Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

Can life-history traits predict the response of forb populations to changes in climate variability?

1.  Climate change will cause changes in average temperature and precipitation as well as increased fluctuations around the mean, yet few studies have considered the impacts of altered climate variability on plant populations. We tested whether life-history traits (expected life span, generation time and seed size) can predict plant responses to increased environmental variability across similar plant species sharing the same habitat.
2.  We combined long-term demographic data on 10 prairie forb species with stochastic demography techniques to estimate the effects of potential changes in matrix element means and variances on the long-term stochastic population growth rate.
3.  For all 10 species, recruitment had higher contribution and elasticity values than survival, meaning that climate change is more likely to influence population growth through effects on recruitment than on survival for these relatively short-lived forbs. Species with longer generation times had lower elasticities to increases in matrix element variability.
4.   Synthesis. Our analysis of a unique, long-term data set suggests that longer-lived plant species will be less vulnerable to the effects of future increases in climate variability. While this relationship was previously reported for diverse taxa from many locations, our results show that it also applies within a guild of short-lived species from a single community. The generality of the pattern demonstrates the potential for using life-history traits to make predictions about which species may be the most vulnerable to climate change.     

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.     

Sensitivity of the population growth rate to demographic variability within and between phases of the disturbance cycle

For species in disturbance-prone ecosystems, vital rates (survival, growth and reproduction) often vary both between and within phases of the cycle of disturbance and recovery; some of this variation is imposed by the environment, but some may represent adaptation of the life history to disturbance. Anthropogenic changes may amplify or impede these patterns of variation, and may have positive or negative effects on population growth. Using stochastic population projection matrix models, we develop stochastic elasticities (proportional derivatives of the long-run population growth rate) to gauge the population effects of three types of change in demographic variability (changes in within- and between-disturbance-phase variability and phase-specific changes). Computing these elasticities for five species of disturbance-influenced perennial plants, we pinpoint demographic rates that may reveal adaptation to disturbance, and we demonstrate that species may differ in their responses to different types of changes in demographic variability driven by climate change.     

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.     

Effects of climate variability on the demography of wild geladas

Nonhuman primates are an essential part of tropical biodiversity and play key roles in many ecosystem functions, processes, and services. However, the impact of climate variability on nonhuman primates, whether anthropogenic or otherwise, remains poorly understood. In this study, we utilized age‐structured matrix population models to assess the population viability and demographic variability of a population of geladas (Theropithecus gelada) in the Simien Mountains, Ethiopia with the aim of revealing any underlying climatic influences. Using data from 2008 to 2019 we calculated annual, time‐averaged, and stochastic population growth rates (λ) and investigated relationships between vital rate variability and monthly cumulative rainfall and mean temperature. Our results showed that under the prevailing environmental conditions, the population will increase (λ _s = 1.021). Significant effects from rainfall and/or temperature variability were widely detected across vital rates; only the first year of infant survival and the individual years of juvenile survival were definitively unaffected. Generally, the higher temperature in the hot‐dry season led to lower survival and higher fecundity, while higher rainfall in the hot‐dry season led to increased survival and fecundity. Overall, these results provide evidence of greater effects of climate variability across a wider range of vital rates than those found in previous primate demography studies. This highlights that although primates have often shown substantial resilience to the direct effects of climate change, their vulnerability may vary with habitat type and across populations.     

Alternative life-history pathways and the elasticity of stochastic matrix models

Loop analysis is a powerful tool for analyzing matrix population models. This note shows that the results of loop analysis, which have been proved for constant matrices only, apply to stochastic matrices as well if elasticity is defined as the effect of a proportional perturbation of both mean and variance. Using the ideas of loop analysis, it is shown that the structure of the stochastic matrix in terms of alternative life-history pathways has important consequences for the effect of stochasticity on elasticities. If the life cycle contains nonoverlapping, alternative life-history pathways, the ranking in terms of elasticity of the most critical vital rates may be reversed in stochastic and the corresponding average environments. This has obvious and important consequences for population management because focusing on a deterministic model would lead to an ineffective or counterproductive management strategy.     

