Evolution of phenotypic plasticity and environmental tolerance of a labile quantitative character in a fluctuating environment.

doi: 10.1111/jeb.12360

Evolution of phenotypic plasticity and environmental tolerance of a labile quantitative character in a fluctuating environment R. LANDE Division of Biology, Imperial College London, Berkshire, UK

Keywords:

Abstract

community ecology; cost of plasticity; development; environmental predictability; environmental variance; niche width; norm of reaction; physiological ecology; temperature; trade-off.

Quantitative genetic models of evolution of phenotypic plasticity are used to derive environmental tolerance curves for a population in a changing environment, providing a theoretical foundation for integrating physiological and community ecology with evolutionary genetics of plasticity and norms of reaction. Plasticity is modelled for a labile quantitative character undergoing continuous reversible development and selection in a fluctuating environment. If there is no cost of plasticity, a labile character evolves expected plasticity equalling the slope of the optimal phenotype as a function of the environment. This contrasts with previous theory for plasticity influenced by the environment at a critical stage of early development determining a constant adult phenotype on which selection acts, for which the expected plasticity is reduced by the environmental predictability over the discrete time lag between development and selection. With a cost of plasticity in a labile character, the expected plasticity depends on the cost and on the environmental variance and predictability averaged over the continuous developmental time lag. Environmental tolerance curves derived from this model confirm traditional assumptions in physiological ecology and provide new insights. Tolerance curve width increases with larger environmental variance, but can only evolve within a limited range. The strength of the trade-off between tolerance curve height and width depends on the cost of plasticity. Asymmetric tolerance curves caused by male sterility at high temperature are illustrated. A simple condition is given for a large transient increase in plasticity and tolerance curve width following a sudden change in average environment.

Introduction Evolutionary biologists have long been interested in phenotypic plasticity and its role in adaptation to geographical and temporal environmental variation (Schlichting & Pigliucci et al., 1998). Ecologists and conservation biologists have increasingly employed environmental tolerance curves to understand niche width and geographical range of species and to predict how these respond to a changing environmental, especially natural environmental fluctuations and anthropogenic global warming (Fordham et al., 2013). These subjects Correspondence: Russell Lande, Division of Biology, Imperial College London, Silwood Park Campus, Ascot, Berkshire SL5 7PY, UK. Tel.: +44 (0)20759 42353; e-mail: [email protected]

have a close theoretical connection because phenotypic plasticity described by norms of reaction (giving the expected phenotype of a genotype vs. the environment) and tolerance curves (giving the expected fitness of a genotype or population vs. the environment) are partial views of the more general relationship of fitness as a function of phenotype and the environment (as suggested by Chevin et al., 2010). This general relationship provides a theoretical foundation for integrating physiological and community ecology with evolutionary genetics of phenotypic plasticity and norms of reaction. Here, I derive and analyse a model of development and evolution of plasticity in labile phenotypes, such as physiological and behavioural characters, that fluctuate continuously during an individual’s lifetime in a changing environment. I then show how the evolution of

ª 2014 THE AUTHOR. J. EVOL. BIOL. JOURNAL OF EVOLUTIONARY BIOLOGY ª 2014 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY

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environmental tolerance curves can be derived from quantitative genetic models of evolution of phenotypic plasticity. Phenotypic plasticity Phenotypic plasticity operates through a variety of mechanisms acting on different time scales. Particularly for species with deterministic growth, morphological characters often have a short critical stage of development early in life during which the environment influences the development of a constant adult phenotype. However, many phenotypic characters of an individual develop and fluctuate continuously and reversibly throughout life in response to environmental change. Physiological and behavioural characters are especially labile in this respect. Some morphological characters also are labile, even in species with determinate growth in body size such as birds and mammals that undergo seasonal changes in characters involved with thermal tolerance and sexual selection. I analyse a model of development and evolution of phenotypic plasticity in labile characters in a fluctuating environment, focusing on quantitative characters, rather than meristic or threshold characters constrained by developmental discontinuities (Chevin & Lande, 2013). Previous theory on the evolution of phenotypic plasticity in quantitative characters almost exclusively concerned plasticity influenced by the environment during a brief critical stage of early development producing a constant adult phenotype. A fundamental result of previous theory involves the predictability of the environment over the time lag between the critical stage of development and selection on the adult phenotype. Assuming a fluctuating autocorrelated environment with a stationary continuous distribution of states, linear norms of reaction and no cost of plasticity, the mean norm of reaction (giving the expected phenotype of a genotype as function of the environment) evolves to have an expected slope (or plasticity) proportional to the slope of the optimal phenotype as a function of the environment, discounted by the predictability of the environment over the time lag between development and selection on adults (Gavrilets & Scheiner, 1993; Lande, 2009). In contrast, I show for labile characters with no cost of plasticity that the expected plasticity always evolves to equal the slope of the optimum phenotype as a function of the environment, regardless of environmental predictability. Another important difference between labile vs. constant individual phenotypes concerns the cost of plasticity. For constant characters, the direct cost of plasticity is typically small or absent (Van Buskirk & Steiner, 2009), but the energy expenditure involved in continuous development of labile behavioural and physiological characters may often involve a substantial cost (Hoffmann, 1995; DeWitt et al., 1998; Angilletta,

