Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.     

EVOLUTION IN FLUCTUATING ENVIRONMENTS: DECOMPOSING SELECTION INTO ADDITIVE COMPONENTS OF THE ROBERTSON–PRICE EQUATION

We analyze the stochastic components of the Robertson–Price equation for the evolution of quantitative characters that enables decomposition of the selection differential into components due to demographic and environmental stochasticity. We show how these two types of stochasticity affect the evolution of multivariate quantitative characters by defining demographic and environmental variances as components of individual fitness. The exact covariance formula for selection is decomposed into three components, the deterministic mean value, as well as stochastic demographic and environmental components. We show that demographic and environmental stochasticity generate random genetic drift and fluctuating selection, respectively. This provides a common theoretical framework for linking ecological and evolutionary processes. Demographic stochasticity can cause random variation in selection differentials independent of fluctuating selection caused by environmental variation. We use this model of selection to illustrate that the effect on the expected selection differential of random variation in individual fitness is dependent on population size, and that the strength of fluctuating selection is affected by how environmental variation affects the covariance in Malthusian fitness between individuals with different phenotypes. Thus, our approach enables us to partition out the effects of fluctuating selection from the effects of selection due to random variation in individual fitness caused by demographic stochasticity.     

Exclusion rules, bottlenecks and the evolution of stochastic phenotype switching

Stochastic phenotype switching--often considered a bet hedging or risk-reducing strategy--can enhance the probability of survival in fluctuating environments. A recent experiment provided direct evidence for an adaptive origin by showing the de novo evolution of switching in bacterial populations propagated under a selective regime that captured essential features of the host immune response. The regime involved strong frequency-dependent selection realized via dual imposition of an exclusion rule and population bottleneck. Applied at the point of transfer between environments, the phenotype common in the current environment was assigned a fitness of zero and was thus excluded from participating in the next round (the exclusion rule). In addition, also at the point of transfer, and so as to found the next bout of selection, a single phenotypically distinct type was selected at random from among the survivors (the bottleneck). Motivated by this experiment, we develop a mathematical model to explore the broader significance of key features of the selective regime. Through a combination of analytical and numerical results, we show that exclusion rules and population bottlenecks act in tandem as potent selective agents for stochastic phenotype switching, such that even when initially rare, and when switching engenders a cost in Malthusian fitness, organisms with the capacity to switch can invade non-switching populations and replace non-switching types. Simulations demonstrate the robustness of our findings to alterations in switching rate, fidelity of exclusion, bottleneck size, duration of environmental state and growth rate. We also demonstrate the relevance of our model to a range of biological scenarios such as bacterial persistence and the evolution of sex.     

An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in σ_e², the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the θ-logistic model, where density dependence of population growth is a function of N^θ, long-term evolution maximizes E[N^θ]=[1−σ_e²/(2r)]K^θ. While σ_e² is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, θ has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.     

EVOLUTIONARY EPIDEMIOLOGY AND THE DYNAMICS OF ADAPTATION

The mean fitness of a population, often equal to its growth rate, measures its level of adaptation to particular environmental conditions. A better understanding of the evolution of mean fitness could thus provide a natural link between evolution and demography. Yet, after the seminal work of Fisher and its renowned "fundamental theorem of natural selection," the dynamics of mean fitness has attracted little attention, and mostly from theoretical population geneticists. Here we analyze the dynamics of mean fitness in the context of host-parasite interactions. We illustrate the potential relevance of this analysis under different scenarios ranging from a simple situation in which a parasite evolves in a homogeneous host population to a more complex one with host-parasite coevolution. In each case, we contrast the effects of natural selection, recurrent mutations, and the change of the biotic environment, on the dynamics of adaptation. Decoupling these three components helps elucidate the interplay between evolutionary and ecological dynamics. In particular, it offers new perspectives on situations leading to evolutionary suicide. As mean fitness is an easily measurable quantity in microbial systems, this analysis provides new ways to track the dynamics of adaptation in experimental evolution and coevolution studies.     

