Incorporating the temporal autocorrelation of demographic rates into structured population models

Population dynamics are typically temporally autocorrelated: population sizes are positively or negatively correlated with past population sizes. Previous studies have found that positive temporal autocorrelation increases the risk of extinction due to ‘inertia’ that prolongs downward fluctuations in population size. However, temporal autocorrelation has not yet been analyzed at the level of life cycle transitions. We developed an R package, colorednoise, which creates stochastic matrix population projections with distinct temporal autocorrelation values for each matrix element. We used it to analyze long-term demographic data on 25 populations from the COMADRE and COMPADRE databases and simulate their stochastic dynamics. We found a broad range of temporal autocorrelation across species, populations and life cycle stages. The number of stage-classes in the matrix strongly affected the temporal autocorrelation of the growth rate. In the plant populations, reproduction transitions had more negative temporal autocorrelation than survival transitions, and matrices dominated by positive temporal autocorrelation had higher extinction risk, while in animal populations transition type was not associated with noise color. Our results indicate that temporal autocorrelation varies across life cycle transitions, even among populations of the same species. We present the colorednoise package for researchers to analyze the temporal autocorrelation of structured demographic rates.     

Environmental Uncertainty and Variable Diapause

We analyze a stage-structured model of a population that displays variable diapause in a randomly varying environment. The ruggedness of the environment is measured by the extent of random variation in per-capita reproductive success. We show how variable diapause and environmental characteristics affect the population′s stochastic growth rate. In rugged unpredictable environments, phenotypes that show some tendency to diapause are found to have a higher growth rate than nondiapausing phenotypes. In harsh rugged environments, some tendency to diapause may be all that permits population persistence. Positive serial autocorrelation causes the optimal diapause fraction to decrease, while negative autocorrelation causes that fraction to increase. The structured model behaves very differently from a scalar model for large diapause fractions even in uncorrelated environments, and in many cases predicts a broad optimum. The difference between models is due to the extreme variability of stage structure in populations subject to even small variability when diapause tendency is high.     

Demography in an increasingly variable world

Recent advances in stochastic demography provide unique insights into the probable effects of increasing environmental variability on population dynamics, and these insights can be substantially different compared with those from deterministic models. Stochastic variation in structured population models influences estimates of population growth rate, persistence and resilience, which ultimately can alter community composition, species interactions, distributions and harvesting. Here, we discuss how understanding these demographic consequences of environmental variation will have applications for anticipating changes in populations resulting from anthropogenic activities that affect the variance in vital rates. We also highlight new tools for anticipating the consequences of the magnitude and temporal patterning of environmental variability.     

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.     

The diversity of population responses to environmental change

The current extinction and climate change crises pressure us to predict population dynamics with ever‐greater accuracy. Although predictions rest on the well‐advanced theory of age‐structured populations, two key issues remain poorly explored. Specifically, how the age‐dependency in demographic rates and the year‐to‐year interactions between survival and fecundity affect stochastic population growth rates. We use inference, simulations and mathematical derivations to explore how environmental perturbations determine population growth rates for populations with different age‐specific demographic rates and when ages are reduced to stages. We find that stage‐ vs. age‐based models can produce markedly divergent stochastic population growth rates. The differences are most pronounced when there are survival‐fecundity‐trade‐offs, which reduce the variance in the population growth rate. Finally, the expected value and variance of the stochastic growth rates of populations with different age‐specific demographic rates can diverge to the extent that, while some populations may thrive, others will inevitably go extinct.     

Habitat dependence and correlations between elasticities of long-term growth rates

In population biology, elasticity is a measure of the importance of a demographic rate on population growth. A relatively small amount of stochasticity can substantially impact the dynamics of a population whose growth is a function of deterministic and stochastic processes. Analyses of natural populations frequently neglect the latter. Even in a population that fluctuates substantially with time, the results of a deterministic perturbation analysis correlated strongly with results of a perturbation analysis of the long-run stochastic growth rate. Population growth was, however, not uniformly sensitive to demographic rates across different environmental conditions. The overall correlation between deterministic and stochastic perturbation analysis may be high, but environmental variability can dramatically alter the contributions of demographic rates in different environmental conditions. This potentially informative detail is neglected by deterministic analysis, yet it highlights one difficulty when extrapolating results from long-term analysis to shorter-term environmental change.     

