How competition affects evolutionary rescue

How competition affects evolutionary rescue. Populations facing novel environments can persist by adapting. In nature, the ability to adapt and persist will depend on interactions between coexisting individuals. Here we use an adaptive dynamic model to assess how the potential for evolutionary rescue is affected by intra- and interspecific competition. Intraspecific competition (negative density-dependence) lowers abundance, which decreases the supply rate of beneficial mutations, hindering evolutionary rescue. On the other hand, interspecific competition can aid evolutionary rescue when it speeds adaptation by increasing the strength of selection. Our results clarify this point and give an additional requirement: competition must increase selection pressure enough to overcome the negative effect of reduced abundance. We therefore expect evolutionary rescue to be most likely in communities which facilitate rapid niche displacement. Our model, which aligns to previous quantitative and population genetic models in the absence of competition, provides a first analysis of when competitors should help or hinder evolutionary rescue. 1. Introduction. Individuals are often adapted to their current environment [1]. When the environment changes individuals may become maladapted, fitness may drop, and population abundances may decline [2]. If the changes in the environment are severe enough, populations may go extinct. But populations can also evolve in response to the stress and thereby return to healthy abundances [3,4]. Why some populations are capable of rescuing themselves from extinction through evolution, while others go extinct, is a central question to both basic evolutionary theory and conservation [5]. Ecological and evolutionary responses to changing environments are contingent on the community in which the change occurs [6–10]. A population's ability to adapt and persist in changing environments will therefore also hinge on the surrounding community [11] (see also [12]). By including the ecological community in a formal theory of adaptation to changing environments, we may better predict the response of natural communities to contemporary stresses, such as invasive species [13,14] and global climate change [15,16]. Competition reduces population abundance [17–20]. Since less abundant populations are more likely to go extinct when exposed to new environments [21,22], competition may therefore lower the potential for evolutionary rescue. But competition can also increase selective pressure [23], speed niche expansion [24–26] and increase rates of evolution [27], possibly allowing populations to adapt to new conditions faster. These potentially contrasting effects may account for the unanticipated population dynamics and patterns of persistence in competitive communities [6] (but see [10]). Currently, most theories on adaptation to abrupt environmental change consider only isolated populations [3,28–33], and many of these studies assume unbounded population growth, thus ignoring intraspecific competition as well. The studies that do consider intraspecific competition, in the form of negative density-dependence, give inconsistent conclusions, stating that density-dependence has no effect [29] or decreases [30,34] persistence. Of the handful of studies that examine the effect of interspecific competition on adaptation to environmental change, nearly all predict slower adaptation and more extinctions (reviewed in [35]). One notable exception suggests that interspecific competition can aid persistence in a continuously changing environment, by adding a selection pressure that effectively ‘pushes’ the more adapted populations in the direction of the moving environment [36]. Here we use the mathematical framework of adaptive dynamics to describe the evolutionary and demographic dynamics of a population experiencing competition and an abrupt change in the environment. Adaptive dynamics allows us to incorporate both intra- and interspecific competition in an evolutionary model while maintaining analytical tractability. We assess the potential for evolution to rescue populations by measuring the ‘time at risk’, i.e. the time a population spends below a critical abundance [3]. First, we derive an expression for the ‘time at risk’ in a population undergoing an abrupt change in isolation. We then compare our results to previous studies and test the robustness of our results by relaxing a number of simplifying assumptions using computer simulations. Finally, we examine how a population's ability to adapt and persist to an abrupt environmental change is impacted by the presence of competing species. 