ðC 5Þ

Equation (C5) determines the direction of selection.Evolution proceeds until gðẑ; z�Þ ¼ 0, in this case whenk’ ¼ C’. The trait values giving gðẑ; z�Þ ¼ 0 are evolutionarilysingular strategies, which we will denote z*c. If z*c maximizesgðẑ; z�Þ, z�c is a fitness maximum; when ẑ ¼ z�c no nearbymutant can invade and the population remains monomorphicwith ẑ ¼ z�c . However, when z*c minimizes gðẑ; z�Þ, z�c is a fitnessminima and evolutionary branching may occur [40]. A singularpoint zc

* is a fitness maximum when

@2

@z2m

1nm

dnmdt

� ��zm¼ẑ¼z�c

¼ �R a00ðk2 � CkÞ þ kðC00 � k00Þ þ ðk0Þ2ðk3 � Ck2 � 1Þ

k2

" #, 0:

ðC 6Þ

To simplify our analysis of evolutionary rescue, we assume thatall singular strategies our population approaches are fitnessmaxima. This assumes, at zm ¼ ẑ ¼ z�c ,

a00ðk2 � CkÞ þ kðC00 � k00Þ þ ðk0Þ2ðk3 � Ck2 � 1Þ . 0: ðC 7Þ

We will also assume the singular strategies areconvergence stable, requiring

@2

@z2m

1nm

dnmdt

� �� zm¼ẑ¼z�c

,@2

@ ẑ21

nm

dnmdt

� �� zm¼ẑ¼z�c

: ðC 8Þ

Appendix DBeginning with equation (2.16), we look to find when inter-specific competition speeds adaptation towards the optimalz* ¼ z*n. Dropping the arguments of the functions anddenoting @=dẑ with prime, equation (2.16) reads

dẑdt¼�ms2m

2½k � C�R k

0 � C0k

� �ðD 1Þ

and

dẑdt¼�ms2mR

2ðk � CÞðk0 � C0Þ

k

� �: ðD 2Þ

Since in the one-population case dẑ=dt ¼ ð�ms2mR=2Þk0(equation (2.6)), competition will speed evolution when

ðk � CÞðk0 � C0Þk

�� . jk0j: ðD 3Þ

Since k and k 2 C must be positive for the population to persist,

jk0 � C0j . kk � C jk

0j; ðD 4Þ

yielding equation (2.17).


12


References


SocB368:20120085

1. Hereford J. 2009 A quantitative survey of localadaptation and fitness trade-offs. Am. Nat. 173,579 – 588. (doi:10.1086/597611)

2. Maynard Smith J. 1989 The causes of extinction.Phil. Trans. R. Soc. Lond. B 325, 241 – 252. (doi:10.1098/rstb.1989.0086)

3. Gomulkiewicz R, Holt R. 1995 When does evolutionby natural selection prevent extinction? Evolution49, 201 – 207. (doi:10.2307/2410305)

4. Gienapp P, Teplitsky C, Alho J, Mills J, Merilä J. 2008Climate change and evolution: disentanglingenvironmental and genetic responses. Mol. Ecol. 17,167 – 178. (doi:10.1111/j.1365-294X.2007.03413.x)

5. Bell G, Gonzalez A. 2009 Evolutionary rescue canprevent extinction following environmental change.Ecol. Lett. 12, 942 – 948. (doi:10.1111/j.1461-0248.2009.01350.x)

6. Tylianakis JM, Didham RK, Bascompte J, Wardle DA.2008 Global change and species interactions interrestrial ecosystems. Ecol. Lett. 11, 1351 – 1363.(doi:10.1111/j.1461-0248.2008.01250.x)

7. Poloczanska E, Hawkins S, Southward A, Burrows M.2008 Modeling the response of populations ofcompeting species to climate change. Ecology 89,3138 – 3149. (doi:10.1890/07-1169.1)

8. Harmon JP, Moran NA, Ives AR. 2009 Speciesresponse to environmental change: impacts of foodweb interactions and evolution. Science 323, 1347 –1350. (doi:10.1126/science.1167396)

