Evolution of size-dependent flowering in a variable environment: partitioning the effects of fluctuating selection

In a stochastic environment, two distinct processes, namely nonlinear averaging and non-equilibrium dynamics, influence fitness. We develop methods for decomposing the effects of temporal variation in demography into contributions from nonlinear averaging and non-equilibrium dynamics. We illustrate the approach using Carlina vulgaris, a monocarpic species in which recruitment, growth and survival all vary from year to year. In Carlina the absolute effect of temporal variation on the evolutionarily stable flowering strategy is substantial (ca. 50% of the evolutionarily stable flowering size) but the net effect is much smaller (ca. 10%) because the effects of temporal variation do not influence the evolutionarily stable strategy in the same direction.     

Stochastic stable population growth in integral projection models: theory and application

Stochastic matrix projection models are widely used to model age- or stage-structured populations with vital rates that fluctuate randomly over time. Practical applications of these models rest on qualitative properties such as the existence of a long term population growth rate, asymptotic log-normality of total population size, and weak ergodicity of population structure. We show here that these properties are shared by a general stochastic integral projection model, by using results in (Eveson in D. Phil. Thesis, University of Sussex, 1991, Eveson in Proc. Lond. Math. Soc. 70, 411-440, 1993) to extend the approach in (Lange and Holmes in J. Appl. Prob. 18, 325-344, 1981). Integral projection models allow individuals to be cross-classified by multiple attributes, either discrete or continuous, and allow the classification to change during the life cycle. These features are present in plant populations with size and age as important predictors of individual fate, populations with a persistent bank of dormant seeds or eggs, and animal species with complex life cycles. We also present a case-study based on a 6-year field study of the Illyrian thistle, Onopordum illyricum, to demonstrate how easily a stochastic integral model can be parameterized from field data and then applied using familiar matrix software and methods. Thistle demography is affected by multiple traits (size, age and a latent "quality" variable), which would be difficult to accommodate in a classical matrix model. We use the model to explore the evolution of size- and age-dependent flowering using an evolutionarily stable strategy (ESS) approach. We find close agreement between the observed flowering behavior and the predicted ESS from the stochastic model, whereas the ESS predicted from a deterministic version of the model is very different from observed flowering behavior. These results strongly suggest that the flowering strategy in O. illyricum is an adaptation to random between-year variation in vital rates.     

Evolution in the real world: stochastic variation and the determinants of fitness in Carlina vulgaris

Empirical studies of life histories often ignore stochastic variation, despite theoretical demonstrations of its potential impact on life-history evolution. Here we use a novel approach to explore the effects of stochastic variation on life-history evolution and estimate the selection pressures operating on the monocarpic perennial Carlina vulgaris, in which flowering may be delayed by up to eight years. The approach is novel in that we use modern theoretical techniques to estimate selection pressures and the fitness landscape from a fully parameterised individual-based model. These approaches take into account temporal variation in demographic rates and density dependence. Analysis of 16 years' data revealed significant temporal variation in growth, mortality, and recruitment in our study population. Flowering was strongly size dependent and, unusually for such a species, also age dependent. Individual-based models of the flowering strategy, parameterized using field data, consistently underestimated the size at flowering, when temporal variation in demographic rates was ignored. In contrast, models that incorporated temporal variation in growth, mortality, and recruitment predicted sizes at flowering not significantly different from those observed in the field. Temporal variation in mortality, which had the largest effect on the flowering strategy, selected for increased size at flowering. An analytical approximation is presented to explain this result, extending the "1-year look-ahead criterion" presented in Rees et al. (2000). A fitness landscape generated by following the fate of rare mutant invaders with a broad range of alternative flowering strategies demonstrated that the observed parameters were adaptive. However, the fitness landscape reveals that approximately equal fitness is achieved by a broad range of strategies, providing a mechanism for the maintenance of genetic variation. To understand how the different parameters that defined our models determine the fitness of rare mutants, we numerically estimated the elasticities and sensitivities of mutant fitness. This demonstrated strong selection on a number of the parameters. Elasticities and sensitivities estimated in constant and random environments were significantly positively correlated, and both were negatively related to the standard error of the parameter. This last result is surprising and, we argue, reflects the genetic and phenotypic responses to selection.     