2009). For labile characters with a cost of plasticity, I show that the expected plasticity that evolves depends on the variance and predictability of the environment averaged over the continuous developmental time lag. A conceptual oversimplification pervading much of the literature on phenotypic plasticity concerns the terminology of plasticity being applied to fitness and its major components, such as (age-specific) viability and fecundity. Plasticity in fitness should not be conceived in the same way as plasticity in morphological, physiological and behavioural characters that determine fitness. Wright (1935) emphasized the distinction between the primary characters that underlie fitness and the secondary characters of fitness and its major components. Although secondary characters can be analysed by biometric and quantitative genetic methods, just as for primary characters, this distinction is especially useful in the context of plasticity studies. The shortcoming of applying the term plasticity to secondary characters is that this may instead be caused by the environmental dependence of selection on primary characters, even if the primary characters show no plasticity. Despite a general understanding, based on voluminous evidence, that natural selection generally changes with the environment in time and space, the attribution of plasticity to fitness and its major components fails to consider this most plausible interpretation, usually because in such studies primary characters and their plasticity are omitted from the analysis (see Fig. 1). This has created considerable confusion in the literature on plasticity, and the related subjects of developmental stability, canalization and environmental tolerance. I therefore restrict the terminology of plasticity to the primary characters of morphology, behaviour and physiology that, along with the environment, determine fitness and its major components. Environmental tolerance An environmental tolerance curve for a population or species gives its fitness as a function of the environment. Environmental tolerance curves are a basic tool in physiological ecology because they describe an aspect of the fundamental niche of a species that in part determines its geographical range and species persistence in response to environmental change (Deutsch et al., 2008; Tewksbury et al., 2008; Chevin et al., 2010; Fordham et al., 2013). Previous theories on the evolution of tolerance curves postulate an arbitrary trade-off between the width and height of tolerance curves, for example, that the area under the tolerance curve remains constant during evolution (Janzen, 1967; Levins, 1968; Lynch & Gabriel, 1987; Gabriel & Lynch, 1992; Ghalambor et al., 2006). Quantitative genetic experiments (Gilchrist, 1996; Gilchrist et al., 1997) and interspecific comparative data (Huey & Kingsolver, 1989; Kingsolver & Huey, 2008; Ara ujo et al., 2013)

ª 2014 THE AUTHOR. J. EVOL. BIOL. doi: 10.1111/jeb.12360 JOURNAL OF EVOLUTIONARY BIOLOGY ª 2014 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY

Phenotypic plasticity in a labile character

Fi Phenotype

Environment

Fig. 1 Fitness as a bivariate function of phenotype and environment, assuming no cost of plasticity. Stabilizing selection towards an optimum phenotype occurs in each environment, and the optimum phenotype depends on the environment. Thick coloured lines on the fitness surface each represent a genotype (or population) that may experience a range of environments: Red line: no plasticity, Blue line: partially adaptive plasticity. Dotted lines are projections of the solid lines of corresponding colour giving norms of reaction on the bottom horizontal plane, and environmental tolerance curves on the back vertical plane.

indicate several constraints on tolerance curve shape, but the assumption of a 1 : 1 trade-off between tolerance curve width and height is not supported by any empirical evidence. The main impediment to deriving a more mechanistic theory of tolerance curve evolution has been the omission of evolution in primary phenotypic characters, and their plasticity, that underly fitness and environmental tolerance. Environmental tolerance curves depicting fitness as a function of the environment, and norms of reaction showing for a given genotype the expected phenotype of a primary character as a function of the environment in which it develops, are both simplifications of a more general relationship in which fitness depends jointly on the environment and the phenotype of a primary character (Fig. 1). Despite this intimate connection, the application of tolerance curves to problems in physiological ecology and conservation biology has been pursued almost independently of work on phenotypic plasticity and norms of reaction in quantitative genetics. That some conceptual link exists between tolerance curves and plasticity has long been understood (Levins, 1968; Kingsolver & Huey, 2008), but never explicitly quantified for primary quantitative characters that

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determine fitness in a fluctuating environment. The general relationship between fitness, environment and phenotype is a necessary ingredient of every model for the evolution of plasticity in primary quantitative characters (Via & Lande, 1985; Gavrilets & Scheiner, 1993; de Jong, 1995; Lande, 2009; Chevin et al., 2010). I derive environmental tolerance curves from a model of evolution of phenotypic plasticity in a labile quantitative character. The results provide the first general theoretical foundation for two postulates in physiological ecology: the width of the tolerance curve increases with larger environmental variance, and a trade-off exists between tolerance curve width and height. I show that with a cost of plasticity, tolerance curve width increases with larger environmental variance and higher predictability of the environment averaged over the continuous developmental time lag. Regardless of the cost of plasticity, tolerance curve width is constrained to evolve within a limited range. I also show that the strength of the trade-off between tolerance curve width and height depends on the cost of plasticity.

