* The value of the mutation rate was picked according to results of a simulation model on the evolution of the mutation rate after 300 generations of a population experiencing stochastic environmental conditions (no climate change, and no plasticity, Romero-Mujalli et al. 2019a). In addition this value is within the range of mutation rates used in other simulation models (reviewed in Romero-Mujalli et al. 2019b).

3. Results

3.1 Implications of assuming limits and flexible development for the modeling of phenotypic plasticity

Considering the degree of adaptation under directional deterministic environmental change (no stochastic noise, typical of laboratory experiments), any form of adaptive phenotypic plasticity was of advantage over non-adaptive plasticity and absence of plasticity (i.e., genetic determinism) (Fig. 5A). When the slope equals one, linear reaction norms yield perfect plasticity, while the “perfect match” of the phenotype with the environmental optimum depends on whether the limits are exceeded or not for those forms of plasticity that account for limits (Fig. 5A).

Fig. 5 (A) Mean degree of adaptation (fitness proxy) in populations with and without adaptive phenotypic plasticity experiencing directional deterministic environmental change (rate of change η = 0.03 , density dependence effect ψ = 1.0 , mutation rate µ = 0.001 ). A scenario of no adaptation (µ = 0 ) was also simulated, for comparison. The simulation was run for 100 generations. (B) Genetic variance (average ± standard error from 100 replicates) present in a population experiencing constant environmental conditions (no climate change, no stochasticity; for example, laboratory conditions) per scenario of phenotypic plasticity. (C) Time series of the genotypic and phenotypic variances present in a population experiencing deterministic directional environmental change (no stochasticity), under the scenarios of adaptive sinusoidal and logistic phenotypic plasticity (upper panel). When the population is locally adapted (initial part), there is less phenotypic than genetic variation (developmental canalization , at the population level, Posadas and Carthew 2014). Under the assumption of adaptive sinusoidal plasticity, as the environment changes and the population is pushed towards its limits of plasticity, cryptic phenotypic variation arises, and the mean fitness (here, degree of adaptation) of the population reduces (lower panel). In linear reaction norms, the genotypic and phenotypic variances are directly proportional.

Interestingly, adaptive sinusoidal and logistic plasticity led to stable functioning phenotypes (close to the optimum) despite of variation at the genetic level (developmental canalization at the population level, Posadas and Carthew 2014) (Figure 5B, C). A consequence of this property is that they maintained higher genetic variance than the alternative methods even when the population was exposed to a constant environment (e.g., laboratory conditions) for several generations (Fig. 5B). Additionally, in the model assuming a sinusoidal reaction norm, cryptic variation appeared in the population when it was pushed towards the limits (Fig. 5C). Cryptic variation refers to genetic variation that normally has little or no effect on phenotypic variation but that under atypical conditions generates phenotypic variation (Paaby and Rockman 2014).

3.2 Phenotypic plasticity: life histories under environmental change

Under scenarios of weak density dependence effect (ψ = 0.5 ), in which breeding females in the population could produce relatively few offspring, adaptive phenotypic plasticity played a major role promoting persistence as compared to organisms with higher ψ (Fig. 6, 7). Particularly for this life history strategy under slow climate change, adaptive phenotypic plasticity became of high importance promoting adaptation under positively autocorrelated environmental stochasticity (red noise). When the rate of change η was too rapid, adaptive phenotypic plasticity (particularly, linear reaction norm and logistic plasticity) became of advantage for all simulated environmental conditions of noise color (Fig. 6).

Fig. 6 The effect of non-adaptive (random) and adaptive (linear, sinusoidal, and logistic) phenotypic plasticity on probability of persistence (100 replicates, 250 generations each) of a population in which breeding females are limited to produce relatively few offspring (ψ = 0.5 , weak density dependence effect). The linear reaction norm had a slope b = 0.5 . A scenario of genetic determinism (narrow sense heritability h² = 1 ) was also simulated, for comparison. The color bar illustrates the color of the stochastic noise.

Fig. 7 Effect of non-adaptive (random) and adaptive (sinusoidal, logistic, and linear) phenotypic plasticity on the probability of persistence (100 replicates, 250 generations each) of a population of intermediate and strong density dependence effect (ψ = 1.8 and 2.5 , respectively). The linear reaction norm had a slopeb = 0.5 . A scenario of genetic determinism (narrow sense heritability h² = 1 ) was also simulated. The color bar illustrates the color of the stochastic noise. Scenarios ofη < 0.03 were not shown, because all treatments led to maximum probability of persistence.