Reproductive Value and Fluctuating Selection in an Age-Structured Population

Fluctuations in age structure caused by environmental stochasticity create autocorrelation and transient fluctuations in both population size and allele frequency, which complicate demographic and evolutionary analyses. Following a suggestion of Fisher, we show that weighting individuals of different age by their reproductive value serves as a filter, removing temporal autocorrelation in population demography and evolution due to stochastic age structure. Assuming weak selection, random mating, and a stationary distribution of environments with no autocorrelation, we derive a diffusion approximation for evolution of the reproductive value weighted allele frequency. The expected evolution obeys an adaptive topography defined by the long-run growth rate of the population. The expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Simulations of the age-structured model verify the accuracy of the diffusion approximation. We develop statistical methods for measuring the expected selection on the reproductive value weighted allele frequency in a fluctuating age-structured population.THE evolutionary dynamics of age-structured populations were formalized by Charlesworth (1980, 1994) and Lande (1982) on the basis of earlier ideas of Fisher (1930, 1958), Medawar (1946, 1952), and Hamilton (1966), showing that the strength of selection on genes affecting the vital rates of survival or fecundity depends on their age of action (reviewed by de Jong 1994; Charlesworth 2000). Fisher defined the reproductive value of individuals of a given age as their expected contribution to future population growth, determined by the age-specific vital rates. This has the property that in a constant environment the total reproductive value in a population always increases at a constant rate. The total population size, however, undergoes transient fluctuations as the stable age distribution is approached, and the total population size only asymptotically approaches a constant growth rate (Caswell 2001).Environmental stochasticity creates continual fluctuations in age structure, producing temporal autocorrelation in population size and in allele frequencies, which seriously complicate demographic and evolutionary analyses. Fisher (1930, 1958, p. 35) suggested for analysis of genetic evolution that individuals should be weighted by their reproductive value to compensate for deviations from the stable age distribution. Here we apply this suggestion to study weak fluctuating selection in an age-structured population in a stochastic environment.One of the central conceptual paradigms of evolutionary biology was described by Wright (1932). His adaptive topography represents a population as a point on a surface of population mean fitness as a function of allele frequencies. Assuming weak selection, random mating, and loose linkage (implying approximate Hardy–Weinberg equilibrium within loci and linkage eqilibrium among loci), natural selection in a constant environment causes the population to evolve uphill of the mean fitness surface (Wright 1937, 1945, 1969; Arnold et al. 2001; Gavrilets 2004). Evolution by natural selection thus tends to increase the mean fitness of a population in a constant environment.Lande (2007, 2008) generalized Wright''s adaptive topography to a stochastic environment, allowing density-dependent population growth but assuming density-independent selection, showing that the expected evolution maximizes the long-run growth rate of the population at low density, . Here r is population growth rate at low density in the average environment and is the environmental variance in population growth rate among years, which are standard parameters in stochastic demography (Cohen 1977, 1979; Tuljapurkar 1982; Caswell 2001; Lande et al. 2003). In this model of stochastic evolution the adaptive topography describing the expected evolution is derived by expressing r and as functions of allele frequencies with parameters being the mean Malthusian fitnesses of the genotypes and their temporal variances and covariances. These results are based on diffusion approximations for the coupled stochastic processes of population size and allele frequencies in a fluctuating environment.Diffusion approximations are remarkably accurate for many problems in evolution and ecology (Crow and Kimura 1970; Lande et al. 2003). Because a diffusion process is subject to white noise with no temporal autocorrelation, the approximation is most accurate if the noise in the underlying biological process is approximately uncorrelated among years. Despite temporal autocorrelation in total population size produced by age-structure fluctuations, the stochastic demography of age-structured populations over timescales of a generation or more can nevertheless be accurately approximated by a diffusion process (Tuljapurkar 1982; Lande and Orzack 1988; Engen et al. 2005a, 2007). The success of the diffusion approximation for total population size occurs because the noise in the total reproductive value is nearly white, with no temporal autocorrelation to first order, and the log of total population size fluctuates around the log of reproductive value with a return time to equilibrium on the order of a few generations (Engen et al. 2007). Hence the diffusion approximation is well suited to describe the stochastic dynamics of total reproductive value as well as total population size.This article extends Lande''s (2008) model of fluctuating selection without age structure by deriving a diffusion approximation for the evolution of an age-structured population in a stochastic environment. Assuming weak selection at all ages, random mating, and a stationary distribution of environments with no temporal autocorrelation, we show that the main results of the model remain valid, provided that the model parameters are expressed in terms of means, variances, and covariances of age-specific vital rates and that allele frequencies are defined by weighting individuals of different age by their reproductive value, as suggested by Fisher (1930, 1958). We perform simulations to verify the accuracy of the diffusion approximation and outline statistical methods for estimating the expected selection acting on the reproductive value weighted allele frequency.     