Demography of Verreaux’s sifaka in a stochastic rainfall environment

In this study, we use deterministic and stochastic models to analyze the demography of Verreaux’s sifaka (Propithecus verreauxi verreauxi) in a fluctuating rainfall environment. The model is based on 16 years of data from Beza Mahafaly Special Reserve, southwest Madagascar. The parameters in the stage-classified life cycle were estimated using mark-recapture methods. Statistical models were evaluated using information-theoretic techniques and multi-model inference. The highest ranking model is time-invariant, but the averaged model includes rainfall-dependence of survival and breeding. We used a time-series model of rainfall to construct a stochastic demographic model. The time-invariant model and the stochastic model give a population growth rate of about 0.98. Bootstrap confidence intervals on the growth rates, both deterministic and stochastic, include 1. Growth rates are most elastic to changes in adult survival. Many demographic statistics show a nonlinear response to annual rainfall but are depressed when annual rainfall is low, or the variance in annual rainfall is high. Perturbation analyses from both the time-invariant and stochastic models indicate that recruitment and survival of older females are key determinants of population growth rate.     

Recruitment rates exhibit high elasticity and high temporal variation in populations of a short-lived perennial herb

Empirical studies for different life histories have shown an inverse relationship between elasticity (i.e. the proportional contribution to population growth rate) and temporal variation in vital rates. It is accepted that this relationship indicates the effect of selective pressures in reducing variation in those life‐history traits with a major impact on fitness. In this paper, we sought to determine whether changes in environmental conditions affect the relationship between elasticity of vital rates and their temporal variation, and whether vital rates with simultaneously large elasticity and temporal variation might represent a characteristic life‐history strategy. We used demographic data on 13 populations of the short‐lived Hypericum cumulicola over 5–6 years, in three time‐since‐fire classes. For each population of each time‐since‐fire, we computed the mean matrix over years and its respective elasticity matrix, and the coefficients of variation in matrix entries over study years as an estimate of temporal variability. We found that mean elasticity negatively significantly correlated with temporal variation in vital rates in populations (overall eight out of 13) included in each time‐since‐fire. However, seedling recruitment exhibited both high elasticity and high temporal variation in almost all study populations. These results indicated that (1) the general relationship between elasticity and temporal variation in vital rates was not modified by environmental changes due to time‐since‐fire, and (2) high elasticity and high temporal variation in seedling recruitment in H. cumulicola is a particular trait of the species' life history. After seed survival in the soil seed bank, seedling recruitment represents the most important life‐history trait influencing H. cumulicola population growth rate (and fitness). The high temporal variability in seedling recruitment suggests that this trait is determined by environmental cues, leading to an increase in population size and subsequent replenishment of the seed bank in favorable years.     

Climatic drivers of pinyon mouse Peromyscus truei population dynamics in a resource-restricted environment

Highly variable patterns in temperature and rainfall events can have pronounced consequences for small mammals in resource-restricted environments. Climatic factors can therefore play a crucial role in determining the fates of small mammal populations. We applied Pradel's temporal symmetry model to a 21-year capture–recapture dataset to study population dynamics of the pinyon mouse (Peromyscus truei) in a semi-arid mixed oak woodland in California, USA. We examined time-, season- and sex-specific variation in realized population growth rate (λ) and its constituent vital rates, apparent survival and recruitment. We also tested the influence of climatic factors on these rates. Overall monthly apparent survival was 0.81 ± 0.004 (estimate ± SE). Survival was generally higher during wetter months (October–May) but varied over time. Monthly recruitment rate was 0.18 ± 0.01, ranging from 0.07 ± 0.01 to 0.63 ± 0.07. Although population growth rate (λ) was highly variable, overall monthly growth rate was close to 1.0, indicating a stable population during the study period (λ ± SE = 0.99 ± 0.01). Average temperature and its variability negatively affected survival, whereas rainfall positively influenced survival and recruitment rates, and thus the population growth rate. Our results suggest that seasonal rainfall and variation in temperature at the local scale, rather than regional climatic patterns, more strongly affected vital rates in this population. Discerning such linkages between species' population dynamics and environmental variability are critical for understanding local and regional impacts of global climate change, and for gauging viability and resilience of populations in resource-restricted environments.     