In contrast, results of strong and very strong density compensation (ψ = 1.8 and 2.5 , respectively) always persisted under conditions of only environmental stochasticity (no directional climate change) and relatively slow to medium rate of environmental change (η = 0, 0.01, 0.02 ), regardless of the type of environmental stochasticity (noise color). For these life history strategies, genetic determinism and all forms of plasticity performed equally well (Fig. 7).

Under scenarios of relatively rapid directional climate change, adaptive linear and logistic phenotypic plasticity performed the best for organisms of intermediate and high density dependence effects (ψ = 1.8 and ψ = 2.5 ) (Fig. 8, η = 0.04 ). The life history strategy with weak density compensation was not included in this analysis, since its populations always went extinct, except for the scenarios of linear and logistic plasticity. On the other hand, adaptive sinusoidal, non-adaptive random phenotypic plasticity, and genetic determinism showed similar performance across all scenarios of rapid rate of directional climate change (Fig. 7).

The performance of random plasticity was notably prominent for organisms of intermediate and high density dependence effects under scenarios of lower mutation rates (μ = 10^-4, Fig. 8). Similar results were observed when considering μ = 10^-3 and mutational effects according to the model of slightly deleterious mutations (Ohta 1973; Eyre-Walker et al. 2002), as in Romero-Mujalli et al. (2019b) (Fig. S2, Appendix B: Supplementary material). When genetic mechanisms producing novel variation are somehow constrained, these organisms can cope with changing environmental conditions relaying on non-adaptive phenotypic plasticity only (Fig. 8). This ability was not observed for organisms of weak density compensation, typical of large mammals and some bird species. However, populations of organisms with strong density compensation (ψ = 2.5 ) were considerably more vulnerable to extinction due to stochastic fluctuations in the carrying capacity (Fig. 9).

Fig. 8 Relative importance of forms of non-adaptive (random) and adaptive (sinusoidal, logistic and linear) phenotypic plasticity affecting persistence of populations differing in levels of density compensation (ψ ) and experiencing moderate rate of directional stochastic environmental change (η = 0.01 , ψ = 0.5 ; η = 0.025 , ψ = 1.8 and ψ = 2.5; 100 replicates, 250 generations each) under scenarios of low mutation rate (μ = 10^-4). A higher η , results for ψ = 0.5 were not shown, because scenarios of random and sinusoidal plasticity, and genetic determinism (narrow sense heritability h² = 1 ) went usually extinct. The color bar illustrates the color of the stochastic noise.

Fig. 9 Probability of persistence (100 replicates, 250 generations each) per life history strategy (levels of ψ ) under stochastic fluctuations in both, the carrying capacity (sd = 10and 20% of K ) and the phenotypic optimum (white noise, no directional climate change).

When the plastic response precedes the environmental change (i.e., change after a sensitive period of plastic trait development), the dynamic can considerably change for those life history strategies of weak density compensation (ψ = 0.5 ) under blue stochastic noise (Fig. S3, Appendix B: Supplementary material). Whether plastic responses precede or follow the environmental change seems to have little impact under life history strategies with stronger density compensation (Fig. S3, Appendix B: Supplementary material).

4. Discussion

The objectives of this study were: (i ) to evaluate the effect of assuming plasticity as a developmentally flexible phenotypic response with limits, and (ii ) to assess the relative importance of adaptive and non-adaptive plasticity for populations of sexual species with different life history strategy experiencing scenarios of directional climate change and environmental stochasticity (noise color). The simulated environmental conditions, though simplified as in every model, mimic realistic expected scenarios of environmental climate change (Björklund et al. 2009; Kopp and Matuszewski 2014; Vincenzi 2014). Our results show that the relative importance of phenotypic plasticity varies among life history strategies. Furthermore, they show that the advantage of non-adaptive and adaptive forms of phenotypic plasticity on population persistence depends on the type of environmental stochasticity (noise color) and the rate of directional climate change. In addition, the advantage of adaptive and non-adaptive forms of plasticity depends on whether plasticity precedes or follows the environmental change. Finally, assuming limits to plasticity lead to genetic accommodation (in the absence of costs to plasticity) and to the appearance of cryptic genetic variation when limits are exceeded. Moreover, assuming plasticity as the result of genotypes responding flexibly to feedback from their environment leads to developmental canalization at the population level (Posadas and Carthew 2014), many-to-one genotype-phenotype map (Wagner 2008; Aguilar-Rodríguez et al. 2018) promoting the coexistence of polymorphism, and higher maintained genetic variation, potentially increasing evolvability, as compared to traditional approaches (e.g., linear reaction norms).