Temporal autocorrelation in host density increases establishment success of parasitoids in an experimental system

Environmental variation is classically expected to affect negatively population growth and to increase extinction risk, and it has been identified as a major determinant of establishment failures in the field. Yet, recent theoretical investigations have shown that the structure of environmental variation and more precisely the presence of positive temporal autocorrelation might alter this prediction. This is particularly likely to affect the establishment dynamics of biological control agents in the field, as host–parasitoid interactions are expected to induce temporal autocorrelation in host abundance. In the case where parasitoid populations display overcompensatory dynamics, the presence of such positive temporal autocorrelation should increase their establishment success in a variable environment. We tested this prediction in laboratory microcosms by introducing parasitoids to hosts whose abundances were manipulated to simulate uncorrelated or positively autocorrelated variations in carrying capacity. We found that environmental variability decreased population size and increased parasitoid population variance, which is classically expected to extinction risk. However, although exposed to significant environmental variation, we found that parasitoid populations experiencing positive temporal autocorrelation in host abundance were more likely to persist than populations exposed to uncorrelated variation. These results confirm that environmental variation is a key determinant of extinction dynamics that can have counterintuitive effects depending on its autocorrelation structure.     

Frequency responses of age-structured populations: Pacific salmon as an example

Increasing evidence of the effects of changing climate on physical ocean conditions and long-term changes in fish populations adds to the need to understand the effects of stochastic forcing on marine populations. Cohort resonance is of particular interest because it involves selective sensitivity to specific time scales of environmental variability, including that of mean age of reproduction, and, more importantly, very low frequencies (i.e., trends). We present an age-structured model for two Pacific salmon species with environmental variability in survival rate and in individual growth rate, hence spawning age distribution. We use computed frequency response curves and analysis of the linearized dynamics to obtain two main results. First, the frequency response of the population is affected by the life history stage at which variability affects the population; varying growth rate tends to excite periodic resonance in age structure, while varying survival tends to excite low frequency fluctuation with more effect on total population size. Second, decreasing adult survival strengthens the cohort resonance effect at all frequencies, a finding that addresses the question of how fishing and climate change will interact.     

Buffering and plasticity in vital rates of oldfield rodents

1. Under the hypothesis of environmental buffering, populations are expected to minimize the variance of the most influential vital rates; however, this may not be a universal principle. Species with a life span <1 year may be less likely to exhibit buffering because of temporal or seasonal variability in vital rate sensitivities. Further, plasticity in vital rates may be adaptive for species in a variable environment with reliable cues. 2. We tested for environmental buffering and plasticity in vital rates using stage-structured matrix models from long-term data sets in four species of grassland rodents. We used periodic matrices to estimate stochastic elasticity for each vital rate and then tested for correlations with a standardized coefficient of variation for each rate. 3. We calculated stochastic elasticities for individual months to test for an association between increased reproduction and the influence of reproduction, relative to survival, on the population growth rate. 4. All species showed some evidence of buffering. The elasticity of vital rates of Peromyscus leucopus (Rafinesque, 1818), Sigmodon hispidus Say & Ord, 1825 and Microtus ochrogaster (Wagner, 1842) was negatively related to vital rate CV. Elasticity and vital rate CV were negatively related in Peromyscus maniculatus (Wagner, 1845), but the relationship was not statistically significant. Peromyscus leucopus and M. ochrogaster showed plasticity in vital rates; reproduction was higher following months where elasticity for reproduction exceeded that of survival. 5. Our results suggest that buffering is common in species with fast life histories; however, some populations that exhibit buffering are capable of responding to short-term variability in environmental conditions through reproductive plasticity.     