2. Model and results. (a) One-population model. We first examine how, in the absence of competitors, an asexual population with density- and frequency-dependent population growth responds to an abrupt change in the environment. We assume that each individual in the population has a trait value z , and that a phenotype's growth rate is determined by both its own trait value as well as the trait value of all other individuals within the population. Population dynamics are described by the logistic equation (eqn 2 in [37]) where n i is the number of individuals with trait value z i , R is the per capita intrinsic growth rate, α ( z i , z j ) is the per capita competitive effect of individuals with trait z j on individuals with trait z i , and k ( z i , z *) is the carrying capacity of individuals with trait z i in an environment where the trait value giving maximum carrying capacity is z *. We describe carrying capacity k as a Gaussian distribution (eqn 1 in [37]) where K is the maximum carrying capacity and σ k >0 is the ‘environmental tolerance’, which describes how strongly carrying capacity varies with z i . For a given deviation from z *, smaller variances σ k 2 mean larger declines in carrying capacity k . We therefore refer to σ k −2 as the strength of stabilizing selection. Data on yeast responses to salt [5,38] fit Gaussian carrying capacity functions, as described in equation (2.2) (see the electronic supplementary material). We do not give a specific form for intraspecific competition α , but instead give requirements that are satisfied by a wide range of functions. First, we assume that individuals with the same trait value compete most strongly, that is (d/dz) α ( z , z ) = 0 and (d 2 /d z 2 ) α ( z,z ) 2 /d z 2 ) α ( z,z ) > σ k −2 the ‘optimal trait value’ z * is both convergence stable (i.e. by small steps the resident trait converges to z *) and evolutionarily stable (i.e. once no other strategies can invade; z * is an ESS, sensu Maynard Smith & Price [41]). We assume (d 2 /d z 2 ) α ( z,z ) > σ k −2 for the remainder of the paper, which means frequency-dependence is weak enough [42]. Our results apply for any function α , as long as z * is both convergence and evolutionarily stable. Let our population begin in a constant environment with optimal trait value . In time, all individuals become perfectly adapted . The population will reach equilibrium abundance , and its growth rate will become zero ( figure 1 ). Let us call this original abundance K 0 . Our initially adapted population is monomorphic for the optimal phenotype in the original environment (grey). When the environment changes, the carrying capacity function shifts (black). The new carrying capacity of our population K n = k ( z * 0 , z * n ) is the height of the intersection of the original trait value z * 0 and the new carrying capacity function. The population evolves towards the new optimal phenotype z * n . The population is at risk of extinction while its abundance is less than N c , or equivalently, while . Suppose then that the environment suddenly changes so that the new optimal trait value is . Our monomorphic population, with trait value , then immediately has equilibrial abundance ( figure 1 ). The environmental change serves to decrease the carrying capacity of the population. The population will initially survive the abrupt change if or, equivalently. Note that setting as the extinction threshold scales population abundance in units of minimal viable population size [37,43]. Because z * n is the new evolutionarily and convergence stable strategy, if the population survives the change it will evolve towards the new optimal trait value, . According to the canonical equation of adaptive dynamics [44], the monomorphic trait value will change at rate. where μ is the per capita per generation mutation rate, σ μ 2 is the mutational variance (mutations symmetrically distributed with mean of parental value) and is the local fitness gradient (appendix A): where n m and z m are a rare mutant's abundance and trait value, respectively, and is the resident trait value [40]. The local fitness gradient describes the slope of the fitness function in the neighbourhood of the parental trait value. Steeper slopes signify greater fitness differences between individuals with similar but unequal trait values [45]. Notice that R / σ k 2 is the strength of stabilizing selection per unit time. The rate of change in trait value is then. We cannot solve equation (2.6) explicitly for , but using a first-order Taylor expansion, we derive an approximate solution, describing evolution and demography following the abrupt change (appendix B): Taking the Taylor expansion about z * 0 − z * n = 0 results in the assumption that the environmental change is small relative to environmental tolerance α k (i.e. a weak ‘initial stress’). Our first-order approximation of the Gaussian k is therefore taken at the maximum z = 0, which is a line with slope zero and height K 0 . This means we assume mutational input μ k is constant at μ K 0 , effectively decoupling the demographic and evolutionary dynamics of the recovering population. Our first-order approximation is the highest-order for which we can obtain an analytical solution. Now, let N c be the abundance below which demographic or environmental stochasticity are likely to cause rapid extinction [3,46]. We use this heuristic N c , in the place of stochastic models, for simplicity. We are interested in the amount of time a population spends below this threshold, i.e. how long the population is at risk of extinction. The population