Stephan B. Munch

Predator-induced phenotypic plasticity within- and across-generations: a challenge for theory?

Much work has shown that the environment can induce non-genetic changes in phenotype that span multiple generations. Theory predicts that predictable environmental variation selects for both increased within- and across-generation responses. Yet, to the best of our knowledge, there are no empirical tests of this prediction. We explored the relationship between within- versus across-generation plasticity by evaluating the influence of predator cues on the life-history traits of Daphnia ambigua. We measured the duration of predator-induced transgenerational effects, determined when transgenerational responses are induced, and quantified the cues that activate transgenerational plasticity. We show that predator exposure during embryonic development causes earlier maturation and increased reproductive output. Such effects are detectable two generations removed from predator exposure and are similar in magnitude in response to exposure to cues emitted by injured conspecifics. Moreover, all experimental contexts and traits yielded a negative correlation between within- versus across-generation responses. That is, responses to predator cues within- and across-generations were opposite in sign and magnitude. Although many models address transgenerational plasticity, none of them explain this apparent negative relationship between within- and across-generation plasticities. Our results highlight the need to refine the theory of transgenerational plasticity.

Determining Individual Variation in Growth and Its Implication for Life-History and Population Processes Using the Empirical Bayes Method

The differences in demographic and life-history processes between organisms living in the same population have important consequences for ecological and evolutionary dynamics. Modern statistical and computational methods allow the investigation of individual and shared (among homogeneous groups) determinants of the observed variation in growth. We use an Empirical Bayes approach to estimate individual and shared variation in somatic growth using a von Bertalanffy growth model with random effects. To illustrate the power and generality of the method, we consider two populations of marble trout Salmo marmoratus living in Slovenian streams, where individually tagged fish have been sampled for more than 15 years. We use year-of-birth cohort, population density during the first year of life, and individual random effects as potential predictors of the von Bertalanffy growth functions parameters k (rate of growth) and (asymptotic size). Our results showed that size ranks were largely maintained throughout marble trout lifetime in both populations. According to the Akaike Information Criterion (AIC), the best models showed different growth patterns for year-of-birth cohorts as well as the existence of substantial individual variation in growth trajectories after accounting for the cohort effect. For both populations, models including density during the first year of life showed that growth tended to decrease with increasing population density early in life. Model validation showed that predictions of individual growth trajectories using the random-effects model were more accurate than predictions based on mean size-at-age of fish.

Predator-driven brain size evolution in natural populations of Trinidadian killifish (Rivulus hartii)

Vertebrates exhibit extensive variation in relative brain size. It has long been assumed that this variation is the product of ecologically driven natural selection. Yet, despite more than 100 years of research, the ecological conditions that select for changes in brain size are unclear. Recent laboratory selection experiments showed that selection for larger brains is associated with increased survival in risky environments. Such results lead to the prediction that increased predation should favour increased brain size. Work on natural populations, however, foreshadows the opposite trajectory of evolution; increased predation favours increased boldness, slower learning, and may thereby select for a smaller brain. We tested the influence of predator-induced mortality on brain size evolution by quantifying brain size variation in a Trinidadian killifish, Rivulus hartii, from communities that differ in predation intensity. We observed strong genetic differences in male (but not female) brain size between fish communities; second generation laboratory-reared males from sites with predators exhibited smaller brains than Rivulus from sites in which they are the only fish present. Such trends oppose the results of recent laboratory selection experiments and are not explained by trade-offs with other components of fitness. Our results suggest that increased male brain size is favoured in less risky environments because of the fitness benefits associated with faster rates of learning and problem-solving behaviour.

Local adaptation in transgenerational responses to predators

Environmental signals can induce phenotypic changes that span multiple generations. Along with phenotypic responses that occur during development (i.e. ‘within-generation’ plasticity), such ‘transgenerational plasticity’ (TGP) has been documented in a diverse array of taxa spanning many environmental perturbations. New theory predicts that temporal stability is a key driver of the evolution of TGP. We tested this prediction using natural populations of zooplankton from lakes in Connecticut that span a large gradient in the temporal dynamics of predator-induced mortality. We reared more than 120 clones of Daphnia ambigua from nine lakes for multiple generations in the presence/absence of predator cues. We found that temporal variation in mortality selects for within-generation plasticity while consistently strong (or weak) mortality selects for increased TGP. Such results provide us the first evidence for local adaptation in TGP and argue that divergent ecological conditions select for phenotypic responses within and across generations.

On estimating the reliability of ecological forecasts.

Recent work has highlighted the utility of nonparametric forecasting methods for predicting ecological time series (Perretti et al., 2013. Proc. Natl. Acad. Sci. U.S.A. 110, 5253-5257). However, one topic that has received considerably less attention is the quantification of uncertainty in nonparametric forecasts. This important topic was brought to the forefront in the recent work by Jabot (2014. J. Theor. Biol.). Here, we add to this emerging discussion by reviewing the available methods for quantifying forecast uncertainty in nonparametric models. We conclude with a demonstration of one such method using the simulation model of Jabot (2014. J. Theor. Biol.). We find that nonparametric forecast error is accurately estimated with as few as 10 observations in the time series.