Interactive life‐history traits predict sensitivity of plants and animals to temporal autocorrelation

Temporal autocorrelation in demographic processes is an important aspect of population dynamics, but a comprehensive examination of its effects on different life‐history strategies is lacking. We use matrix population models from 454 plant and animal populations to simulate stochastic population growth rates (log λ_s) under different temporal autocorrelations in demographic rates , using simulated and observed covariation among rates. We then test for differences in sensitivities, or changes of log λ_s to changes in autocorrelation among two major axes of life‐history strategies, obtained from phylogenetically informed principal component analysis: the fast‐slow and reproductive‐strategy continua. Fast life histories exhibit highest sensitivities to simulated autocorrelation in demographic rates across reproductive strategies. Slow life histories are less sensitive to temporal autocorrelation, but their sensitivities increase among highly iteroparous species. We provide cross‐taxonomic evidence that changes in the autocorrelation of environmental variation may affect a wide range of species, depending on complex interactions of life‐history strategies.     

Autocorrelated environmental variation and the establishment of invasive species

Environmental parameters such as temperature and rainfall have a positively autocorrelated variance structure which makes it likely that runs of good or bad conditions will occur. It has previously been demonstrated that such autocorrelated environmental variance can increase the probability of extinction in small populations, in much the same way that increased variance without autocorrelation can increase extinction risk. As a result, it has also been suggested that positive autocorrelation will decrease the probability that a species will establish in a novel location. We suggest that describing the probability of invasion success as the probability of indefinite persistence may be an inappropriate definition of risk. Economic or ecological damage may be associated with a population that initially reaches high densities before going extinct in the new location. In addition, such populations may spread to new locations before extirpation. We use a modeling approach to examine the effect of positively autocorrelated conditions on the probability that small populations will reach large size before extinction. We find that where variance is high and the geometric mean of the population growth rate is low, autocorrelation increases the risk that a population will pass a an upper threshold density, even when extinction probability is unaffected. Therefore species classified as having low probability of invasion risk on the basis of population growth rates measured in low variance environments may actually have quite a substantial probability of establishing a large population for a period of time. The mechanism behind the effect is the disproportionate influence of short runs of good conditions initially following introduction.     

Growth autocorrelation and animal size variation

It has long been recognized that variability in animal size is affected by how individual growth rate is autocorrelated in time. Earlier studies have attributed the mechanism generating the autocorrelation primarily to size‐dependent growth rate and autocorrelation in resource abundance. All of these studies have shown that positive autocorrelation in individual growth rate always translates into increased variability in size. We show that energy reserves in individuals induce growth autocorrelation by acting like a low pass filter between the resource and the internal energy that is available for metabolism, growth and reproduction. However, the reserve also reduces the variance in growth rate. Consequently, reserve‐induced growth autocorrelation has relatively little effect on size variability in the population, contradicting existing ideas about the relationship between the growth autocorrelation and size variability.     

Stochastic LTRE analysis of the effects of herbivory on the population dynamics of a perennial grassland herb

Herbivores can have strong deleterious effects on vital rates (growth, reproduction, and survival) and thus negatively impact the population dynamics of plant species. In practice, however, these effects might be strongly correlated, for example as a result of tradeoffs between vital rates. To get better insights into the effects of herbivory on the population dynamics of the long‐lived grassland plant Primula veris population projection matrices were constructed from demographic data collected between 1999 and 2008 (nine annual transitions). Data were collected in two large grassland populations, each of which was subjected to two treatments (grazing by cattle versus a mowing treatment), yielding a total of 36 matrices. We applied a lower‐level vital rate life table response experiment (LTRE) using the small noise approximation (SNA) of the stochastic population growth rate to disentangle the contributions of changes in mean vital rates, variability in vital rates, correlations between vital rates and vital rate elasticities to the difference in the stochastic growth rate. Stochastic growth rates (a= log λ_S) were significantly lower in grazed than in mown plots (a= 0.0185 and 0.1019, respectively). SNA LTRE analysis showed that contributions of mean vital rates by far made the largest contribution to the observed difference in a between grazed and control plots. In particular, changes in sexual reproduction rates made the largest contributions to lower the stochastic growth rate in grazed plots: both adult flowering probabilities and flower and seed production were importantly lower in grazed populations, but these negative effects were largely buffered by increased establishment and seedling survival rates. Among the stochastic terms of the SNA decomposition, contributions of covariance and correlations between vital rates had the largest impact, whereas contributions of elasticities were smaller. The strongest correlation driver was the association between adult survival and seedling establishment, suggesting that environmental conditions favouring adult survival also are beneficial for seedling establishment. Overall, our results show that herbivory had a strong negative effect on the long‐term population growth rate of P. veris that was primarily mediated by differences in fecundity (flower and seed production) and germination.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.     