Model and results Phenotypic development in a fluctuating environment To derive a theory for evolution of plasticity in labile characters undergoing continual development and selection through time, we require an explicit model of developmental dynamics describing how the phenotype of an individual changes during its lifetime in response to environmental fluctuations. The phenotype of an individual at time t is denoted as zt and represented as the sum of a constant additive genetic effect a, a constant micro-environmental contribution that varies among individuals (including nonadditive genetic effect and developmental noise), e, plus a term ft describing labile changes in the individual phenotype in response to the same macro-environment experienced by all individuals in the population at any given time zt ¼ a þ e þ ft

(1)

Hereafter, the macro-environment et is referred to simply as the environment. The developmental dynamics are assumed for simplicity to be governed by a linear differential equation dft ¼ k½ft /ðet Þ dt

(2a)

where the rate of development k is assumed to be a constant independent of the genotype and environment, and /(e) denotes the (possibly nonlinear) dependence of the genotypic norm of reaction on the environment e. Time dependence of the environment and other variables is indicated by subscripts, which are


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sometimes suppressed for brevity. The environment is measured as a deviation from its average value, so that e ¼ 0, and is assumed to have a stationary distribution with temporal variance r2e and autocorrelation function qu ¼ E½et etu =r2e . Starting at time t = 0, the labile part of the individual phenotype through time has the olution Z t ft ¼ f0 ekt þ k ekðtsÞ /ðes Þds (2b) 0

To facilitate the evolutionary analysis, I make several assumptions. 1 Linear norm of reaction: the expected phenotype that a genotype develops in a constant controlled environment is a linear function of the environment, a + /(e) = a + be with slope b measuring the plasticity. 2 Labile developmental changes in the individual phenotype occur on a time scale that is short compared with the population generation time, 1/k T. This entails that juvenile growth and development are short compared with the average adult lifespan, so that selection acts primarily on labile adult phenotypes. It is therefore justified to let t?∞ in eqn (2b), so that by changing variables, the individual phenotype can be approximated as an integral over past environments with an exponentially diminishing contribution from longer time lags u, Z zt ¼ a þ e þ bk

1

eku etu du

optimum phenotype in the average (or reference) environment, e = 0, and B giving the slope of the optimum phenotype as a function of the environment. 5 The population mates randomly, and among genotypes, the reaction norm elevation a (measured in the reference or average environment) and slope b are normally distributed with constant additive genetic variances, Gaa and Gbb and additive genetic covariance Gab (Gavrilets & Scheiner, 1993; Lande, 2009). Micro-environmental effects on individual phenotypes have an identical normal distribution with variance r2e and mean zero, e ¼ 0 in every environment. All individuals in the population experience the same (macro)environment at any given time, so that the phenotype also has a normal distribution. Averaging the individual Malthusian fitness over the phenotype distribution at any time gives the mean fitness as a function of the mean phenotype and mean plasticity, c zt ; et Þ ¼ mmax ðet Þ ðzt A Bet Þ2 þ r2z mð 2 (3b) c 2 þ Gbb Þ b ðb 2 The mean phenotype in the population is (from eqn 2c) Z 1 zt ¼ aþ bk eku etu du (4a) 0

(2c)

0

Phenotypic selection and genetic variation Additional assumptions necessary to derive the evolutionary dynamics concern genetic variation in reaction norm elevation and slope, and the functional form and environmental dependence of natural selection. 3 Stabilizing selection acts continuously in time towards an optimal phenotype ht, with individual Malthusian fitness described by the quadratic function c c mðzt ; et Þ ¼ mmax ðet Þ ðzt ht Þ2 b b2 : (3a) 2 2 The strength of stabilizing selection is given by the curvature of the fitness function, c (Lande & Arnold, 1983), which has units of Malthusian fitness per character-squared. The last term models the cost of plasticity by stabilizing selection acting directly against the reaction norm slope, b, favoring no plasticity, distinct from stabilizing selection acting directly on the plastic phenotype, zt. Selection is both density and frequency independent. 4 The optimum phenotype is a linear function of the environment, ht = A +B et, with A being the

and the phenotypic variance is Z 1 eku etu du r2z ¼ Gaa þ 2Gab k 0 Z 1 2 eku etu du þr2e þ Gbb k