will never be at risk of extinction if its equilibrial abundance remains above the critical abundance N c . In this model, equilibrial abundance strictly increases in evolutionary time in a constant environment. Abundance is therefore at a minimum immediately following the abrupt shift in the environment. The population will avoid all chance of extinction if N c 2 and the strength of stabilizing selection per unit time R / σ k 2 . Time at risk t r is a unimodal function of environmental tolerance σ k , 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 greater abundances for a given degree of maladaptation. Adaptation following an abrupt change in the environment. ( a ) Population trait value evolves towards the new optimal z * n (equation (2.7)). The time it takes to evolve a trait value z N c , which gives a critical abundance N c , is the expected ‘time at risk’ t r (equation (2.10)). ( b ) Population abundance 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 computer simulation, with dark common and white rare. The thick dashed line is total abundance at each time step in simulation. The observed time at risk is denoted t r obs . ( a ) Time at risk t r (equation (2.10)) increases monotonically with the magnitude of environmental change . Magnitudes of change smaller than Δ z ** are not large enough to put the population at risk of extinction (equation (2.9)) and magnitudes of change larger than Δ z * cause immediate extinction (equation (2.3)). ( b ) Time at risk t r increases as the critical abundance N c approaches maximum abundance K . As the critical abundance approaches the maximum abundance, N c / K → 1, the ratio has a stronger effect on the time at risk. ( c ) Time at risk t r is a unimodal function of ‘environmental tolerance’ σ k , where extinction is most likely at intermediate values. We must have σ k > σ * k for the population to survive the initial change in the environment and σ k 2 is the trait heritability and P is the constant phenotypic variance [3]. We derive a qualitatively similar trajectory (equation (2.7)), in continuous time, from adaptive dynamics. Adaptive dynamics provides greater ecological context by including intrinsic growth rate and maximum carrying capacity as parameters in the evolutionary trajectory. The trajectories are identical when. Gomulkiewicz & Holt [3] refer to equation (2.12) as the evolutionary ‘inertia’ of a trait. Inertia is bounded between zero and unity in both models. When inertia is unity there is no evolution. In Gomulkiewicz & Holt [3], evolution halts when trait heritability h 2 or phenotypic variance P is zero. In our model, inertia is determined by mutational input μ K 0 , and evolution halts when there are no mutations. For a given w and h 2 ≠0, inertia is minimized and evolution proceeds at a maximum rate in Gomulkiewicz & Holt [3] as phenotypic variance goes to infinity P → ∞. In our model, for a given strength of stabilizing selection per unit time R / σ k 2 , inertia approaches zero and the rate of evolution is maximized as mutational input goes to infinity μ K 0 → ∞. Note that to maintain analytical tractability both models assume the material which selection acts upon (phenotypic variance P or mutational input μ K 0 ) is constant. Both models will therefore be more accurate when the environmental change is relatively small. Large changes in the environment are likely to cause strong selection and large variation in abundance, which could greatly alter phenotypic variance and mutational input [30]. Since phenotypic variance and mutational input are expected to decline under strong stabilizing selection and reduced abundance [47], respectively, the analytical results of both models will tend to underestimate a population's time at risk. Our evolutionary trajectory aligns even more closely with that of Chevin and Lande (eqn 10 in [34]; also see eqn 18a in [48]), who incorporated both density-dependence and phenotypic plasticity. The two trajectories are identical when there is constant plasticity φ = 0, additive genetic variance is equivalent to the supply rate of beneficial mutations multiplied by mutational size σ a 2 = μ σ μ 2 K 0 /2, and the two measures of stabilizing selection strength per unit time are the same γ * = R / σ k 2 . Although our evolutionary trajectory aligns closely with those of Gomulkiewicz & Holt [3] and Chevin & Lande [34], we uncover an analytical approximation for the time at risk t r by assuming a time-scale separation between demographics and evolution. Gomulkiewicz & Holt [3] and Chevin & Lande [34] do not assume such a time-scale separation, leading to more complex population dynamics and the need to calculate t r numerically. This makes a quantitative comparison with our time at risk approximation impossible. However, Gomulkiewicz & Holt [3] agree that the time at risk t r should increase with initial maladaptation (i.e. magnitude of environmental change) and that at high degrees of maladaptation the relationship with time at risk should be close to linear ( figure 3 ; fig. 5 A in [3]). In addition, in both Gomulkiewicz & Holt [3] and Chevin & Lande [34] strengthening