9. Collins S. 2011 Competition limits adaptation andproductivity in a photosynthetic alga at elevatedCO2. Proc. R. Soc. B 278, 247 – 255. (doi:10.1098/rspb.2010.1173)

10. Low-Décarie E, Fussmann GF, Bell G. 2011 The effectof elevated CO2 on growth and competition inexperimental phytoplankton communities. GlobalChange Biol. 17, 2525 – 2535. (doi:10.1111/j.1365-2486.2011.02402.x)

11. Zhang Q-G, Buckling A. 2011 Antagonistic coevolutionlimits population persistence of a virus in a thermallydeteriorating environment. Ecol. Lett. 14, 282 – 288.(doi:10.1111/j.1461-0248.2010.01586.x)

12. Kovach-Orr C, Fussmann GF. 2012 Evolutionaryand plastic rescue in multitrophic modelcommunities. Phil. Trans. R. Soc. B 368, 20120084.(doi:10.1098/rstb.2012.0084)

13. Mooney Ha, Cleland EE. 2001 The evolutionaryimpact of invasive species. Proc. Natl Acad. Sci. USA98, 5446 – 5451. (doi:10.1073/pnas.091093398)

14. Gurevitch J, Padilla DK. 2004 Are invasive species amajor cause of extinctions? Trends Ecol. Evol. 19,470 – 474. (doi:10.1016/j.tree.2004.07.005)

15. Lavergne S, Mouquet N, Thuiller W, Ronce O. 2010Biodiversity and climate change: integratingevolutionary and ecological responses of species andcommunities. Annu. Rev. Ecol. Evol. Syst. 41, 321 –350. (doi:10.1146/annurev-ecolsys-102209-144628)

16. Hoffmann AA, Sgrò CM. 2011 Climate change andevolutionary adaptation. Nature 470, 479 – 485.(doi:10.1038/nature09670)

17. Gause G, Witt A. 1935 Behavior of mixedpopulations and the problem of natural selection.Am. Nat. 69, 596 – 609. (doi:10.1086/280628)

18. Ullyett GC. 1950 Competition for food and alliedphenomena in sheep-blowfly populations. Phil.Trans. R. Soc. Lond. B 234, 77 – 174. (doi:10.1098/rstb.1950.0001)

19. Ayala FF. 1969 Experimental invalidation of theprinciple of competitive exclusion. Nature 224,1076 – 1079. (doi:10.1038/2241076a0)

20. Martin P, Martin T. 2001 Ecological and fitnessconsequences of species coexistence: a removalexperiment with wood warblers. Ecology 82,189 – 206. (doi:10.1890/0012-9658(2001)082[0189:EAFCOS]2.0.CO;2)

21. Bengtsson J. 1989 Interspecific competitionincreases local extinction rate in a metapopulationsystem. Nature 340, 713 – 715. (doi:10.1038/340713a0)

22. Willi Y, Hoffmann AA. 2009 Demographic factorsand genetic variation influence populationpersistence under environmental change. J. Evol.Biol. 22, 124 – 133. (doi:10.1111/j.1420-9101.2008.01631.x)

23. Martin MJ, Pérez-tomé JM, Toro MA. 1988Competition and genotypic variability in Drosophilamelanogaster. Heredity 60, 119 – 123. (doi:10.1038/hdy.1988.17)

24. Antonovics J. 1976 The nature of limits to naturalselection. Ann. Missouri Bot. Garden 63, 224 – 247.(doi:10.2307/2395303)

25. Bolnick DI. 2001 Intraspecific competition favoursniche width expansion in Drosophila melanogaster.Nature 410, 463 – 466. (doi:10.1038/35068555)

26. Agashe D, Bolnick DI. 2010 Intraspecific geneticvariation and competition interact to influenceniche expansion. Proc. R. Soc. B 277, 2915 – 2924.(doi:10.1098/rspb.2010.0232)

27. Birch L. 1955 Selection in Drosophila pseudoobscurain relation to crowding. Evolution 9, 389 – 399.(doi:10.2307/2405474)

28. Holt RD, Gomulkiewicz R. 1996 The evolution ofspecies’ niches: a population dynamic perspective.In Case studies in mathematical modeling (edsHG Othmer, fr Adler, MA Lewis, JC Dallon),pp. 25 – 50. Saddle River, NJ: Prentice-Hall.