The evolution of bet-hedging adaptations to rare scenarios

When faced with a variable environment, organisms may switch between different strategies according to some probabilistic rule. In an infinite population, evolution is expected to favor the rule that maximizes geometric mean fitness. If some environments are encountered only rarely, selection may not be strong enough for optimal switching probabilities to evolve. Here we calculate the evolution of switching probabilities in a finite population by analyzing fixation probabilities of alleles specifying switching rules. We calculate the conditions required for the evolution of phenotypic switching as a form of bet-hedging as a function of the population size N, the rate theta at which a rare environment is encountered, and the selective advantage s associated with switching in the rare environment. We consider a simplified model in which environmental switching and phenotypic switching are one-way processes, and mutation is symmetric and rare with respect to the timescale of fixation events. In this case, the approximate requirements for bet-hedging to be favored by a ratio of at least R are that sN>log(R) and thetaN>square root R .     

On measuring selection in experimental evolution

Distributions of mutation fitness effects from evolution experiments are available in an increasing number of species, opening the way for a vast array of applications in evolutionary biology. However, comparison of estimated distributions among studies is hampered by inconsistencies in the definitions of fitness effects and selection coefficients. In particular, the use of ratios of Malthusian growth rates as ‘relative fitnesses’ leads to wrong inference of the strength of selection. Scaling Malthusian fitness by the generation time may help overcome this shortcoming, and allow accurate comparison of selection coefficients across species. For species reproducing by binary fission (neglecting cellular death), ln2 can be used as a correction factor, but in general, the growth rate and generation time of the wild-type should be provided in studies reporting distribution of mutation fitness effects. I also discuss how density and frequency dependence of population growth affect selection and its measurement in evolution experiments.     

Population effects of increased climate variation

Global circulation models predict and numerous observations confirm that anthropogenic climate change has altered high-frequency climate variability. However, it is not yet well understood how changing patterns of environmental variation will affect wildlife population dynamics and other ecological processes. Theory predicts that a population's long-run growth rate is diminished and the chance of population extinction is increased as environmental variation increases. This results from the fact that population growth is a multiplicative process and that long-run population growth rate is the geometric mean of growth rates over time, which is always less than the arithmetic mean. However, when population growth rates for unstructured populations are related nonlinearly to environmental drivers, increasing environmental variation can increase a population's long-run growth rate. This suggests that patterns of environmental variation associated with different aspects of climate change may affect population dynamics in different ways. Specifically, increasing variation in rainfall might result in diminished long-run growth rates for many animal species while increasing variation in temperature might result in increased long-run growth rates. While the effect of rainfall is theoretically well understood and supported by data, the hypothesized effect of temperature is not. Here, I analyse two datasets to study the effect of fluctuating temperatures on growth rates of zooplankton. Results are consistent with the prediction that fluctuating temperatures should increase long-run growth rates and the frequency of extreme demographic events.     

GENETIC CONSTRAINTS ON ADAPTATION TO A CHANGING ENVIRONMENT

Genetic correlations between traits can constrain responses to natural selection. To what extent such correlations limit adaptation depends on patterns of directional selection. I derive the expected rate of adaptation (or evolvability) under randomly changing selection gradients. When directional selection gradients have an arbitrary covariance matrix, the average rate of adaptation depends on genetic correlations between traits, contrary to the isotropic case investigated in previous studies. Adaptation may be faster on average with more genetic correlation between traits, if these traits are selected to change jointly more often than the average pair of traits. However, natural selection maximizes the long‐term fitness of a population, not necessarily its rate of adaptation. I therefore derive the average lag load caused by deviations of the mean phenotype from an optimum, under several forms of environmental changes typically experienced by natural populations, both stochastic and deterministic. Simple formulas are produced for how the G matrix affects long‐term fitness in these contexts, and I discuss how their parameters can be estimated empirically.     

Predicting population genetic change in an autocorrelated random environment: Insights from a large automated experiment

Most natural environments exhibit a substantial component of random variation, with a degree of temporal autocorrelation that defines the color of environmental noise. Such environmental fluctuations cause random fluctuations in natural selection, affecting the predictability of evolution. But despite long-standing theoretical interest in population genetics in stochastic environments, there is a dearth of empirical estimation of underlying parameters of this theory. More importantly, it is still an open question whether evolution in fluctuating environments can be predicted indirectly using simpler measures, which combine environmental time series with population estimates in constant environments. Here we address these questions by using an automated experimental evolution approach. We used a liquid-handling robot to expose over a hundred lines of the micro-alga Dunaliella salina to randomly fluctuating salinity over a continuous range, with controlled mean, variance, and autocorrelation. We then tracked the frequencies of two competing strains through amplicon sequencing of nuclear and choloroplastic barcode sequences. We show that the magnitude of environmental fluctuations (determined by their variance), but also their predictability (determined by their autocorrelation), had large impacts on the average selection coefficient. The variance in frequency change, which quantifies randomness in population genetics, was substantially higher in a fluctuating environment. The reaction norm of selection coefficients against constant salinity yielded accurate predictions for the mean selection coefficient in a fluctuating environment. This selection reaction norm was in turn well predicted by environmental tolerance curves, with population growth rate against salinity. However, both the selection reaction norm and tolerance curves underestimated the variance in selection caused by random environmental fluctuations. Overall, our results provide exceptional insights into the prospects for understanding and predicting genetic evolution in randomly fluctuating environments.     