selection 1/ ω → ∞ increases the rate of adaptation while decreasing abundance (through a decline in mean fitness). Time at risk should therefore be minimized at an intermediate selection strength, as in our model ( figure 3 c ), although they do not explore this explicitly. Gomulkiewicz & Holt [3] also argue that the time at risk t r should decrease with the abundance before environmental change, since the population declines geometrically beginning at this abundance. In our model, time at risk also decreases with abundance before environmental change K 0 , but for a different reason. Recall that because of our first-order approximation we assume a small initial stress and hence a small change in abundance. This allows us to assume that mutations are supplied at a constant rate μ K 0 , where μ is the per capita mutation rate and K 0 is the abundance before environmental change. A greater abundance before environmental change K 0 therefore causes faster evolution resulting in less time at risk. 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 − C (solid curve minus dashed curve). Populations can persist in communities only when they have positive population size (region of persistence; solid line higher than the dashed line). The selection pressure in the new environment is proportional to the selection for carrying capacity (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 slopes of the two curves are equal . 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 same direction. ( b ) Competition reduces selection pressure. Competition increases as carrying capacity increases, meaning carrying capacity and competition exert opposing selection pressures. Note that if the competition curve was steeper than carrying capacity competition could reverse the direction of evolution. ( c ) Competition affects all phenotypes equally, and therefore has no effect on selection pressure. ( d ) Competition increases or decreases selection pressure. When competition and carrying capacity exert opposing selection pressures. When competition and carrying capacity select in the same direction, towards . (c) Simulations. Adaptive dynamics assumes mutations are rare enough such that, on the time-scale of evolution, the population remains monomorphic (i.e. a mutation fixes or is lost before the next arises [49]) and at demographic equilibrium (i.e. demography is faster than evolution), and that mutations are small enough to allow local stability analyses to determine evolutionary stability [40,45]. Our approximation of time at risk t r (equation (2.10)) also rests on the assumption that the initial stress is weak. We therefore performed computer simulations to examine how well our analytical result (time at risk t r ) holds when we relax these assumptions. To do this, we varied (i) mutation rate μ and maximum carrying capacity K , (ii) mutational variance σ μ 2 , and (iii) the strength of the initial stress . Computer simulations allow multiple phenotypes to coexist and introduces stochasticity in mutation rate and size. Simulations describe the numerical integration of equation (2.1), using a fourth-order Runge–Kutta algorithm with adaptive step size, and stochastic mutations. Mutations occur in a phenotype with probability μ n Δ t , where μ is the per capita per time mutation rate, n is the abundance of the phenotype and Δ t is the realized time step. For each mutation occurring in a phenotype with trait value z , one individual is given a new trait value, randomly chosen from a normal distribution with mean z and standard deviation σ μ . Trait values are rounded to the third decimal to prevent the accumulation of overly similar phenotypes. Phenotypes with abundance below unity were declared extinct. Simulations began with the population at maximum carrying capacity K and all individuals optimally adapted with trait value z = z 0 * . At the time-step 500, the optimal trait value instantaneously shifted to z n * ≠ z 0 * . Simulations were terminated at time-step 50 000. Code available upon request; implemented in R [50]. Parameter values for μ , K and were chosen in the range of those observed for yeast exposed to increased salt concentration [5]. We estimated σ k from fig. S1 in Bell & Gonzalez [5] (see the electronic supplementary material). In all simulations, the population evolved towards z n * , and, if successful in reaching z n * , remained there. Likewise, population size always approached carrying capacity, as expected ( figure 2 ). The transient dynamics, however, showed varying degrees of congruence with our prediction (equations (2.7) and (2.8); figure 4 ). In simulations, the amount of standing phenotypic variance increases with mutation rate μ multiplied by population size. Our time-scale assumption, which implies zero phenotypic variance, is thought to become unrealistic as μ K log( K ) approaches unity [51]. The threshold of μ K log( K ) is obtained because μ K is the mutational input and log( K ) is the typical time of fixation for a successful mutant when the population is well adapted [51]. Over our parameter range ( μ = , K = ) μ K log( K ) seemed to be an excellent