(4b)

0

The phenotypic variance fluctuates through time independently of the mean reaction norm parameters a and b, and hence does not affect the expected selection gradient. The expected phenotypic variance averaged over the stationary distribution of the environment is þ r2e (for any value of Gab), E½r2z ¼ Gaa þRGbb r2e q 1 ¼ k 0 eku qu du is the predictability of the where q environment averaged over the developmental time lag (see Appendix). Evolution of labile plasticity in a fluctuating environment Following Lande (1982, 2009), the continuous time evolution of the mean elevation and slope of norms of reaction in the population is d dt

1 Gaa a ¼ b T Gba

@=@ a Gab where r ¼ rm (5) Gbb @=@ b



Applying the selection gradient operator (eqn 5) to the mean Malthusian fitness (eqn 3), using eqn (4a), produces 1 0 R 1 ku ¼ cðzt A Bet Þ rm cb k 0 e etu du b (6a) which can be employed for stochastic simulations. The expected evolution of the mean reaction norm is governed by the expected selection gradient averaged over the stationary distribution of the environment. Taking expectations, using the Appendix, gives a A 0 ¼ c E½rm (6b) cb 2 ½b Bre q b , the predictability of Note here the reappearance of q the environment averaged over the developmental time lag (see Appendix). The mean reaction norm in the population is therefore expected to evolve towards a ‘stochastic equilibrium’ with elevation and slope a ¼ A and b ¼

B 1 þ crb2=cq

(7)

e

With no cost of plasticity (cb = 0), the mean reaction norm has the stochastic equilibrium with elevation and slope a ¼ A and b ¼ B identical to line for the optimum phenotype as a function of the environment (assumption 4). Although environmental autocorrelation does affect the evolutionary dynamics, it has no impact on the stochastic equilibrium of the norm of reaction for labile characters in the absence of a cost to plasticity. The evolution of plasticity in labile characters therefore differs qualitatively from that of characters with deterministic growth governed by a critical early stage of development (Gavrilets & Scheiner, 1993; Lande, 2009) for which the reaction norm slope is reduced by the imperfect predictability of the environment across the developmental time lag. Reasons for this surprising discrepancy are considered in the Discussion. The evolution of plasticity in a labile character depends critically on a single dimensionless ratio of two quantities (eqn 7). The first is itself a ratio of the strengths of stabilizing selection on plasticity to that on the phenotype, cb/c, and the denominator is the environmental variance times the predictability of the . The parameter cb describes the environment, r2e q strength of stabilizing selection against plasticity (Chevin & Lande, 2010) and has units of Malthusian fitness per plasticity-squared. As plasticity itself has units of character per environment, the ratio of selection intensities cb/c thus has units of environment-squared, and hence, its ratio to the environmental variance , produces a dimensionless times predictability, r2e q number.

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With a small cost of plasticity and/or high environ, the mental variance and predictability cb =c r2e q expected plasticity b is nearly as large as B, the slope of the optimum phenotype as a function of the environment. With a large cost of plasticity and/or low envi, the ronmental variance and predictability, cb =c r2e q expected plasticity is a small fraction of B. Environmental tolerance An environmental tolerance curve describes genotypic or population mean fitness as a function of the environment. Much of the theory on the evolution of niche width and the evolution of generalist vs. specialist species postulates a trade-off between the height and width of tolerance curves among populations or species adapted to different amounts of environmental variance (Levins, 1968; Huey & Kingsolver, 1989; van Tienderen, 1991, 1997; Gilchrist, 1995, 1996). For example, Levins (1968), Lynch & Gabriel (1987), and Gabriel & Lynch (1992) arbitrarily assume that the area under the tolerance curve must remain constant during evolution. Here, I show that the existence and magnitude of the basic constraint on the evolution on tolerance curves can be derived from first principles and depends on the cost of plasticity (as suggested by Chevin et al., 2010). Tolerance curves traditionally have a minimum fitness approaching or equal to 0 in extreme environments and therefore implicitly employ (Wrightian) absolute fitness, which is the exponential of (Fisherian) Malthusian fitness. To clarify the basic constraint on tolerance curve evolution, I derive and compare the environmental tolerance curves for a genotype and for a population, assuming for simplicity that the maximum individual fitness is independent of the environment, mmax(e) = mmax. (This assumption is later relaxed.) The Malthusian fitness functions in the present model are then quadratic and become transformed to Gaussian functions on the scale of absolute fitness. Denoting the constant coefficients of the zeroth-order and second-order terms in the environment as H and J/2, the width p and ffiffiffi height of a Gaussian tolerance curve are then 1= J and eH. An empirical environmental tolerance curve for a genotype can be constructed by experimentally growing replicate individuals of the genotype from zygote to adult across a range of constant environments. A theoretical genotypic tolerance curve can be derived from the individual Malthusian fitness function (eqn 3a) by averaging over the distribution of e, micro-environmental variation and developmental noise in the phenotype among individuals with the same genotype in a constant environment, r2e . A genotype with norm of reaction parameters a and b in a constant environment e has expected phenotype ~z ¼ a þ be. Substituting this into eqn (3a) produces the environmental tolerance curve for a genotype,


