Yngvild Vindenes

Individual Heterogeneity in Vital Parameters and Demographic Stochasticity

Most population models assume that individuals have equal opportunities for survival and reproduction, although many natural populations consist of individuals with different vital parameters that remain different over time. Individual heterogeneity in vital parameters, which may depend on age or stage, can alter many population characteristics compared with a homogeneous population, affecting both deterministic and stochastic properties of the population process. Demographic variance is an important parameter influenced by heterogeneity. However, whether heterogeneity leads to increased or decreased demographic variance has been an unresolved question, except for special cases. Here, we present a general stochastic matrix model for a heterogeneous population that allows us to examine effects of heterogeneity on population dynamics, even when the degree of heterogeneity depends on age. Using this model, we found that the demographic variance may increase, decrease, or remain unaltered compared with a homogeneous comparison model, depending on the vital parameter values and on how these are distributed among individuals at each time step. Furthermore, if the reproductive value is the same for all individuals, heterogeneity has no effect on the demographic variance. Thus, we provide a general theoretical framework for analyzing how individual heterogeneity caused by different biological mechanisms affects fluctuations of especially small populations.

Individual heterogeneity in life histories and eco‐evolutionary dynamics

Individual heterogeneity in life history shapes eco-evolutionary processes, and unobserved heterogeneity can affect demographic outputs characterising life history and population dynamical properties. Demographic frameworks like matrix models or integral projection models represent powerful approaches to disentangle mechanisms linking individual life histories and population-level processes. Recent developments have provided important steps towards their application to study eco-evolutionary dynamics, but so far individual heterogeneity has largely been ignored. Here, we present a general demographic framework that incorporates individual heterogeneity in a flexible way, by separating static and dynamic traits (discrete or continuous). First, we apply the framework to derive the consequences of ignoring heterogeneity for a range of widely used demographic outputs. A general conclusion is that besides the long-term growth rate lambda, all parameters can be affected. Second, we discuss how the framework can help advance current demographic models of eco-evolutionary dynamics, by incorporating individual heterogeneity. For both applications numerical examples are provided, including an empirical example for pike. For instance, we demonstrate that predicted demographic responses to climate warming can be reversed by increased heritability. We discuss how applications of this demographic framework incorporating individual heterogeneity can help answer key biological questions that require a detailed understanding of eco-evolutionary dynamics.

Effects of Climate Change on Trait-Based Dynamics of a Top Predator in Freshwater Ecosystems

Predicted universal responses of ectotherms to climate warming include increased maximum population growth rate and changes in body size through the temperature-size rule. However, the mechanisms that would underlie these predicted responses are not clear. Many studies have focused on proximate mechanisms of physiological processes affecting individual growth. One can also consider ultimate mechanisms involving adaptive explanations by evaluating temperature effects on different vital rates across the life history and using the information in a population dynamical model. Here, we combine long-term data for a top predator in freshwater ecosystems (pike; Esox lucius) with a stochastic integral projection model to analyze concurrent effects of temperature on vital rates, body size, and population dynamics. As predicted, the net effect of warming on population growth rate (fitness) is positive, but the thermal sensitivity of this rate is highly size- and vital rate–dependent. These results are not sensitive to increasing variability in temperature. Somatic growth follows the temperature-size rule, and our results support an adaptive explanation for this response. The stable length structure of the population shifts with warming toward an increased proportion of medium-sized but a reduced proportion of small and large individuals. This study highlights how demographic approaches can help reveal complex underlying mechanisms for population responses to warming.

Poor environmental tracking can make extinction risk insensitive to the colour of environmental noise.

The relative importance of environmental colour for extinction risk compared with other aspects of environmental noise (mean and interannual variability) is poorly understood. Such knowledge is currently relevant, as climate change can cause the mean, variability and temporal autocorrelation of environmental variables to change. Here, we predict that the extinction risk of a shorebird population increases with the colour of a key environmental variable: winter temperature. However, the effect is weak compared with the impact of changes in the mean and interannual variability of temperature. Extinction risk was largely insensitive to noise colour, because demographic rates are poor in tracking the colour of the environment. We show that three mechanisms—which probably act in many species—can cause poor environmental tracking: (i) demographic rates that depend nonlinearly on environmental variables filter the noise colour, (ii) demographic rates typically depend on several environmental signals that do not change colour synchronously, and (iii) demographic stochasticity whitens the colour of demographic rates at low population size. We argue that the common practice of assuming perfect environmental tracking may result in overemphasizing the importance of noise colour for extinction risk. Consequently, ignoring environmental autocorrelation in population viability analysis could be less problematic than generally thought.

Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.

FIXATION OF SLIGHTLY BENEFICIAL MUTATIONS: EFFECTS OF LIFE HISTORY

Recent studies of rates of evolution have revealed large systematic differences among organisms with different life histories, both within and among taxa. Here, we consider how life history may affect the rate of evolution via its influence on the fixation probability of slightly beneficial mutations. Our approach is based on diffusion modeling for a finite, stage‐structured population with stochastic population dynamics. The results, which are verified by computer simulations, demonstrate that even with complex population structure just two demographic parameters are sufficient to give an accurate approximation of the fixation probability of a slightly beneficial mutation. These are the reproductive value of the stage in which the mutation first occurs and the demographic variance of the population. The demographic variance also determines what influence population size has on the fixation probability. This model represents a substantial generalization of earlier models, covering a large range of life histories.