Evolution of Size-Dependent Flowering in Onopordum illyricum: A Quantitative Assessment of the Role of Stochastic Selection Pressures

We explore the evolution of delayed, size-dependent reproduction in the monocarpic perennial Onopordum illyricum, using a range of mathematical models, parameterized with long-term field data. Analysis of the long-term data indicated that mortality, flowering, and growth were age and size dependent. Using mixed models, we estimated the variance about each of these relationships and also individual-specific effects. For the field populations, recruitment was the main density-dependent process, although there were weak effects of local density on growth and mortality. Using parameterized growth models, which assume plants grow along a deterministic trajectory, we predict plants should flower at sizes approximately 50% smaller than observed in the field. We then develop a simple criterion, termed the "1-yr look-ahead criterion," based on equating seed production now with that of next year, allowing for mortality and growth, to determine at what size a plant should flower. This model allows the incorporation of variance about the growth function and individual-specific effects. The model predicts flowering at sizes approximately double that observed, indicating that variance about the growth curve selects for larger sizes at flowering. The 1-yr look-ahead approach is approximate because it ignores growth opportunities more than 1 yr ahead. To assess the accuracy of this approach, we develop a more complicated dynamic state variable model. Both models give similar results indicating the utility of the 1-yr look-ahead criterion. To allow for temporal variation in the model parameters, we used an individual-based model with a genetic algorithm. This gave very accurate prediction of the observed flowering strategies. Sensitivity analysis of the model suggested that temporal variation in the parameters of the growth equation made waiting to flower more risky, so selected for smaller sizes at flowering. The models clearly indicate the need to incorporate stochastic variation in life-history analyses.     

Seed dormancy and delayed flowering in monocarpic plants: selective interactions in a stochastic environment

We explore the effects of temporal variation in multiple demographic rates on the joint evolution of delayed reproduction and seed dormancy using integral projection models (IPMs). To do this, we extend the standard IPM to include a discrete state variable representing the number of seeds in the seed bank, density-dependent recruitment, and temporal variation in demography. Parameter estimates for Carlina vulgaris and Carduus nutans are obtained from long-term studies. Carlina is relatively long lived and has a short-lived seed bank, whereas most Carduus plants flower in their first year and the seed bank is long lived. Using the evolutionarily stable strategy (ESS) approach, we predict the observed flowering and germination strategies. There is excellent agreement between the predictions and the field observations. The effects of temporal variation on the joint ESS are partitioned into components arising from nonlinear averaging (systematic changes in the mean resulting from the interaction between variability and nonlinearity) and nonequilibrium dynamics (fluctuations in fitness caused by temporal variation). This shows that temporal variation can have substantial effects on the observed flowering and germination strategies and that covariance between demographic processes is important. We extend the models to include spatial population structure and assess the robustness of the results from the nonspatial models.     

Evolution of flowering strategies in Oenothera glazioviana: an integral projection model approach

The timing of reproduction is a key determinant of fitness. Here, we develop parameterized integral projection models of size-related flowering for the monocarpic perennial Oenothera glazioviana and use these to predict the evolutionarily stable strategy (ESS) for flowering. For the most part there is excellent agreement between the model predictions and the results of quantitative field studies. However, the model predicts a much steeper relationship between plant size and the probability of flowering than observed in the field, indicating selection for a 'threshold size' flowering function. Elasticity and sensitivity analysis of population growth rate lambda and net reproductive rate R(0) are used to identify the critical traits that determine fitness and control the ESS for flowering. Using the fitted model we calculate the fitness landscape for invading genotypes and show that this is characterized by a ridge of approximately equal fitness. The implications of these results for the maintenance of genetic variation are discussed.     

Individualism in plant populations: using stochastic differential equations to model individual neighbourhood-dependent plant growth

We study individual plant growth and size hierarchy formation in an experimental population of Arabidopsis thaliana, within an integrated analysis that explicitly accounts for size-dependent growth, size- and space-dependent competition, and environmental stochasticity. It is shown that a Gompertz-type stochastic differential equation (SDE) model, involving asymmetric competition kernels and a stochastic term which decreases with the logarithm of plant weight, efficiently describes individual plant growth, competition, and variability in the studied population. The model is evaluated within a Bayesian framework and compared to its deterministic counterpart, and to several simplified stochastic models, using distributional validation. We show that stochasticity is an important determinant of size hierarchy and that SDE models outperform the deterministic model if and only if structural components of competition (asymmetry; size- and space-dependence) are accounted for. Implications of these results are discussed in the context of plant ecology and in more general modelling situations.     