predictor of accuracy; our predictions were much more accurate when μ K log( K ) 1, we greatly underestimated the time at risk (triangles in figure 4 ). Accuracy of analytical prediction, in the one-population case. Each point represents the mean±s.e. for 10 replicated simulation runs. Solid line is 1 : 1 line; points falling on line represent perfect predictions of time at risk t r . Squares, μ K log( K ) ≤ 0.1; circles, μ K log( K ) ≤ 1; triangles, μ K log( K ) > 1; black, ; grey, . Parameters: μ = , K = , σ μ = , R = 1, σ k = 1, σ α = 1.5 and N c is 1000 greater than the minimum abundance of each run. Mutational variance σ μ 2 seemed to have little effect on the accuracy of our predictions, at least over the range of parameter space explored here ( σ μ = ; figure 4 ). However, our analytical prediction did perform consistently better when the initial stress was small, for all parameter combinations (compare black = 1.2 and grey = 2.1 points in figure 4 ). (d) Competition. We now introduce interspecific competition. Let the population dynamics of the focal population be described by the logistic growth equation. where C ( z i , t ) ≥ 0 is the effect of interspecific competition on individuals in the focal population with trait value z i at time t . We do not model the coevolution of the competitors explicitly; we instead keep interspecific competition C ( z i , t ) as general as possible, allowing it to depend on focal trait value z i and vary in time t with any other biotic or abiotic factor (including the trait values and abundance of the focal and competing populations). For evolutionary rescue of the focal population, the only relevant dependency is with z i . Our formulation allows competition C to encompass all possible types of coevolution feedback. In fact, C could even be interpreted as an abiotic selection pressure. However, for brevity, we limit our discussion to C as the effect of a competitor. Previous studies have explicitly modelled the coevolution of competing species in a constant environment [37,52,53], at the expense of analytical results. All other variables in equation (2.13) are defined as in the one-population case. We again assume that mutations are rare, so that our focal population remains monomorphic with trait value and equilibrial abundance . In the presence of competition, equilibrium abundance of the focal population is. Comparison with the one-population case, where , shows how competition reduces abundance. Now, let the competing populations coexist in a constant environment with z * = z * 0 . The equilibrial abundance ñ of the focal population is not necessarily maximized at z * 0 , but at a ‘competitive optimal’ z * c,0 (appendix C). Assuming z * c,0 is a fitness maximum (appendix C), the focal population will eventually evolve to the competitive optimal . We then let the competitive optimal change abruptly, to new trait value z * c,n ≠ z * c,0 . This change could arise from a shift in competition C or in the optimal trait value z * = z * n . The abundance of the focal population is now k ( z * c,0 , z * n ) − C ( z * c,0 , t ). The amount of competition a population feels immediately following the environmental change C ( z c,0 * , t ) will depend on the type of environmental change as well as the response of the competitors. Competition may be close to negligible if resources remain plentiful but the abundance of competitors are greatly reduced (e.g. when a pollutant causes severe mortality in the competitor). However, competition may be exceptionally strong if the change in environment is a shift in available resources, so that the supply of resources is limiting (e.g. seed size changes on an island supporting multiple species of finch [54]). Persistence requires k ( z c,0 * , z n * ) − C ( z c,0 * , t ) ≥ 1, and therefore persistence following environmental change is more likely when competition C ( z c,0 * , t ) is weak. In appendix C, we derive the local fitness gradient of the focal population. In the new environment, with z * = z * n , it can be written as. The population evolves in a direction that increases abundance k − C until , which occurs when the population reaches the competitive optimal in the new environment ( 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-population case (equation (2.5)), competition can alter the strength and direction of selection, depending on how competition changes with trait value ( figure 5 ). Competition increases the strength of selection when . This is will always occur when competition selects in the same direction as carrying capacity (i.e. and are of different signs). Competition decreases selection when , which will occur when competition weakly selects in the opposite direction to carrying capacity (i.e. and are of the same sign and is small). When competition selects in the opposite direction as carrying capacity and has a stronger selective effect , it will reverse the direction of selection and the population will evolve away from . Competition has no effect on selection when it is independent of trait value . Combining equations (2.14) and (2.15), we compute the rate of adaptation, as