29. Boulding EG, Hay T. 2001 Genetic anddemographic parameters determining populationpersistence after a discrete change in the environment.Heredity 86, 313 – 324. (doi:10.1046/j.1365-2540.2001.00829.x)

30. Orr HA, Unckless RL. 2008 Population extinction andthe genetics of adaptation. Am. Nat. 172,160 – 169. (doi:10.1086/589460)

31. Gomulkiewicz R, Holt RD, Barfield M, Nuismer SL.2010 Genetics, adaptation, and invasion in harshenvironments. Evol. Appl. 3, 97 – 108. (doi:10.1111/j.1752-4571.2009.00117.x)

32. Chevin L-M, Lande R, Mace GM. 2010 Adaptation,plasticity, and extinction in a changing

environment: towards a predictive theory. PLoS Biol.8, e1000357. (doi:10.1371/journal.pbio.1000357)

33. Uecker H, Hermisson J. 2011 On the fixation processof a beneficial mutation in a variable environment.Genetics 188, 915 – 930. (doi:10.1534/genetics.110.124297)

34. Chevin L-M, Lande R. 2010 When do phenotypicplasticity and genetic evolution prevent extinctionof a density-regulated population? Evolution64, 1143 – 1150. (doi:10.1111/j.1558-5646.2009.00875.x)

35. Urban MC, De Meester L, Vellend M, Stoks R,Vanoverbeke J. 2012 A crucial step toward realism:responses to climate change from an evolvingmetacommunity perspective. Evol. Appl. 5, 154 –167. (doi:10.1111/j.1752-4571.2011.00208.x)

36. Jones AG. 2008 A theoretical quantitative geneticstudy of negative ecological interactions andextinction times in changing environments. BMCEvol. Biol. 8, 119. (doi:10.1186/1471-2148-8-119)

37. Taper M, Case T. 1992 Models of characterdisplacement and the theoretical robustness oftaxon cycles. Evolution 46, 317 – 333. (doi:10.2307/2409853)

38. Samani P, Bell G. 2010 Adaptation of experimentalyeast populations to stressful conditions in relationto population size. J. Evol. Biol. 23, 791 – 796.(doi:10.1111/j.1420-9101.2010.01945.x)

39. Smith T. 1987 Bill size polymorphism andintraspecific niche utilization in an African finch.Nature 329, 717 – 719. (doi:10.1038/329717a0)

40. Geritz S, Kisdi E, Meszena G, Metz J. 1998Evolutionarily singular strategies and the adaptivegrowth and branching of the evolutionary tree. Evol.Ecol. 12, 35 – 57. (doi:10.1023/A:1006554906681)

41. Maynard Smith J, Price G. 1973 The logic of animalconflict. Nature 246, 15 – 18. (doi:10.1038/246015a0)

42. Doebeli M, Dieckmann U. 2000 Evolutionarybranching and sympatric speciation caused bydifferent types of ecological interactions. Am. Nat.156, S77 – S101. (doi:10.1086/303417)

43. Rummel J, Roughgarden J. 1985 A theory of faunalbuildup for competition communities. Evolution 39,1009 – 1033. (doi:10.2307/2408731)

44. Dieckmann U, Law R. 1996 The dynamical theory ofcoevolution: a derivation from stochastic ecologicalprocesses. J. Math. Biol. 34, 579 – 612. (doi:10.1007/BF02409751)

45. Waxman D, Gavrilets S. 2005 20 questions onadaptive dynamics. J. Evol. Biol. 18, 1139 – 1154.(doi:10.1111/j.1420-9101.2005.00948.x)