Trade-offs between accuracy and interpretability in von Bertalanffy random-effects models of growth

Better understanding of variation in growth will always be an important problem in ecology. Individual variation in growth can arise from a variety of processes; for example, individuals within a population vary in their intrinsic metabolic rates and behavioral traits, which may influence their foraging dynamics and access to resources. However, when adopting a growth model, we face trade-offs between model complexity, biological interpretability of parameters, and goodness of fit. We explore how different formulations of the von Bertalanffy growth function (vBGF) with individual random effects and environmental predictors affect these trade-offs. In the vBGF, the growth of an organism results from a dynamic balance between anabolic and catabolic processes. We start from a formulation of the vBGF that models the anabolic coefficient (q) as a function of the catabolic coefficient (k), a coefficient related to the properties of the environment (γ) and a parameter that determines the relative importance of behavior and environment in determining growth (ψ). We treat the vBGF parameters as a function of individual random effects and environmental variables. We use simulations to show how different functional forms and individual or group variability in the growth functions parameters provide a very flexible description of growth trajectories. We then consider a case study of two fish populations of Salmo marmoratus and Salmo trutta to test the goodness of fit and predictive power of the models, along with the biological interpretability of vBGFs parameters when using different model formulations. The best models, according to AIC, included individual variability in both k and γ and cohort as predictor of growth trajectories, and are consistent with the hypothesis that habitat selection is more important than behavioral and metabolic traits in determining lifetime growth trajectories of the two fish species. Model predictions of individual growth trajectories were largely more accurate than predictions based on mean size-at-age of fish. Our method shares information across individuals, and thus, for both fish populations investigated, allows using a single measurement early in the life of individual fish or cohort to obtain accurate predictions of lifetime individual or cohort size-at-age.

The intrinsic predictability of ecological time series and its potential to guide forecasting

Successfully predicting the future states of systems that are complex, stochastic and potentially chaotic is a major challenge. Model forecasting error (FE) is the usual measure of success; however model predictions provide no insights into the potential for improvement. In short, the realized predictability of a specific model is uninformative about whether the system is inherently predictable or whether the chosen model is a poor match for the system and our observations thereof. Ideally, model proficiency would be judged with respect to the systems’ intrinsic predictability – the highest achievable predictability given the degree to which system dynamics are the result of deterministic v. stochastic processes. Intrinsic predictability may be quantified with permutation entropy (PE), a model-free, information-theoretic measure of the complexity of a time series. By means of simulations we show that a correlation exists between estimated PE and FE and show how stochasticity, process error, and chaotic dynamics affect the relationship. This relationship is verified for a dataset of 461 empirical ecological time series. We show how deviations from the expected PE-FE relationship are related to covariates of data quality and the nonlinearity of ecological dynamics. These results demonstrate a theoretically-grounded basis for a model-free evaluation of a system’s intrinsic predictability. Identifying the gap between the intrinsic and realized predictability of time series will enable researchers to understand whether forecasting proficiency is limited by the quality and quantity of their data or the ability of the chosen forecasting model to explain the data. Intrinsic predictability also provides a model-free baseline of forecasting proficiency against which modeling efforts can be evaluated. Glossary Active information: The amount of information that is available to forecasting models (redundant information minus lost information; Fig. 1). Forecasting error (FE): A measure of the discrepancy between a model’s forecasts and the observed dynamics of a system. Common measures of forecast error are root mean squared error and mean absolute error. Entropy: Measures the average amount of information in the outcome of a stochastic process. Information: Any entity that provides answers and resolves uncertainty about a process. When information is calculated using logarithms to the base two (i.e. information in bits), it is the minimum number of yes/no questions required, on average, to determine the identity of the symbol (Jost 2006). The information in an observation consists of information inherited from the past (redundant information), and of new information. Intrinsic predictability: the maximum achievable predictability of a system (Beckage et al. 2011). Lost information: The part of the redundant information lost due to measurement or sampling error, or transformations of the data (Fig. 1). New information, Shannon entropy rate: The Shannon entropy rate quantifies the average amount of information per observation in a time series that is unrelated to the past, i.e., the new information (Fig. 1). Nonlinearity: When the deterministic processes governing system dynamics depend on the state of the system. Permutation entropy (PE): permutation entropy is a measure of the complexity of a time series (Bandt & Pompe, 2002) that is negatively correlated with a system’s predictability (Garland et al. 2015). Permutation entropy quantifies the combined new and lost information. PE is scaled to range between a minimum of 0 and a maximum of 1. Realized predictability: the achieved predictability of a system from a given forecasting model. Redundant information: The information inherited from the past, and thus the maximum amount of information available for use in forecasting (Fig. 1). Symbols, words, permutations: symbols are simply the smallest unit in a formal language such as the letters in the English alphabet i.e., {“A”, “B”,…, “Z”}. In information theory the alphabet is more abstract, such as elements in the set {“up”, “down”} or {“1”, “2”, “3”}. Words, of length m refer to concatenations of the symbols (e.g., up-down-down) in a set. Permutations are the possible orderings of symbols in a set. In this manuscript, the words are the permutations that arise from the numerical ordering of m data points in a time series. Weighted permutation entropy (WPE): a modification of permutation entropy (Fadlallah et al., 2013) that distinguishes between small-scale, noise-driven variation and large-scale, system-driven variation by considering the magnitudes of changes in addition to the rank-order patterns of PE.