described by the canonical equation [44]: The rate the focal population adapts depends on how competition affects abundance relative to selection. Owing to the added complexity of competition we are unable to solve equation (2.16) for trait value as a function of time and are therefore unable to compute a time at risk t r , as we did in the one-population case. However, we can show when competition will help or hinder adaptation, and therefore when competition has the potential to increase or decrease the likelihood of evolutionary rescue. Rearranging equation (2.16) and comparing with the one-population case (equation (2.6)) show that competition will increase the rate of adaptation when (appendix D) and decrease the rate of adaptation when the inequality is reversed. Competition will tend to speed adaptation when competition C is weak and gets much weaker as the focal population evolves towards z c,n * (dotted-dashed curve in figure 6 ). Note that although competition may increase the rate of adaptation, and therefore cause a greater rate of increase in abundance, abundance will still be depressed by competition. Competition's effect on evolutionary rescue (the time at risk t r ) will therefore depend on both its effect on adaptation and the abundance k − C relative to critical abundance N c ( figure 6 c ). As maximal abundance [ k − C ] ẑ = z * c,n approaches the critical value N c evolutionary rescue becomes less likely, and regardless of the rate of adaptation, when [ k − C ] ẑ = z * c,n ≤ N c evolutionary rescue is impossible. Competition can help or hinder evolutionary rescue. ( a ) Carrying capacity k (solid curve) as a function of trait value and two competition C scenarios: complete niche overlap (dashed curve) or partial niche overlap (dotted-dashed curve). ( b ) With complete niche overlap (dashed curve) competition increases as the population adapts, and the population therefore adapts slower than it would without competition (solid curve). With partial niche overlap (dot-dashed curve) competition decreases as the population adapts, and the population adapts faster. ( c ) The time a population spends at risk of extinction (the time abundance is below critical abundance N c ) depends on competition's effect on abundance and evolution as well as on the value of the critical abundance. For instance, when the critical abundance is low N c,low both competition scenarios increase the time at risk relative to when there is no competition (solid curve) because they depress the focal population's abundance. However, when the critical abundance is high N c,high partial niche overlap (dotted-dashed curve) decreases the time at risk relative to the no competition case (solid curve) because it sufficiently increases the rate of adaptation. 3. Discussion. In nature, population abundance cannot increase indefinitely [55]. One of the main ‘checks of increase’ [56] is competition for resources [17,19,57–59]. Because populations with lower abundances are more likely to go extinct [46], any factor which limits abundance is likely to hinder persistence, especially when the environment changes [22]. However, when we consider that populations can persist in new environments by adapting [3,5], competition has a second effect, in addition to lowering population size, which could potentially help populations persist in novel environments. Since the rate a population adapts depends on the strength of selection it experiences [44,60], competition which increases the strength of selection may speed up adaptation [61] possibly increasing the chances of persistence in the face of change. Intraspecific competition often has relatively little impact on selective pressures [58,62] (but see [63]), and therefore the effect it has on evolutionary rescue will often be determined primarily by the effect it has on abundance. Previous computer simulations have suggested that negative density-dependence will have little effect on population persistence because survival depends on the dynamics of populations which are well below carrying capacity [29]. More recent analytical work has come to a different conclusion, showing that, relative to the density-independent case, density-dependence can increase the rate at which abundance declines as well as decrease the rate abundance recovers, therefore increasing the time a population spends at risk of extinction [34]. The conflicting results are due to the different types of density-dependence used in the two studies. In Boulding & Hay [29], density-dependence is linear (i.e. per capita growth rate declines linearly with abundance) while in Chevin & Lande [34] density-dependence is stronger than linear at low abundances (the per capita growth rate declines logarithmically with abundance). Since it is the effect of density-dependence at low abundances that is critical for population persistence, this explains why Chevin & Lande [34] claim density-dependence increases the chances of extinction. A similar trend is expected in biological invasions, where populations experiencing strong density-dependence at low 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 