Individual size variation and population stability in a seasonal environment: a discrete-time model and its calibration using grasshoppers

Much recent literature is concerned with how variation among individuals (e.g., variability in their traits and fates) translates into higher-level (i.e., population and community) dynamics. Although several theoretical frameworks have been devised to deal with the effects of individual variation on population dynamics, there are very few reports of empirically based estimates of the sign and magnitude of these effects. Here we describe an analytical model for size-dependent, seasonal life cycles and evaluate the effect of individual size variation on population dynamics and stability. We demonstrate that the effect of size variation on the population net reproductive rate varies in both magnitude and sign, depending on season length. We calibrate our model with field data on size- and density-dependent growth and survival of the generalist grasshopper Melanoplus femurrubrum. Under deterministic dynamics (fixed season length), size variation impairs population stability, given naturally occurring densities. However, in the stochastic case, where season length exhibits yearly fluctuations, size variation reduces the variance in population growth rates, thus enhancing stability. This occurs because the effect of size variation on net reproductive rate is dependent on season length. We discuss several limitations of the current model and outline possible routes for future model development.     

Age-specific fitness components and their temporal variation in the barn owl

Theory predicts that temporal variability plays an important role in the evolution of life histories, but empirical studies evaluating this prediction are rare. In constant environments, fitness can be measured by the population growth rate lambda, and the sensitivity of lambda to changes in fitness components estimates selection on these traits. In variable environments, fitness is measured by the stochastic growth rate lambda(S), and stochastic sensitivities estimate selection pressure. Here we examine age-specific schedules for reproduction and survival in a barn owl population (Tyto alba). We estimated how temporal variability affected fitness and selection, accounting for sampling variance. Despite large sample sizes of old individuals, we found no strong evidence for senescence. The most variable fitness components were associated with reproduction. Survival was less variable. Stochastic simulations showed that the observed variation decreased fitness by about 30%, but the sensitivities of lambda and lambda(S) to changes in all fitness components were almost equal, suggesting that temporal variation had negligible effects on selection. We obtained these results despite high observed variability in the fitness components and relatively short generation time of the study organism, a situation in which temporal variability should be particularly important for natural selection and early senescence is expected.     

Flowering Frequency and Plant Performance and their Relation to Age in the Perennial Orchid Spiranthes spiralis (L.) Chevall.

Abstract: Long-term demographic data have been analyzed to establish possible costs of flowering in the terrestrial orchid Spiranthes spiralis (L.) Chevall. in The Netherlands. Costs of flowering can be expressed as individual plant performance and flowering frequency in relation to the generative or vegetative status in the following year. Flowering in individuals of S. spiralis in a given year (t) is followed by a non-flowering phase in the next growing season (t + 1) in more than 80 % of the plants. The decline in flowering frequency is not a result of the age structure of the population involved because individual plants do not show signs of senescence after 10 - 15 years of aerial presence as an autotrophic plant. Rosettes have a smaller leaf area in the year of flowering (t), compared to the previous (t - 1) and following year (t + 1), due to the allocation of the limited underground resources to both flowering stalk and rosette at the beginning of its growing season. Generative reproduction in S. spiralis has a significant negative impact on both flowering frequency in subsequent years and on rosette size in the year of flowering. The flowering frequency and rosette size in relation to the life history, characterized by the yearly replacement of the underground tuber, is discussed. Better understanding of the life-history strategy, including costs of reproduction, may contribute to the creation of sustainable environmental conditions for growth of S. spiralis, e.g., optimal conditions for photosynthesis during the aboveground stage of the tiny wintergreen rosettes.     