4.1 On the limits and the model implementations of adaptive phenotypic plasticity

Our work explicitly accounts for limits to plasticity. Though a thorough analysis of limits to plasticity was beyond the scope of this study, there are some aspects worth consideration. Linear reaction norms have no limits and their evolution can theoretically result in organisms displaying perfectly adapted phenotypes under all environmental contexts, if no cost is imposed, which is unrealistic. Plastic responses certainly have energetic demands (i.e., costs), but also limits to plasticity – in the absence of costs – can result from the properties and characteristics of the underlying machinery producing the phenotypic response. Therefore, linear adaptive phenotypic plasticity can overestimate the probability of persistence. In our simulations this was prevented by having a slope of 0.5 instead of 1. Even so, linear adaptive phenotypic plasticity had the best performance (in terms of increasing population persistence) in comparison to the other methods of adaptive plasticity. In the model, it took approximately 80, 60 and 40 generations or years to cross the limits of plasticity for a rate of directional change of 0.02, 0.03 and 0.04, respectively. Thus, for the population to persist the 250 generations under such scenarios of environmental change, genetic changes need to necessarily follow to enable survival beyond plasticity limits. Only linear adaptive phenotypic plasticity, as simulated in the model, could – for some life history strategies, ψ = 1.8 and 2.5 – lead to persistence of the population with minimal, if any, genetic changes. Thus, the existence of limits to plasticity (without the consideration of costs; i.e., when costs are negligible, see Murren et al. 2015) can lead to a transition from an “environmentally induced” to a “genetically encoded” state of a trait (genetic accommodation, Sommer 2020). The process of genetic accommodation occurs when phenotypic variants that are environmentally induced, become genetically determined by natural selection (West-Eberhard 2003; Schlichting and Wund 2014), and has been observed in nature (Schlichting and Wund 2014; Kulkarni et al. 2017). Moreover, our model shows that when the environmental change pushes towards the limits of plasticity, cryptic phenotypic variation may arise. This means higher phenotypic than genotypic variance, a ratio that could be measured in nature. This feature comes along with a reduction in mean fitness of the population (in the model, degree of adaptation) and could potentially be used as an early warning signal for the inability of a population to sustain environmental change (Boettiger et al. 2013).

In our model, forms of phenotypic plasticity with limits are assumed to enable diverse functional responses through flexible interactions of the genotype with the environment (Laland et al. 2015; van Gestel and Weissing 2016). This assumption leads to, first, stable functioning of the phenotype (close to the optimum) despite the variation at the genetic level (developmental canalization at the population level, Posadas and Carthew 2014). Second, multiple genotypes can have the same phenotype (many-to-one genotype-phenotype map, Ahnert 2017; Aguilar-Rodríguez et al. 2018), which does not occur under linear reaction-norms, thus enabling coexistence of polymorphism when different genotypes have the same fitness. Third, genotypes with the same phenotype are mutationally interconnected, such that small mutations can transform these genotypes into one another without altering the phenotype (robustness), or lead to new phenotypes (evolvability), at the same time (Wagner 2008; Ahnert 2017; Aguilar-Rodríguez et al. 2018; Payne and Wagner 2019). Moreover, it can act as a mechanism that maintains genetic variation, potentially increasing evolvability (van Gestel and Weissing 2016). In our implementation of adaptive plasticity this is evidenced by the sinusoidal and logistic plasticity scenarios resulting in the highest maintained genetic variation – higher than linear reaction norms and genetic determinism – even when the population is exposed to a constant environment (e.g., laboratory conditions) for several generations. This agrees with observations on the maintenance of adaptive capacity in populations (often clonal populations) kept in enclosed environments (e.g., chemostats) for several generations (Maharjan et al. 2006; Fussmann et al. 2007). Modelling approaches have been limited by the tendency of the trait variance to unrealistically decline to zero over time (Merico et al. 2014; Romero-Mujalli et al. 2019b). Our study shows that a simulation of trait dynamics more in line with developmental systems responding flexibly to internal (genotype) and external inputs (environment) enabling highly diverse functional responses, as done for logistic and sinusoidal plasticity, could help to overcome this limitation. So far, our implementation offers a phenomenological alternative only. Future theoretical work should focus on unraveling the mechanisms that make the above-discussed phenotypic response (trait dynamics) possible. Furthermore, and in contrast to other forms of plasticity, flexible development of the phenotype can provide an adaptive response in a novel environment that need not to have been pre-screened by earlier selection, which is particularly true for learning (Laland et al. 2015; Watson and Szathmáry 2016; Romero-Mujalli et al. 2017).