is adapted; populations below carrying capacity will increase in abundance without evolving, and hence may not require evolutionary rescue if their carrying capacity is large enough. In our model, it is the maximum carrying capacity that affects the potential, and need, for evolutionary rescue. Since abundance asymptotically approaches maximum carrying capacity in evolutionary time ( figure 2 ), maximum carrying capacity will have a larger effect on the time at risk as it approaches the critical abundance ( figure 3 ). Notice that maximum carrying capacity plays both a demographic and evolutionary role; for a given environmental change, larger values keep populations at larger abundances ( K in equation (2.8)) and, following the change, increase the rate of evolution ( K 0 in equation (2.7)). Here we assume greater abundances lead to faster evolution because they cause greater mutational inputs. In previous models [3,34], where the rate of evolution is determined by additive genetic variation instead of mutational input, the relationship between population size and the rate of evolution can be weaker (reviewed in [65]). Although non-additive genetic effects, such as epistasis and dominance, and temporal fluctuations in abundance (leading to lower effective population sizes) can weaken the relationship between population size and the rate of evolution [66], they do not qualitatively alter our results, but merely lead to a slower rate of evolution than predicted. Given the differences between quantitative genetics and adaptive dynamics [51], our results are surprisingly consistent with previous quantitative genetic models of evolutionary rescue [3,34]. We derive a similar evolutionary trajectory and agree with Gomulkiewicz & Holt [3] on how time at risk should increase with initial maladaptation and decrease with abundance before environmental change. There is, however, one major difference between our approach and previous models of evolutionary rescue. All previous models assume the environmental change affects intrinsic growth rate, and that it is the intrinsic growth rate that must evolve fast enough to allow persistence. In our model, intrinsic growth rate R has no effect on abundance since populations are assumed to remain at demographic equilibrium, which is independent of R . In particular, the environmental change might affect R with no effect on abundance (so long as R > 0). Intrinsic growth rate is therefore irrelevant for evolutionary rescue in our model. Here rescue depends on the effect of the environmental change on carrying capacity k , and the evolution of k . Past models describe evolutionary rescue under r -selection while we describe evolutionary rescue under K -selection [67,68]. Hence, our model is more applicable to situations where density-dependence remains strong following the environmental change, during subsequent adaptation. Density-dependence will remain strong when the demand for resources continues to equal the supply. Obviously, density-dependence will remain strong when an environmental change acts only to reduce the supply of resources. This describes how a population of Darwin's finches has responded to drought [54]. The drought lowered the supply of seeds the finches ate, causing a rapid decline in finch abundance. Competition for small seeds intensified following drought and the finch population remained at carrying capacity, a carrying capacity which had been reduced by decreased food supply. Density-dependence can also be maintained when an environmental change leaves the supply of resources unaffected but increases the per capita demands. For instance, if stress tolerance requires increased energetic demands, a population exposed to a stress may continue to experience strong density-dependence despite a decline in abundance and unaffected resources. This may describe the situation observed in recent experiments of evolutionary rescue in yeast populations exposed to salt, where glucose concentration was unaffected [5,38]. Simulations indicate that our analytical approximations are sensitive to mutational input and the fixation times of new beneficial mutations. When mutations are too frequent or fixation times are too long, we consistently underestimate the time at risk ( figure 4 ). The underestimate probably arises from the adaptive dynamic assumption that fixation occurs instantaneously and the population remains monomorphic. In simulations that permit greater polymorphism, less fit phenotypes compete with those closer to the adaptive optimum, imposing a demographic load on the population. The continued existence of less fit phenotypes slows the increase of carrying capacity, causing populations to remain at risk of extinction for longer than expected. This is similar to what, in microbial evolution, is referred to as ‘clonal interference’ [69]. However, many populations should conform to our low mutation input assumption. For instance, the mutations rate of Saccharomyces cerevisiae salt tolerance is approximately μ = 10 −7 mutations per genome per generation [5]. Since our analytical approximations are accurate when μ K log( K ) 2 ,