The biennial life strategy in a random environment

A discrete-time population model with two age classes is studied which describes the growth of biennial plants in a randomly varying environment. A fraction of the oldest age class delays its flowering each year. The solution of the model involves products of random matrices. We calculate the exact mean and variance of the long-run geometric growth rate assuming a gamma distribution for the random number of offspring per flowering plant after one year. It is shown, both by analytical calculation and numerical examples, that it is profitable for the population to delay its flowering, in the sense that the average growth rate increases and the extinction probability decreases. The optimal values of the flowering fraction depend upon the environmental and model parameters.     

Flowering phenology in Solidago altissima: adaptive strategies against temporal variation in temperature

The evolution of flowering phenology is the result of a trade-off that balances many factors, including growth, reproductive capacity, and temporal overlap with pollinators. When there is large temporal variation in temperature, particularly in the onset of frost, the optimum flowering strategy will vary from year to year. In Duluth, MN, USA, the end of the growing season can vary by more than 30 days. In this study, we observed flowering phenology and pollinator abundance on 15 genotypes of Solidago altissima in Duluth, MN. We predicted that temporal variation in temperature would lead to a range of flowering strategies in the S. altissima population; some genotypes flower early and in synchrony, some ‘hedge their bets’ by flowering over a range of dates, and others have an intermediate strategy. Our results indicate that genotypes vary in mean flowering date and duration of flowering and, for the two observed years, pollinator abundance was highest for early-flowering genotypes.     

What drives selection on flowering time? An experimental manipulation of the inherent correlation between genotype and environment

The optimal timing of the seasonal switch from somatic growth to reproduction can depend on an individual's condition at reproduction, the quality of the environment in which it will reproduce, or both. In annual plants, vegetative size (a function of age at flowering) affects resources available for seed production, whereas exposure to mutualists, antagonists, and abiotic stresses in the environment (functions of Julian date of flowering) influences success in converting resources into offspring. The inherent tight correlation between age, size, and environment obscures their independent fitness contributions. We isolated the fitness effects of these factors by experimentally manipulating the correlation between age at flowering and date of flowering in Brassica rapa. We staggered the planting dates of families with differing ages at flowering to produce experimental populations in which age at flowering and date of flowering were positively, negatively, or uncorrelated. In all populations, plants with an early date of flowering produced more seed than those flowering late, regardless of age or size at flowering onset. The temporal environment was thus the principal driver of selection on flowering time, but its importance relative to that of age and size varied with the presence/absence of herbivores and seed predators.     

Changes in size and age at maturity in a population of kokanee Oncorhynchus nerka during a period of declining growth conditions

Trends in size distributions and age at maturity of spawning kokanee Oncorhynchus nerka during a 5 year period of declining growth conditions at Bucks Lake, California, U.S.A. were consistent with the hypothesis that reductions in growth rates in successive cohorts induce a shift to an older age at maturity. This forestalls decreases in size at maturity during a transitional period characterized by an increasing proportion of individuals that delay maturation. During the course of the study, kokanee first began declining in size at maturity, and then shifted from a 3 year to a 4 year egg to adult cycle. Individuals that spawned during their fourth year (age 3 years) were significantly larger, on average, than members of their cohort that spawned during their third year (age 2 years). This difference was greatest when age 2 year adults were smallest. The shift to an older age at maturity prevented a steady decline in size at maturity, even though age‐specific size was steadily declining over time. Size at maturity, however, began to decline again once the transition to a 4 year cycle was complete. In addition, there was a general trend of decreasing length‐specific mass. The data indicate that there is a range of growth trajectories over which delayed maturity can prevent a temporal pattern of decreasing size at maturity as growth rates decline.     

Evolution of a plastic quantitative trait in an age-structured population in a fluctuating environment

We analyze weak fluctuating selection on a quantitative character in an age-structured population not subject to density regulation. We assume that early in the first year of life before selection, during a critical state of development, environments exert a plastic effect on the phenotype, which remains constant throughout the life of an individual. Age-specific selection on the character affects survival and fecundity, which have intermediate optima subject to temporal environmental fluctuations with directional selection in some age classes as special cases. Weighting individuals by their reproductive value, as suggested by Fisher, we show that the expected response per year in the weighted mean character has the same form as for models with no age structure. Environmental stochasticity generates stochastic fluctuations in the weighted mean character following a first-order autoregressive model with a temporally autocorrelated noise term and stationary variance depending on the amount of phenotypic plasticity. The parameters of the process are simple weighted averages of parameters used to describe age-specific survival and fecundity. The "age-specific selective weights" are related to the stable distribution of reproductive values among age classes. This allows partitioning of the change in the weighted mean character into age-specific components.     