4.2 The relative importance of adaptive and non-adaptive phenotypic plasticity differs between life history strategies

According to our model, any form of adaptive phenotypic plasticity is of advantage over non-adaptive plasticity under controlled systematic changes of environmental conditions, as it is the case of laboratory experiments. However, under stochastic environmental fluctuations typical of natural habitats, adaptive phenotypic plasticity is particularly of advantage when facing positively autocorrelated environmental fluctuations (i.e., red noise) and slow directional environmental change; and for a broader spectrum of climatic fluctuations when there is a faster trend in the mean environmental optimum. Particularly, all forms of adaptive phenotypic plasticity were superior to lack of plasticity for life history strategies with relatively slow growth rate (weak density compensation). This life history strategy resembles characteristics of many vertebrate species (e.g., birds and mammals). In contrast, for life history strategies with strong density dependence effects and therefore, fast growth rate, genetic determinism and all forms of plasticity performed equally well. For these life history strategies, only under relatively high rates of directional climate change, some forms of adaptive phenotypic plasticity (i.e., linear and logistic phenotypic plasticity) and, interestingly, random non-adaptive plasticity, were of advantage. The difference in the relative importance of adaptive and non-adaptive plasticity among life history strategies was even more evident under scenarios of genetic constraints (in our model, scenarios of relatively low mutation rate, and of slightly deleterious mutations fitness effects). Under such scenarios, life history strategies with strong density compensation could cope with the changing environmental conditions relying on random non-adaptive plasticity only. This was not the case for life history strategies with weak density compensation, which strongly depended on adaptive plasticity for their adaptation to the local environment. Therefore, if organisms with different life history strategies, as simulated in this model, would experience equivalent environmental fluctuations and rates of directional climate change, those where breeding females are limited to few offspring are expected to experience stronger selection for the development of mechanisms of adaptive phenotypic plasticity, and stronger selection to expand the limits of their plasticity mechanisms. Populations of organisms with faster growth rate can rely on non-adaptive plasticity (translating into bet-hedging, Donaldson-Matasci et al. 2013) for a broader range of environmental fluctuations. As linear and logistic phenotypic plasticity does not generally outperform non-adaptive plasticity, these adaptive plastic responses are not expected to evolve for life history strategies producing relatively large numbers of offspring, unless they experience relatively rapid rates of directional environmental change. A further life history parameter promoting the evolution of adaptive phenotypic plasticity is longevity/generation time, leading to a limited genetic response (Forsman 2014, not tested in our study).

The importance of adaptive phenotypic plasticity for organisms experiencing directional change of the mean environmental optimum, as inferred in our simulations, may equally apply to dispersing and sessile organisms. Dispersing organisms may experience gradual changes in the mean environmental optimum and will benefit from developing adaptive forms of phenotypic plasticity as they expand their range, especially if density dependence is weak, as it occurs in mammal and bird species. Similarly, sessile organisms exposed to seasonal changes in the mean environmental optimum will also benefit from adaptive forms of phenotypic plasticity, particularly if their longevity is high (Borges 2008). Examples are plant species inhabiting temperate regions (Chmielewski and Rötzer 2001), as well as plants experiencing regular yearly cycles of rain, drought and fire at the equator (Fajardo et al. 2005).

Despite of their good performance across scenarios of directional stochastic environmental change (as tested in this study), populations with strong density dependence are susceptible to extinction under stochastic fluctuations in environmental quality (i.e., stochastic carrying capacity). Fluctuations in the carrying capacity can result from fire, resource contamination, or human land use, among other factors (Anderson et al. 2015). In contrast, populations with weak density compensation are less impacted by fluctuations in the carrying capacity of their environment. This would, for example, suggest that insect populations are more impacted by fluctuations in habitat quality (e.g., because of land use) than mammals. It remains however to be investigated, in how far this observation depends on the specific model implementation.