46. Lande R. 1993 Risks of population extinction fromdemographic and environmental stochasticity andrandom catastrophes. Am. Nat. 142, 911 – 927.(doi:10.1086/285580)

47. Fisher R. 1930 The genetical theory of naturalselection. London, UK: Clarendon Press.

48. Lande R. 1976 Natural selection and randomgenetic drift in phenotypic evolution. Evolution 30,314 – 334. (doi:10.2307/2407703)

http://dx.doi.org/doi:10.1086/597611http://dx.doi.org/doi:10.1098/rstb.1989.0086http://dx.doi.org/doi:10.1098/rstb.1989.0086http://dx.doi.org/doi:10.2307/2410305http://dx.doi.org/doi:10.1111/j.1365-294X.2007.03413.xhttp://dx.doi.org/doi:10.1111/j.1461-0248.2009.01350.xhttp://dx.doi.org/doi:10.1111/j.1461-0248.2009.01350.xhttp://dx.doi.org/doi:10.1111/j.1461-0248.2008.01250.xhttp://dx.doi.org/doi:10.1890/07-1169.1http://dx.doi.org/doi:10.1126/science.1167396http://dx.doi.org/doi:10.1098/rspb.2010.1173http://dx.doi.org/doi:10.1098/rspb.2010.1173http://dx.doi.org/doi:10.1111/j.1365-2486.2011.02402.xhttp://dx.doi.org/doi:10.1111/j.1365-2486.2011.02402.xhttp://dx.doi.org/doi:10.1111/j.1461-0248.2010.01586.xhttp://dx.doi.org/doi:10.1098/rstb.2012.0084http://dx.doi.org/doi:10.1073/pnas.091093398http://dx.doi.org/doi:10.1016/j.tree.2004.07.005http://dx.doi.org/doi:10.1146/annurev-ecolsys-102209-144628http://dx.doi.org/doi:10.1038/nature09670http://dx.doi.org/doi:10.1086/280628http://dx.doi.org/doi:10.1098/rstb.1950.0001http://dx.doi.org/doi:10.1098/rstb.1950.0001http://dx.doi.org/doi:10.1038/2241076a0http://dx.doi.org/doi:10.1890/0012-9658(2001)082[0189:EAFCOS]2.0.CO;2http://dx.doi.org/doi:10.1890/0012-9658(2001)082[0189:EAFCOS]2.0.CO;2http://dx.doi.org/doi:10.1038/340713a0http://dx.doi.org/doi:10.1038/340713a0http://dx.doi.org/doi:10.1111/j.1420-9101.2008.01631.xhttp://dx.doi.org/doi:10.1111/j.1420-9101.2008.01631.xhttp://dx.doi.org/doi:10.1038/hdy.1988.17http://dx.doi.org/doi:10.1038/hdy.1988.17http://dx.doi.org/doi:10.2307/2395303http://dx.doi.org/doi:10.1038/35068555http://dx.doi.org/doi:10.1098/rspb.2010.0232http://dx.doi.org/doi:10.2307/2405474http://dx.doi.org/doi:10.1046/j.1365-2540.2001.00829.xhttp://dx.doi.org/doi:10.1046/j.1365-2540.2001.00829.xhttp://dx.doi.org/doi:10.1086/589460http://dx.doi.org/doi:10.1111/j.1752-4571.2009.00117.xhttp://dx.doi.org/doi:10.1111/j.1752-4571.2009.00117.xhttp://dx.doi.org/doi:10.1371/journal.pbio.1000357http://dx.doi.org/doi:10.1534/genetics.110.124297http://dx.doi.org/doi:10.1534/genetics.110.124297http://dx.doi.org/doi:10.1111/j.1558-5646.2009.00875.xhttp://dx.doi.org/doi:10.1111/j.1558-5646.2009.00875.xhttp://dx.doi.org/doi:10.1111/j.1752-4571.2011.00208.xhttp://dx.doi.org/doi:10.1186/1471-2148-8-119http://dx.doi.org/doi:10.2307/2409853http://dx.doi.org/doi:10.2307/2409853http://dx.doi.org/doi:10.1111/j.1420-9101.2010.01945.xhttp://dx.doi.org/doi:10.1038/329717a0http://dx.doi.org/doi:10.1023/A:1006554906681http://dx.doi.org/doi:10.1038/246015a0http://dx.doi.org/doi:10.1086/303417http://dx.doi.org/doi:10.2307/2408731http://dx.doi.org/doi:10.1007/BF02409751http://dx.doi.org/doi:10.1007/BF02409751http://dx.doi.org/doi:10.1111/j.1420-9101.2005.00948.xhttp://dx.doi.org/doi:10.1086/285580http://dx.doi.org/doi:10.2307/2407703http://rstb.royalsocietypublishing.org/