Population responses of a conifer-dwelling aphid to seasonal changes in its host

Abstract.  1. Current evidence suggests that seasonal changes in spruce needle sap nutrients have a decisive influence on green spruce aphid ( Elatobium abietinum ) population density, but the mechanisms of population change, the roles of development rate, fertility and mortality, and the existence of density-dependent processes, are not clearly understood.
2. Experimental studies of aphid populations were conducted in controlled environments to estimate seasonal patterns in aphid mean relative growth rate, prenatal development, fertility, and mortality. Studies were also made of the effect of aphid crowding on vital rates.
3. Independent of the degree of aphid crowding, seasonal changes in the amino acid concentration of needle sap were tracked by aphid growth rate, fertility (and adult size), but not by rates of aphid mortality. The most pronounced change in vital rates, and the one most likely to drive seasonal population change, was in fertility. Prenatal development time actually became shorter in periods when nutrients were scarce, but the resulting adult aphids were smaller and less fertile than during periods of improved nutrition.
4. Density dependence of vital rates was only observed during mid-summer when nutrients were least available. Mortality, growth rate, and prenatal development were the most strongly density-dependent processes. In contrast, there was no evidence that fertility rates were likely to respond to crowding.
5. There were no important differences between populations reared on small, potted spruce trees and those on plantation trees aged 25 years. This gives confidence that demographic data from a variety of field and laboratory sources could be used to compile data appropriate for population models.     

The demographic impact of extreme events: stochastic weather drives survival and population dynamics in a long-lived seabird

1. Most scenarios for future climate change predict increased variability and thus increased frequency of extreme weather events. To predict impacts of climate change on wild populations, we need to understand whether this translates into increased variability in demographic parameters, which would lead to reduced population growth rates even without a change in mean parameter values. This requires robust estimates of temporal process variance, for example in survival, and identification of weather covariates linked to interannual variability. 2. The European shag Phalacrocorax aristotelis (L.) shows unusually large variability in population size, and large-scale mortality events have been linked to winter gales. We estimated first-year, second-year and adult survival based on 43 years of ringing and dead recovery data from the Isle of May, Scotland, using recent methods to quantify temporal process variance and identify aspects of winter weather linked to survival. 3. Survival was highly variable for all age groups, and for second-year and adult birds process variance declined strongly when the most extreme year was excluded. Survival in these age groups was low in winters with strong onshore winds and high rainfall. Variation in first-year survival was not related to winter weather, and process variance, although high, was less affected by extreme years. A stochastic population model showed that increasing process variance in survival would lead to reduced population growth rate and increasing probability of extinction. 4. As in other cormorants, shag plumage is only partially waterproof, presumably an adaptation to highly efficient underwater foraging. We speculate that this adaptation may make individuals vulnerable to rough winter weather, leading to boom-and-bust dynamics, where rapid population growth under favourable conditions allows recovery from periodic large-scale weather-related mortality. 5. Given that extreme weather events are predicted to become more frequent, species such as shags that are vulnerable to such events are likely to exhibit stronger reductions in population growth than would be expected from changes in mean climate. Vulnerability to extreme events thus needs to be accounted for when predicting the ecological impacts of climate change.     

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.     