Selection in a fluctuating environment leads to decreased genetic variation and facilitates the evolution of phenotypic plasticity

Changes in the environment are expected to induce changes in the quantitative genetic variation, which influences the ability of a population to adapt to environmental change. Furthermore, environmental changes are not constant in time, but fluctuate. Here, we investigate the effect of rapid, continuous and/or fluctuating temperature changes in the seed beetle Callosobruchus maculatus, using an evolution experiment followed by a split-brood experiment. In line with expectations, individuals responded in a plastic way and had an overall higher potential to respond to selection after a rapid change in the environment. After selection in an environment with increasing temperature, plasticity remained unchanged (or decreased) and environmental variation decreased, especially when fluctuations were added; these results were unexpected. As expected, the genetic variation decreased after fluctuating selection. Our results suggest that fluctuations in the environment have major impact on the response of a population to environmental change; in a highly variable environment with low predictability, a plastic response might not be beneficial and the response is genetically and environmentally canalized resulting in a low potential to respond to selection and low environmental sensitivity. Interestingly, we found greater variation for phenotypic plasticity after selection, suggesting that the potential for plasticity to evolve is facilitated after exposure to environmental fluctuations. Our study highlights that environmental fluctuations should be considered when investigating the response of a population to environmental change.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

EVOLUTION AND EXTINCTION IN A CHANGING ENVIRONMENT: A QUANTITATIVE-GENETIC ANALYSIS

Because of the ubiquity of genetic variation for quantitative traits, virtually all populations have some capacity to respond evolutionarily to selective challenges. However, natural selection imposes demographic costs on a population, and if these costs are sufficiently large, the likelihood of extinction will be high. We consider how the mean time to extinction depends on selective pressures (rate and stochasticity of environmental change, and strength of selection), population parameters (carrying capacity, and reproductive capacity), and genetics (rate of polygenic mutation). We assume that in a randomly mating, finite population subject to density-dependent population growth, individual fitness is determined by a single quantitative-genetic character under Gaussian stabilizing selection with the optimum phenotype exhibiting directional change, or random fluctuations, or both. The quantitative trait is determined by a finite number of freely recombining, mutationally equivalent, additive loci. The dynamics of evolution and extinction are investigated, assuming that the population is initially under mutation-selection-drift balance. Under this model, in a directionally changing environment, the mean phenotype lags behind the optimum, but on the average evolves parallel to it. The magnitude of the lag determines the vulnerability to extinction. In finite populations, stochastic variation in the genetic variance can be quite pronounced, and bottlenecks in the genetic variance temporarily can impair the population's adaptive capacity enough to cause extinction when it would otherwise be unlikely in an effectively infinite population. We find that maximum sustainable rates of evolution or, equivalently, critical rates of environmental change, may be considerably less than 10% of a phenotypic standard deviation per generation.     

EVOLUTION OF PHENOTYPE–ENVIRONMENT ASSOCIATIONS BY GENETIC RESPONSES TO SELECTION AND PHENOTYPIC PLASTICITY IN A TEMPORALLY AUTOCORRELATED ENVIRONMENT

Covariation between population‐mean phenotypes and environmental variables, sometimes termed a “phenotype–environment association” (PEA), can result from phenotypic plasticity, genetic responses to natural selection, or both. PEAs can potentially provide information on the evolutionary dynamics of a particular set of populations, but this requires a full theoretical characterization of PEAs and their evolution. Here, we derive formulas for the expected PEA in a temporally fluctuating environment for a quantitative trait with a linear reaction norm. We compare several biologically relevant scenarios, including constant versus evolving plasticity, and the situation in which an environment affects both development and selection but at different time periods. We find that PEAs are determined not only by biological factors (e.g., magnitude of plasticity, genetic variation), but also environmental factors, such as the association between the environments of development and of selection, and in some cases the level of temporal autocorrelation. We also describe how a PEA can be used to estimate the relationship between an optimum phenotype and an environmental variable (i.e., the environmental sensitivity of selection), an important parameter for determining the extinction risk of populations experiencing environmental change. We illustrate this ability using published data on the predator‐induced morphological responses of tadpoles to predation risk.     