SocB368:20120085

13


49. Kopp M, Hermisson J. 2007 Adaptation of aquantitative trait to a moving optimum.Genetics 176, 715 – 719. (doi:10.1534/genetics.106.067215)

50. R Development Core Team. 2011 R: a language andenvironment for statistical computing. Vienna,Austria: R Foundation for Statistical Computing.

51. Champagnat N, Ferrière R, Méléard S. 2006 Unifyingevolutionary dynamics: from individual stochasticprocesses to macroscopic models. Theor.Popul. Biol. 69, 297 – 321. (doi:10.1016/j.tpb.2005.10.004)

52. Case T, Taper M. 2000 Interspecific competition,environmental gradients, gene flow, and thecoevolution of species’ borders. Am. Nat. 155,583 – 605. (doi:10.1086/303351)

53. Goldberg E, Lande R. 2006 Ecological andreproductive character displacement of anenvironmental gradient. Evolution 60, 1344 – 1357.

54. Boag PT, Grant PR. 1981 Intense natural selection ina population of Darwin’s finches (Geospizinae) inthe Galápagos. Science 214, 82 – 85. (doi:10.1126/science.214.4516.82)

55. Malthus T. 1798 An essay on the principle ofpopulation. London, UK: J. Johnson.

56. Darwin C. 1859 On the origin of species. London, UK:John Murray. (Reprinted by Random House 1998.)

57. Crowell K. 1961 The effects of reduced competitionin birds. Proc. Natl Acad. Sci. USA 47, 240 – 243.(doi:10.1073/pnas.47.2.240)

58. Futuyma D. 1970 Variation in genetic response tointerspecific competition in laboratory populationsof Drosophila. Am. Nat. 104, 239 – 252. (doi:10.1086/282658)

59. Bridle JR, Vines TH. 2007 Limits to evolution atrange margins: when and why does adaptation fail?Trends Ecol. Evol. 22, 140 – 147. (doi:10.1016/j.tree.2006.11.002)

60. Falconer D, MacKay T. 1996 Introduction toquantitative genetics. Essex, UK: Longman.

61. Schluter D, Price TD, Grant PR. 1985 Ecological characterdisplacement in Darwin’s finches. Science 227,1056 – 1059. (doi:10.1126/science.227.4690.1056)

62. Bell G. 2008 Selection. Oxford, UK: Oxford UniversityPress.

63. Seaton A, Antonovics J. 1967 Population inter-relationships. I. Evolution in mixtures of Drosophilamutants. Heredity 22, 19 – 33. (doi:10.1038/hdy.1967.2)

64. Filin I, Holt RD, Barfield M. 2008 The relation ofdensity regulation to habitat specialization,evolution of a species’ range, and the dynamics ofbiological invasions. Am. Nat. 172, 233 – 247.(doi:10.1086/589459)

65. Holt RD. 1997 Rarity and evolution: sometheoretical considerations. In The biology of rarity(eds WE Kunin, KJ Gaston), pp. 209 – 234. London,UK: Chapman and Hall.