Fluctuating selection on reproductive timing in Digitalis purpurea

Despite recent, strong interest in the modelling of monocarpic perennial flowering strategies, little is known about how variation in demographic rates affects selection on optimal timing of flowering. Temporal variation may yield fluctuating selective pressures, or, if individuals experience time trends, selection for phenotypic plasticity. Here we report the results of a 3-year study in a large field population of the facultative biennial herb Digitalis purpurea , where we use field data on size-dependent growth, survival and fecundity to parameterize an existing optimisation model. We compare results from models using either deterministic or individually varying demographic rates to address the degree of fluctuating selection on the flowering strategy. In addition, we explore whether recent growing conditions influence the size-specific liability to flower. Model results differed widely between years; immediate onset of reproduction was predicted in 1999, strongly delayed reproduction in 2000. This reflected large differences in both growth and survival rates between years. Observed flowering sizes also varied between years, but were larger in 1999 than in 2000, contrary to model predictions. Incorporating individual variation in growth increased predicted optimal flowering sizes compared to models using deterministic growth, whereas the inclusion of individual survival variation had little effect. There was no significant effect of recent growth rate on flowering probability. Taken together, these results indicate highly fluctuating selection on the flowering strategy in D. purpurea , but no evidence of adaptive plasticity in response to current growing conditions. Fluctuating selection may contribute to maintain genetic variation for threshold size for flowering, and may partly explain the large within-season size-variation in flowering individuals found in natural populations of D. purpurea .     

Demographic stochasticity and temporal autocorrelation in the dynamics of structured populations

Populations can show temporal autocorrelation in the dynamics arising from different mechanisms, including fluctuations in the demographic structure. This autocorrelation is often treated as a complicating factor in the analyses of stochastic population growth and extinction risk. However, it also reflects important information about the demographic structure. Here, we consider how temporal autocorrelation is related to demographic stochasticity in structured populations. Demographic stochasticity arises from inherent randomness in the demographic processes of individuals, like survival and reproduction, and the resulting impact on population growth is measured by the demographic variance. Earlier studies have shown that population structure have positive or negative effects on the demographic variance compared to a model where the structure is ignored. Here, we derive a new expression for the demographic variance of a structured population, using the temporal autocorrelation function of the population growth rate. We show that the relative difference in demographic variance when the structure is included or ignored (the effect of structure on demographic variance) is approximately twice the sum of the autocorrelations. We demonstrate the result for a simple hypothetical example, as well as a set of empirical examples using age‐structured models of 24 mammals from the demographic database COMADRE. In the empirical examples, the sum of the autocorrelation function was negative in all cases, indicating that age structure generally has a negative effect on the demographic variance (i.e. the demographic variance is lower compared to that of a model where the structure is ignored). Other kinds of structure, such as spatial heterogeneity affecting fecundity, can have positive effects on the demographic variance, and the sum of the autocorrelations will then be positive. These results yield new insights into the complex interplay between population structure, demographic variance, and temporal autocorrelation, that shapes the population dynamics and extinction risk of populations.     

A stochastic model for extinction and recurrence of epidemics: estimation and inference for measles outbreaks

Epidemic dynamics pose a great challenge to stochastic modelling because chance events are major determinants of the size and the timing of the outbreak. Reintroduction of the disease through contact with infected individuals from other areas is an important latent stochastic variable. In this study we model these stochastic processes to explain extinction and recurrence of epidemics observed in measles. We develop estimating functions for such a model and apply the methodology to temporal case counts of measles in 60 cities in England and Wales. In order to estimate the unobserved spatial contact process we suggest a method based on stochastic simulation and marginal densities. The estimation results show that it is possible to consider a unified model for the UK cities where the parameters depend on the city size. Stochastic realizations from the dynamic model realistically capture the transitions from an endemic cyclic pattern in large populations to irregular epidemic outbreaks in small human host populations.