Demographic strategies of plant invaders in temporally varying environments

Plant populations may have evolved different demographic strategies to cope with temporal environmental variation. According to the demographic buffering hypothesis, vital rates that are most critical to population persistence are buffered against environmental variation and vary little over time, whereas the demographic lability hypothesis suggests that populations may track and benefit from environmental variation. While the hypotheses of demographic strategies have been widely tested in plant and animal species, they have not been explicitly examined for invasive plants, or in relation to different modelling methods (deterministic vs. stochastic). Here, we tested the demographic buffering and lability hypotheses for 23 populations of eight invasive plant species in relation to life form (woody vs. herbaceous species) and population growth rate using deterministic and stochastic modelling methods, and absolute and relative scales. We found that conclusions of demographic strategies depended on scale, with an absolute scale resulting in stronger negative correlations between the variability and importance of vital rates (i.e., buffering) than a relative scale. Conclusions of demographic strategies were also affected by life form that interacted with method. The populations of woody invaders exhibited buffering regardless of the method used, while for the populations of herbaceous species, deterministic calculations suggested buffering and stochastic calculations suggested lability. Overall, our findings emphasise the role of life form and methodological issues that need to be considered when exploring demographic strategies in fluctuating environments.     

Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1.  Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity.
2.  We analyse a stochastic environment model of the red kangaroo ( Macropus rufus ), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates.
3.  Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4.  Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates.
5.  Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c . 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c . 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.     

Patchy populations in stochastic environments: critical number of patches for persistence

We introduce a model for the dynamics of a patchy population in a stochastic environment and derive a criterion for its persistence. This criterion is based on the geometric mean (GM) through time of the spatial-arithmetic mean of growth rates. For the population to persist, the GM has to be >/=1. The GM increases with the number of patches (because the sampling error is reduced) and decreases with both the variance and the spatial covariance of growth rates. We derive analytical expressions for the minimum number of patches (and the maximum harvesting rate) required for the persistence of the population. As the magnitude of environmental fluctuations increases, the number of patches required for persistence increases, and the fraction of individuals that can be harvested decreases. The novelty of our approach is that we focus on Malthusian local population dynamics with high dispersal and strong environmental variability from year to year. Unlike previous models of patchy populations that assume an infinite number of patches, we focus specifically on the effect that the number of patches has on population persistence. Our work is therefore directly relevant to patchily distributed organisms that are restricted to a small number of habitat patches.     

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.     

Small variance in growth rates in annual plants has large effects on genetic drift

? Premise of the Study: Effective population size (N(e)) is a critical index of the evolutionary capacity of populations. Low N(e) indicates that standing genetic diversity is susceptible to loss via stochastic processes (and inbreeding) and is, therefore, unavailable for natural selection to act upon. Reported N(e) in plant populations is often quite low. What biological and ecological factors might produce such low N(e) ? Methods: We conducted a simulation model to test the effect of randomly assigned and autocorrelated growth rates of annual plants on plant-size distributions at the end of the growing season. Because plant size is directly correlated with reproductive output in annual plants, variation in plant size reflects variation in reproduction, and thus our modeled size distributions can be used to estimate N(e). ? Key Results: Randomly assigned growth rates had a negligble effect on N(e)/N. Autocorrelated growth rates decreased N(e)/N as the length of the growing season increased. This was the case even when the variance in growth rates was as low as 0.1% of the mean. ? Conclusions: While intrinsic plant biology can affect the degree of growth autocorrelation, ecological factors such as competition, herbivory, and abiotic stress can increase or decrease levels of growth autocorrelation. Ecological factors that increase growth autocorrelation can have significant effects on genetic drift within populations.