66. Lande R, Barrowclough G. 1987 Effective populationsize, genetic variation, and their use in population

management. In Viable populations for conservation(ed. ME Soulé), pp. 86 – 99. Cambridge, UK:Cambridge University Press.

67. MacArthur R, Levins R. 1967 The limiting similarity,convergence, and divergence of coexistingspecies. Am. Nat. 101, 377 – 385. (doi:10.1086/282505)

68. Pianka ER. 1970 On r- and K-selection. Am. Nat.104, 592 – 597. (doi:10.1086/282697)

69. Gerrish PJ, Lenski RE. 1998 The fate of competingbeneficial mutations in an asexual population.Genetica 102 – 103, 127 – 144. (doi:10.1023/A:1017067816551)

70. Van Valen L. 1973 A new evolutionary law. Evol.Theory 1, 1 – 30.

71. Grant PR, Grant BR. 2006 Evolution of characterdisplacement in Darwin’s finches. Science 313,224 – 226. (doi:10.1126/science.1128374)

72. Price TD, Kirkpatrick M. 2009 Evolutionarily stablerange limits set by interspecific competition.Proc. R. Soc. B 276, 1429 – 1434. (doi:10.1098/rspb.2008.1199)

73. de Mazancourt C, Johnson E, Barraclough TG. 2008Biodiversity inhibits species’ evolutionary responsesto changing environments. Ecol. Lett. 11, 380 – 388.(doi:10.1111/j.1461-0248.2008.01152.x)

74. Levins R. 1968 Evolution in changing environments.Princeton, NJ: Princeton University Press.

75. Brown WL, Wilson EO. 1956 Character displacement.Syst. Zool. 5, 49 – 64. (doi:10.2307/2411924)

http://dx.doi.org/doi:10.1534/genetics.106.067215http://dx.doi.org/doi:10.1534/genetics.106.067215http://dx.doi.org/doi:10.1016/j.tpb.2005.10.004http://dx.doi.org/doi:10.1016/j.tpb.2005.10.004http://dx.doi.org/doi:10.1086/303351http://dx.doi.org/doi:10.1126/science.214.4516.82http://dx.doi.org/doi:10.1126/science.214.4516.82http://dx.doi.org/doi:10.1073/pnas.47.2.240http://dx.doi.org/doi:10.1086/282658http://dx.doi.org/doi:10.1086/282658http://dx.doi.org/doi:10.1016/j.tree.2006.11.002http://dx.doi.org/doi:10.1016/j.tree.2006.11.002http://dx.doi.org/doi:10.1126/science.227.4690.1056http://dx.doi.org/doi:10.1038/hdy.1967.2http://dx.doi.org/doi:10.1038/hdy.1967.2http://dx.doi.org/doi:10.1086/589459http://dx.doi.org/doi:10.1086/282505http://dx.doi.org/doi:10.1086/282505http://dx.doi.org/doi:10.1086/282697http://dx.doi.org/doi:10.1023/A:1017067816551http://dx.doi.org/doi:10.1023/A:1017067816551http://dx.doi.org/doi:10.1126/science.1128374http://dx.doi.org/doi:10.1098/rspb.2008.1199http://dx.doi.org/doi:10.1098/rspb.2008.1199http://dx.doi.org/doi:10.1111/j.1461-0248.2008.01152.xhttp://dx.doi.org/doi:10.2307/2411924http://rstb.royalsocietypublishing.org/

How competition affects evolutionary rescueIntroductionModel and resultsOne-population modelComparison of one-population model to previous workSimulationsCompetition

DiscussionWe thank Helene Weigang, Ophélie Ronce, Peter Jackson, Robert D. Holt and an anonymous reviewer for helpful comments on the manuscript. M.M.O. was funded by a Alexander Graham Bell Canada Graduate Scholarship from the National Sciences and Engineering Research Council of Canada, the Quebec Centre for Biodiversity Science, and the Dr Neal Simon Memorial Scholarship. C.d.M. acknowledges a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.Appendix AAppendix BAppendix CAppendix DReferences

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