R. LANDE

c c 2 re þ ½a A þ ðb BÞe2 b b2 2 2 (8a)

Tolerance curve width and height for a genotype can be expressed most simply by assuming that the species from which they are sampled has adapted to a given environmental variance, so that parameters of the expected norm of reaction at the stochastic equilibrium can be applied. The tolerance curve for the average genotype (with the optimal norm of reaction in eqn 7) then has 1 r2e q 1 width ¼ ¼ 1 þ (8b) pffiffiffi pffiffiffi jB bj c cb =c B c n o c c height ¼ exp mmax r2e b b2 (8c) 2 2 This reveals that adaptation to a more variable environment by evolution of higher plasticity increases the width of the tolerance curve for the average genotype. Note that the width of the genotypic tolerance curve is pffiffiffi proportional to 1= c, the width of the individual phenotypic fitness function within environments (eqn 3a). Chevin et al. (2013) derived a formula for tolerance curve width similar to the first form of eqn (8b), but did not consider its evolution. Tolerance curve width for the average genotype pffiffiffi evolves to a minimum of 1=ðB cÞ in a nearly constant environment and increases for larger environmental variance (if the population persists). A genotype with perfect plasticity (b = B) produces an infinitely wide environmental tolerance for the average genotype, which can evolve only in the absence of a cost of plasticity. With a cost of plasticity, a more variable environment produces a larger width but a smaller height of the genotypic tolerance curve. The height of the tolerance curve is less sensitive than the width to changes in the environmental variance, unless the cost of plasticity is large. Empirical tolerance curves for a population are usually constructed by ignoring genetic variation among individuals, experimentally raising samples from the population across a range of constant environments. Theoretical tolerance curves for a population in the present model can be constructed by averaging over the distribution of genotypic tolerance curves. The phenotypic mean and variance (eqns 4a,b) that develop for a labile character in a constant environment e are identical to those for previous models with plasticity determined by the environment at a critical stage of development (Gavrilets & Scheiner, 1993; Lande, 2009 eqns 1b,c), z ¼ a þ be and r2z ¼ Gaa þ 2Gab e þ Gbb e2 þ r2e . The mean Malthusian fitness (eqn 3b) then produces the environmental tolerance curve for a population mðeÞ ¼ mmax ðeÞ

c ½a A þ ðb BÞe2 þ r2z 2

cb 2 ðb þ Gbb Þ 2

(9a)

Again assuming that the population is adapted to a given environmental variance (eqn 7) with mmax(e) = mmax independent of the environment, and now also that population is canalized (has minimum phenotypic variance) in the average environment, so that Gab = 0 (Lande, 2009), the width and height of the population tolerance curve are 1 (9b) width ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ðB bÞ2 þ Gbb n o c c 2 b þ Gbb Þ (9c) height ¼ exp mmax ðGaa þ r2e Þ b ð 2 2 Comparison of eqns (9b,c) with the tolerance curve width and height for the average genotype at the expected evolutionary equilibrium (eqns 8b,c) reveals that genetic variance in plasticity Gbb reduces the width, whereas genetic variance in reaction norm elevation and slope, Gaa and Gbb, both reduce the height of the population tolerance curve. Genetic variance in plasticity reduces the maximum width of the tolerance curve in paffiffiffiffiffiffiffiffiffi highly variable environment from infinity to 1= cGbb . Figure 2 (left) compares environmental tolerance curves for populations adapted to low, medium and high environmental variance for a canalized characters with a fairly strong cost of plasticity. Figure 2 (right) shows how the height and width of tolerance curves change with environmental variance over the possible range from a nearly constant to a highly variable environment and how the trade-off between the height and width depends on the cost of plasticity.