4.3 Phenotypic plasticity and environmental noise: when adaptive plasticity hinders evolution

To date, most studies of phenotypic plasticity (and its evolution) under stochastic environmental fluctuations have been focused on the level of correlation among the environmental optimum and the environmental cues sensed by organisms (Reed et al. 2010; Ashander et al. 2016; Ergon and Ergon 2016). They show that adaptive phenotypic plasticity can only evolve under positive correlation of cues with the environmental optimum (environmental predictability). It is important to note that this type of predictability is not the same as the predictable (red noise) year-to-year pattern of our simulations. Specifically, this study adds that adaptive forms of phenotypic plasticity can decrease extinction risk under positively autocorrelated environmental stochasticity (red noise), while extinction risk is expected to be high in the absence of plasticity (Mustin et al. 2013). However, if the environment changes after the sensitive period when the trait is susceptible to be modified by plasticity, those traits initially developed by adaptive plasticity can lead to a disadvantage under negatively autocorrelated environmental stochasticity (blue noise), which agrees with results from evolution experiments (Hallsson and Björklund 2012). This applies for constant characters, i.e., phenotypic traits plastic early in ontogeny, resulting in mature phenotypes which cannot be further modified by the environment (Lande 2014; Romero-Mujalli et al. 2019b). In our model, such disadvantage of adaptive plasticity under blue noise occurred only for life history strategies with weak density compensation. Whether the plastic response precedes or follows the environmental change seems to have little impact under life history strategies with stronger density compensation. Thus, for life history strategies with weak density compensation under blue noise, adaptive phenotypic plasticity amplifies phenotype-environment mismatches due to the negative autocorrelation in the environmental stochasticity and, therefore, hinders evolution in the long run, whilst non-adaptive phenotypic plasticity and genetic determinism increase population persistence under such scenarios. Therefore, life history strategies with weak density compensation relying on any form of adaptive phenotypic plasticity under blue stochastic environmental conditions are expected to be more vulnerable – than those with stronger density compensation – unless they are able to adjust their plasticity strategy. However, adaptive plasticity (particularly, linear and logistic phenotypic plasticity) becomes of advantage for all conditions of stochastic noise – even when the environmental change occurs after the sensitive period – when the rate of environmental change is relatively rapid. In our model, rapid rates of directional change of 0.03 and 0.04 change the environment beyond the range captured by the stochasticity (considering a two-sigma-effect, 95%) of the original reference environment after 100 and 75 generations (or years), respectively. A “fast” rate of environmental change is not necessarily an absolute measurement. It should be related to characteristics – life history strategy – of the population, as shown in this study, and to the strength of selection (Burger and Lynch 1995).

5 Conclusions

Adaptive phenotypic plasticity promotes population persistence under positively autocorrelated (red noise) environmental stochasticity and slow climate change, and for a broader range of fluctuations when the rate of directional change is faster. This form of plasticity is particularly important for life history strategies in which breeding females have a limited number of offspring (low fecundity, and hence slow population growth rate, typical of many vertebrate species: birds and mammals). Organisms producing more offspring per female may cope with environmental fluctuations relying only on genetic changes or random plasticity, unless the rate of environmental change is relatively high. Whenever plasticity precedes the environmental change, typical of constant characters, those life history strategies with weak density compensation will experience high risk of extinction in bluish habitats, unless they can adjust their plasticity into a bet-hedging strategy.

Models employing linear reaction norms may overestimate persistence, if they do not consider limits of plasticity. Furthermore, limits to plasticity lead to genetic accommodation when costs are negligible and to the exposure of cryptic genetic variation when the plastic response is pushed towards the limits of plasticity. In addition, the modelling of plasticity as developmental systems relying on genotypes interacting flexibly with their environment enables coexistence of polymorphisms and highly diverse functional responses, leading to high maintained genetic variation, many-to-one genotype-phenotype map, and to developmental canalization at the population level.

In this work, the mechanisms that shape the limits of adaptive plasticity were not explicitly modeled. Empirical work is needed to unravel molecular mechanisms that may dictate the limits of plastic responses.

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