Effects of demographic structure on key properties of stochastic density-independent population dynamics

The development of stochastic demography has largely been based on age structured populations, although other types of demographic structure, especially permanent and dynamic heterogeneity, are likely common in natural populations. The combination of stochasticity and demographic structure is a challenge for analyses of population dynamics and extinction risk, because the population structure will fluctuate around the stable structure and the population size shows transient fluctuations. However, by using a diffusion approximation for the total reproductive value, density-independent dynamics of structured populations can be described with only three population parameters: the expected population growth rate, the environmental variance and the demographic variance. These parameters depend on population structure via the state-specific vital rates and transition rates. Once they are found, the diffusion approximation represents a substantial reduction in model complexity. Here, we review and compare the key population parameters across a wide range of demographic structure, from the case of no structure to the most general case of dynamic heterogeneity, and for both discrete and continuous types. We focus on the demographic variance, but also show how environmental stochasticity can be included. This study brings together results from recent models, each considering a specific type of population structure, and places them in a general framework for structured populations. Comparison across different types of demographic structure reveals that the reproductive value is an essential concept for understanding how population structure affects stochastic dynamics and extinction risk.

Fitness consequences of early life conditions and maternal size effects in a freshwater top predator

Conditions experienced in early life stages can be an important determinant of individual life histories. In fish, environmental conditions are known to affect early survival and growth, but recent studies have also emphasized maternal effects mediated by size or age. However, the relative sensitivity of the mean fitness (population growth rate λ) to different early life impacts remains largely unexplored. Using a female-based integral projection model (IPM) parameterized from unique long-term demographic data for pike (Esox lucius), we evaluated the relative fitness consequences of different early life impacts, including (i) maternal effects of length on egg weight, potentially affecting offspring (first year) survival, and (ii) effects of temperature on offspring growth and survival. Of the seven vital rates defining the model, offspring survival could not be directly estimated and four scenarios were defined for this rate. Elasticity analyses of the IPM were performed to calculate (i) the total contribution from different lengths to the elasticity of λ to the projection kernel, and (ii) the elasticity of λ to underlying variables of female current length, female offspring length at age 1, and temperature. These elasticities were decomposed into contributions from different vital rates across length. Egg weight increased with female length, as expected, but the effect leveled off for the largest females. However, λ was largely insensitive to this effect, even when egg weight was assumed to have a strong effect on offspring survival. In contrast, λ was sensitive to early temperature conditions through growth and survival. Among mature females, the total elasticity of λ to the projection kernel generally increased with length. The results were robust to a wide range of assumptions. These results suggest that environmental conditions experienced in early life represent a more important driver of mean population growth and fitness of pike than maternal effects of size on offspring survival. We discuss two general mechanisms underlying the weak influence of this maternal effect, suggesting that these may be general for long-lived and highly fecund fishes. This model and results are relevant for the management of long-lived top predators, including many commercially important fish species.

Individual heterogeneity and early life conditions shape growth in a freshwater top predator

Body size can have profound impacts on survival, movement, and reproductive schedules shaping individual fitness, making growth a central process in ecological and evolutionary dynamics. Realized growth is the result of a complex interplay between life history schedules, individual variation, and environmental influences. Integrating all of these aspects into growth models is methodologically difficult, depends on the availability of repeated measurements of identifiable individuals, and consequently represents a major challenge in particular for natural populations. Using a unique 30-yr time series of individual length measurements inferred from scale year rings of wild brown trout, we develop a Bayesian hierarchical model to estimate individual growth trajectories in temporally and spatially varying environments. We reveal a gradual decrease in average juvenile growth, which has carried over to adult life and contributed to decreasing sizes observed at the population level. Commonly studied environmental drivers like temperature and water flow did not explain much of this trend and overall persistent and among-year individual variation dwarfed temporal variation in growth patterns. Our model and results are relevant to a wide range of questions in ecology and evolution requiring a detailed understanding of growth patterns, including conservation and management of many size-structured populations.

Study mortality with hazard rates, not probabilities

Mortality is a key process in ecology and evolution, and much effort is spent on statistical and theoretical modelling of this process. Mortality hazard rates describe individuals’ instantaneous ability to survive at different times, whereas survival probabilities are defined for time-intervals of a given length. In this commentary, we argue that it is often more meaningful to model and interpret interval-specific time-averaged mortality hazard rates than survival probabilities, also when using discrete-time models. We discuss three topics where the concept of hazard rates is essential for sound biological inference, but nevertheless often not used: (i) modelling of covariate effects on survival probabilities or mortality hazard rates, (ii) modelling of multiple sources of mortality and competing risks, and (iii) elasticity analyses of population growth rate to demographic parameters. To facilitate estimation of cause-specific mortality hazard rates, we provide amendments to R package ‘marked’. By focusing on modelling mortality hazard rates, one avoids several inherent problems of comparing survival or mortality probabilities. In particular, interpretations about relative differences between mortality hazard rates (measurements on a ratio scale), or effects of relative changes in mortality hazard rates on population growth (elasticities), are more meaningful than similar interpretations involving survival (or mortality) probabilities or odds (measurements on an absolute scale). The concept of hazard rates is essential for understanding ecological and evolutionary processes and we give an intuitive explanation for this, using several examples. We provide some practical guidelines and suggestions for further methods developments.