0.01 0.2 0.5 Height

~ b; eÞ ¼ mmax ðeÞ mða;

Mean fitness

6

1

σ2ε = 1000 100 1 0

−50

γb = 2

0

Environment

50

0

0 min max 50

Width

Fig. 2 Left: Environmental tolerance curves depicting mean fitness in a population as a function of the environment, after evolution of plasticity to its stochastic equilibrium under environmental variance r2e (eqn 9a). Mean fitness shown on an absolute scale is the exponential of mean Malthusian fitness. The cost of plasticity is measured by the strength of stabilizing selection on the reaction norm slope, cb = 0.5. Right: trade-off between tolerance curve width and height for different costs of plasticity. The possible range of tolerance curve width is shown: the narrowest for a nearly constant environment and the widest for an extremely variable ¼ 0:5, Gaa = 1, environment (eqn 9b,c). Other parameters: q Gbb = 0.1, Gab = 0, r2e ¼ 1, c = 0.01, B = 1.



Mean fitness

If the population is not canalized in the average environment, then the slope and elevation of genotypic norms of reaction are correlated, Gab 6¼ 0, and the phenotypic variance (above eqn 9a) is not minimized in the average environment (Lande, 2009). This does not affect the width of the tolerance curve, but alters the height and position of its mode, as shown in Fig. 3 (left). The mode of tolerance curve evolves to an equilibrium deviating from the average environment by Gab times the squared width of the tolerance curve. Asymmetry of the tolerance curve also causes the optimal environment to differ from the average environment (Martin & Huey, 2008). Empirical tolerance curves often are rather asymmetric, as for temperature tolerance, which generally displays a strong negative skew (Huey & Kingsolver, 1989; Deutsch et al., 2008; Tewksbury et al., 2008; Angilletta, 2009; Amarasekare & Savage, 2012; Ara ujo et al., 2013). Asymmetric tolerance curves can be produced by nonlinearity in the norms of reaction norm or in the optimum phenotype as a function of the environment, asymmetry of the individual fitness function acting on phenotypes and/or asymmetry in any of the following as functions of the environment: the stationary distribution of environments, strength of phenotypic stabilizing selection c depending on the environment or asymmetry of the maximum individual fitness as a function of the environment mmax(e) (Chevin et al., 2010). For example, in ectothermic species, high temperature generally reduces male fertility by disrupting sperm production, and above a threshold temperature emax, males become sterile (Rohmer et al., 2004; Sturup et al., 2013). Temperature also affects female fecundity through plasticity in body size, with high temperature accelerating development and typically reducing adult size of both sexes. Temperature has compensatory effects on female fitness because smaller female body size decreases female fecundity but faster development reduces the demographic generation time that is

2

2

σε = 1000 100 1 0

−50

Environment

50

0

−50

primarily determined by females in polygamous species (Kingsolver & Huey, 2008; Lande et al., 2003). For illustrative purposes, I therefore ignore temperature effects on females and focus only on male fertility and the threshold temperature for sterility. This can be incorporated in the model by making the maximum individual Malthusian fitness a hyperbolic function of the environment, mmax(e) = mmaxc/(emaxe), where mmax, emax and c are constants. Because mmax(e) is assumed to be the same for all genotypes, representing a hard limit on fitness imposed by the environment, it does not affect selection within environments. This can therefore produce a large asymmetry in the tolerance curve, causing the optimal environment to deviate greatly from the average environment, without affecting the evolution of plasticity. Figure 3 (right) shows that with sufficient temporal environmental variance for the population to experience temperatures causing partial male sterility with substantial frequency, the tolerance curve attains the strongly left-skewed form typical for temperature tolerance in ectotherms, with larger environmental variance increasing the asymmetry. Although this type of hard physical environmental constraint on population growth does not affect selection within environments, in a spatially distributed population with geographical variation in the environment, it can greatly affect the evolution of a population by altering its geographical range through differential population growth among localities experiencing different environments, thus altering the distribution of environments in which it is selected. Microgeographical environmental variation and behavioural habitat selection by individuals could also preclude local extinction even when the average ambient temperature exceeds the limit for male sterility. By also including genetic variation in the maximum temperature tolerance, and additional underlying characters, it would be possible to jointly model evolution of the threshold temperature for male sterility, which has been shown experimentally and in comparative data to undergo gradual evolutionary increase of limited extent in populations maintained at high temperature (Rohmer et al., 2004; Gilchrist et al., 1997; Ara ujo et al., 2013). Sudden extreme change in average environment

σε = 1000 100 1

0

7

0 15

Environment

Fig. 3 Left: environmental tolerance curves for populations with incomplete canalization, Gab = 0.3, causing the optimal environment in which mean fitness is maximized to differ from the average environment. Right: temperature tolerance curves as in Fig. 2, and incorporating in the mean Malthusian fitness (eqns 3b and 9a) an additional term describing a threshold temperature for male sterility, with c = 5 and emax = 15. Other parameters as in Fig. 2 (left).