Quantification and decomposition of environment‐selection relationships

In nature, selection varies across time in most environments, but we lack an understanding of how specific ecological changes drive this variation. Ecological factors can alter phenotypic selection coefficients through changes in trait distributions or individual mean fitness, even when the trait‐absolute fitness relationship remains constant. We apply and extend a regression‐based approach in a population of Soay sheep (Ovis aries) and suggest metrics of environment‐selection relationships that can be compared across studies. We then introduce a novel method that constructs an environmentally structured fitness function. This allows calculation of full (as in existing approaches) and partial (acting separately through the absolute fitness function slope, mean fitness, and phenotype distribution) sensitivities of selection to an ecological variable. Both approaches show positive overall effects of density on viability selection of lamb mass. However, the second approach demonstrates that this relationship is largely driven by effects of density on mean fitness, rather than on the trait‐fitness relationship slope. If such mechanisms of environmental dependence of selection are common, this could have important implications regarding the frequency of fluctuating selection, and how previous selection inferences relate to longer term evolutionary dynamics.     

Stochastic gene expression in fluctuating environments

Stochastic mechanisms can cause a group of isogenic bacteria, each subject to identical environmental conditions, to nevertheless exhibit diverse patterns of gene expression. The resulting phenotypic subpopulations will typically have distinct growth rates. This behavior has been observed in several contexts, including sugar metabolism and pili phase variation. Under fixed environmental conditions, the net growth rate of the population is maximized when all cells are of the fastest growing phenotype, so it is unclear what fitness advantage is conferred by population heterogeneity. However, unlike ideal laboratory conditions, natural environments tend to fluctuate, either periodically or randomly. Here we use a stochastic population model to show that, during growth in such fluctuating environments, a dynamically heterogenous bacterial population can sometimes achieve a higher net growth rate than a homogenous one. By using stochastic mechanisms to sample several distinct phenotypes, the bacteria are able to anticipate and take advantage of sudden changes in their environment. However, this heterogeneity is beneficial only if the bacterial response rate is sufficiently low. Our results could be useful in the design of artificial evolution experiments and in the optimization of fermentation processes.     

The limits to parapatric speciation: Dobzhansky-Muller incompatibilities in a continent-island model

How much gene flow is needed to inhibit speciation by the accumulation of Dobzhansky–Muller incompatibilities (DMIs) in a structured population? Here, we derive these limits in a classical migration–selection model with two haploid or diploid loci and unidirectional gene flow from a continent to an island. We discuss the dependence of the maximum gene-flow rate on ecological factors (exogeneous selection), genetic factors (epistasis, recombination), and the evolutionary history. Extensive analytical and numerical results show the following: (1) The maximum rate of gene flow is limited by exogeneous selection. In particular, maintenance of neutral DMIs is impossible with gene flow. (2) There are two distinct mechanisms that drive DMI evolution in parapatry, selection against immigrants in a heterogeneous environment and selection against hybrids due to the incompatibility. (3) Depending on the mechanism, opposite predictions result concerning the genetic architecture that maximizes the rate of gene flow a DMI can sustain. Selection against immigrants favors evolution of tightly linked DMIs of arbitrary strength, whereas selection against hybrids promotes the evolution of strong unlinked DMIs. In diploids, the fitness of the double heterozygotes is the decisive factor to predict the pattern of DMI stability.     

The temporal distribution of directional gradients under selection for an optimum

Temporal variation in phenotypic selection is often attributed to environmental change causing movements of the adaptive surface relating traits to fitness, but this connection is rarely established empirically. Fluctuating phenotypic selection can be measured by the variance and autocorrelation of directional selection gradients through time. However, the dynamics of these gradients depend not only on environmental changes altering the fitness surface, but also on evolution of the phenotypic distribution. Therefore, it is unclear to what extent variability in selection gradients can inform us about the underlying drivers of their fluctuations. To investigate this question, we derive the temporal distribution of directional gradients under selection for a phenotypic optimum that is either constant or fluctuates randomly in various ways in a finite population. Our analytical results, combined with population‐ and individual‐based simulations, show that although some characteristic patterns can be distinguished, very different types of change in the optimum (including a constant optimum) can generate similar temporal distributions of selection gradients, making it difficult to infer the processes underlying apparent fluctuating selection. Analyzing changes in phenotype distributions together with changes in selection gradients should prove more useful for inferring the mechanisms underlying estimated fluctuating selection.