Crucial moments that may determine the long-term persistence and evolution of a species occur occasionally in its existence, such as during colonization of a new habitat, invasion of a new species or abrupt change in in situ physical environment. For plasticity determined by the environment at a critical stage early in development to a constant adult phenotype, Lande (2009) showed that in response to a sudden major change in average environment, beyond the normal range of stationary background environmental fluctuations, phenotypic plasticity can undergo a rapid evolu-


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tionary increase followed by a slow decline resembling genetic assimilation. I now give an analogous condition for a sudden change in average environment to elicit a similar pattern of transient evolution of plasticity and tolerance curve width in labile phenotypes. Consider a sudden change in the average environment from e ¼ 0 to d, and assume that a long history of stabilizing selection around average environment has produced phenotypic canalization in environment 0, such that reaction norm slope and elevation measured there are genetically uncorrelated (Lande, 2009). From the eigenvalues of the stability matrix for this model, the evolutionary dynamics display a fast time scale for transient evolutionary increase in plasticity, and environmental tolerance, followed by a slow time scale for its decay when þ cb =c d2 r2e q

(10)

Thus, the magnitude of sudden change in the average environment must not only be extreme relative to predictable changes in the normal background environmen , but it must also exceed the cost of tal fluctuations, r2e q plasticity in relation to stabilizing selection on the character, cb/c. The relative magnitude of the latter two quantities, along with B itself, governs the potential magnitude of the transient increase in plasticity after a sudden change in the average environment (see eqn 7).

Discussion Many ecologically important physiological and behavioural characters, and some morphological characters, are labile, undergoing continuous reversible development within an individual lifetime in response to environmental change. For a labile quantitative character in a fluctuating environment, the evolution of phenotypic plasticity differs qualitatively from that of characters influenced by the environment only at a brief critical stage of development that determines a constant adult phenotype subject to selection. In previous models of the evolution of plasticity in constant adult phenotypes, with no cost of plasticity, the expected plasticity that evolves in a population is discounted from B, the slope of the optimal phenotype as a function of the environment, because of the imperfect predictability of the environment over the within-generation time lag between development and selection (Gavrilets & Scheiner, 1993; Lande, 2009). In contrast, I have shown for labile characters that, with no cost of plasticity, the expected plasticity that evolves in a population equals B regardless of environmental predictability. This surprising discrepancy arises because, despite developmental time lags occurring in both models, selection and development in labile characters occur simultaneously and continuously in time, rather than at different times within an individual lifespan as in previous models of plasticity. If a discrete time lag is

introduced into the present model of continuous development in a labile character (eqn 2a, changing et to ets), the expected plasticity that evolves also is reduced from B by imperfect environmental predictability over the discrete time lag s (see Appendix). Thus, the difference between the present and previous models of evolution of plasticity is not caused simply by continuity vs. discontinuity of development. Instead, in the absence of a cost to plasticity, the present model produces perfect plasticity equal to B because the direction of development and selection both are determined simultaneously by the current environment, rather than at different times by different environments. A cost of plasticity produces greater conformity between the evolution of plasticity in labile and constant characters. With a cost of plasticity, the expected plasticity that evolves in labile characters does depend on environmental predictability. The cost of plasticity for constant adult phenotypes is usually small (Van Buskirk & Steiner, 2009) in part because their development is restricted in time. For labile characters, the cost of plasticity is likely to be substantial due to the continual expenditure of resources and energy in their development, especially for behavioural characters involving individual movement and physiological characters regulating energy production and/or body temperature, such as endothermic heat production, shivering, panting and sweating (Hoffmann, 1995; DeWitt et al., 1998; Angilletta, 2009). The expected plasticity that evolves in labile characters depends on a ratio: the cost of plasticity relative to the strength of stabilizing selection on the character, divided by the product of environmental variance and environmental predictability averaged over the developmental time lag (eqn 7). In relation to the slope of the optimal phenotype as a function of the environment, the expected plasticity increases in more highly variable and predictable environments and decreases with larger relative cost of plasticity. With a small cost of plasticity and/or high environmental variance and predictability, the expected plasticity is close to the slope of the optimum phenotype as a function of the environment. With a large cost of plasticity and/or low environmental variance and predictability, the expected plasticity is small. Environmental tolerance curves, giving fitness as a function of the environment, are a fundamental tool in physiological ecology. However, the omission of primary phenotypes that determine fitness in tolerance curves has inhibited progress in understanding many central questions in physiological and community ecology. Tolerance curves and norms of reaction, giving the expected phenotype of a genotype as a function of the environment, are different projections of the more general relationship of fitness as a function of both the primary phenotype and the environment (Fig. 1). This general relationship is the basis for all models of the evolution of plasticity in primary characters (eqn 3b)



and therefore constitutes a foundation for integrating physiological and community ecology with evolutionary genetics of plasticity and norms of reaction. Here, I have derived environmental tolerance curves from quantitative genetic models of evolution of plasticity in primary characters that determine fitness. The result, which depends on a cost of plasticity, confirms a traditional postulate in physiological ecology that broader environmental tolerance should evolve in more variable environments (eqns 8 and 9); such concepts also feature in considerations of intraspecific niche width and geographical range, as well as species packing and community structure along spatial environmental gradients such as those occurring with latitude and elevation (Janzen, 1967; Levins, 1968; Huey & Kingsolver, 1989; van Tienderen, 1991, 1997; Ghalambor et al., 2006). With a cost of plasticity, tolerance curve width can only evolve within a limited range (eqns 8 and 9, Figs 1 and 2). The minimum width, which evolves in a nearly constant environment (and with small genetic variance in plasticity), equals the width of the individual phenotypic fitness function within environments divided by the slope of the optimal phenotype as a function of the environment. The maximum width that evolves in a highly variable environment is limited by genetic variance in plasticity. Another traditional postulate in physiological ecology concerns a trade-off between the height and width of environmental tolerance curves, summarized in the rubric ‘A jack of all trades [or temperatures] is a master of none’. Limited intraspecific genetic and interspecific comparative evidence exists to support this assumption (Huey & Kingsolver, 1989; Gilchrist, 1996; Gilchrist et al., 1997 Kingsolver & Huey, 2008; Ara ujo et al., 2013), and previous theory on the evolution of tolerance curves postulated an ad hoc constraint of a constant area under the tolerance curve (Levins, 1968; Lynch & Gabriel, 1987; Gabriel & Lynch, 1992). This trade-off may partially underly and be confounded with the classical trade-off between r- and K-selection in life-history evolution (MacArthur & Wilson, 1967; Pianka, 1970), which also depends on the magnitude of environmental variance (Lande et al., 2009; Engen et al., 2013). The present model demonstrates that, separated from life-history evolution, the strength of the trade-off between the height and width of the tolerance curve also depends on the cost of plasticity (eqns 8 and 9, Figs 1 and 2), as suggested by Chevin et al. (2010). A major challenge for future empirical and theoretical research in ecology and evolution will be to incorporate evolution of plasticity in the primary quantitative characters underlying fitness into analysis of environmental tolerance and species responses to changing environments. Although recent work, including the present article and citations herein, has begun this endeavour, remaining issues of central importance include the coupling of demographic and evolutionary models and empirical

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estimation of their genetic and ecological parameters, especially the environmental dependence of selection (Chevin et al., 2010; Chevin & Lande, 2010, 2011), and different types of costs of plasticity if these can be separated from selection directly on the phenotypes produced by plasticity (DeWitt et al., 1998). Considering multiple genetically correlated characters with both constant and labile phenotypes, based on more complex and realistic models of development, will likely introduce additional constraints on the evolution of plasticity, tolerance curves, species ranges and community structure.

Acknowledgments I thank L.-M. Chevin, E.E. Goldberg, C. Botero and the reviewers for discussions or comments on the manuscript. This work was supported by a Royal Society Research Professorship.

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Appendix To derive the expected selection gradient and the expected phenotypic variance (averaged over the stationary distribution of the environment), it is necessary to calculate the following expectation Z 1 2 Z 1Z 1 E k eku etu du ¼ k2 ekðuþvÞ E½etu etv dudv 0 0 0 Z 1Z 1 ekðuþvÞ quv dudv ¼ r2e k2 0 0 Z 1 Z 1 ¼ 2r2e k2 ekv eku quv du dv v Z0 1 Z 1 2 2 2kv e dv eky qy dy ¼ 2re k 0

0

¼ r2e q R1 ¼ k 0 eky qy dy. In the second line, the intewhere q grand is symmetrical with respect to interchange of u and v, because the environmental autocorrelation function is symmetrical with respect to positive and negative time lag, quv = qvu (with q0 = 1). The third line therefore splits the integral over the upper right quadrant into two equivalent integrals over mutually symmetric triangular regions spanning from the positive u and v axes to the line of symmetry u = v. The inner integral in brackets in the third line is evaluated using the change in variable y = uv for a given v. The expected selection gradient appears in eqn (6b). If a discrete time lag s is introduced into the continuous developmental dynamics, replacing et with ets in eqn (2a), the same change carries over to the phenotype in eqn (2c)R and the expected plasticity that 1 evolves is b ¼ Bðk= qÞ 0 eky qy þ dy. Assuming qu is a nonnegative decreasing function of u, so that b \ B, the discrete time lag reduces the expected plasticity. With an exponential environmental autocorrelation function, qt = ekt, the expected plasticity reduces to b ¼ q B, identical to previous theory of plasticity exerted by the environment at a brief critical stage of development early in life. Received 11 December 2013; revised 16 February 2014; accepted 20 February 2014


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