OPTIMAL REPRODUCTIVE EFFORT IN STOCHASTIC,DENSITY-DEPENDENT ENVIRONMENTS

The amount of effort organisms should put into reproducing at any given time has been a matter of debate for many years. Early models suggested a simple rule of thumb: iteroparity should be favored when juvenile survival is relatively variable and semelparity when adult survival is relatively variable. When more mathematically complex models were developed, these simple conclusions were found to be special cases. Variability can select toward iteroparity or semelparity depending on a number of factors irrespective of relative adult/juvenile survival (e.g, the density-independent models of Orzack and Tuljapurkar). Using new techniques, we estimate the ESS reproductive effort for stage-structured models in density-dependent and stochastic conditions. We find that variability causes significant changes in reproductive effort, these changes are often small (± 10% of determinstic ESS effort, but up to 50% change in some instances), and the amount that effort increases or decreases depends on many factors (e.g., the deterministic population dynamics, the vital rates affected by density, the amount of variation, the correlations between the vital rates, the distribution from which the variation is drawn, and the deterministic ESS effort). In a variable environment, semelparity is the ESS in only 3.5% of cases; iteroparity is the rule.     

Survival-variation within and between functional categories of the African multimammate rat

1. By identifying ecological factors specific to functional categories of individuals, it may be possible to understand the mechanisms underlying life-history evolution and population dynamics. While empirical analyses within the field of population biology have focused on changes in population size, theoretical models assuming differential sensitivities of population growth rate or fitness to demographic parameters have mostly been untested, particularly against data on small mammals.
2. Statistical modelling of capture–mark–recapture data on the multimammate rat ( Mastomys natalensis ) from Tanzania shows that: (i) females survive slightly better than males and subadults survive much better than adults; (ii) temporal variation of survival of all individuals is similarly related to the rainfall of the month; (iii) subadults exhibit a strongly density-dependent low persistence rate in the population immediately after their first capture; (iv) subadults survival in later months is, however, positively related to density; and (v) adult survival shows negative density-dependence.
3. Both density-dependent and density-independent factors simultaneously determine stage-dependent survival variation of the multimammate rat. Whereas environmental factors in this population seem to affect survival rates of all individuals in a similar manner, density-dependent relationships are more complex.
4. The patterns of survival variation in small mammals may be different from those observed in large mammals.
5. Further studies of demography in small mammals should aim at understanding how much of the variability in population growth rate is accounted for by the variability of the demographic rates resulting from limiting (density-independent) and regulating (density-dependent) factors, respectively. This study emphasizes the use of robust and accurate statistical methods as well as stage- or age-structured population modelling.     

Deer herbivory and habitat type influence long-term population dynamics of a rare wetland plant

Pinpointing the factors that alter the population viability of long-lived organisms, such as perennial plants, is especially useful for informing conservation management policies for threatened and endangered species. In this study, I used 4 years of demographic data on rare plant Polemonium vanbruntiae (Eastern Jacob’s ladder, Polemoniaceae) to determine how white-tailed deer herbivory and habitat type (wet meadow and forest seep) affect long-term population viability. I incorporated these factors into matrix population models to estimate the deterministic and stochastic growth rates (λ and λs, respectively), stable stage distribution (SSD), the reproductive value for each stage class, the cumulative probability of extinction, and the elasticity values for all vital rates under each browsing and habitat scenario. Population growth rates of P. vanbruntiae in wet meadow sites are expected to increase at a slightly faster rate than at forest seep sites. Herbivory significantly decreased the predicted population growth rate under stochastic conditions. However, P. vanbruntiae ramets are expected to increase in the future as the population growth rate (λ) > 1 under both “browse” and “no browse” scenarios, but deer herbivory increased the extinction risk to a detectable level. Deer preferentially browsed vegetative and reproductive adult ramets over yearlings and seedlings, and browsing significantly reduced fertility of reproductive ramets and increased the probability of stasis for small and large vegetative ramets. Browsing shifted the elasticity values of vital rates and changed the potential for younger life histories stages, such as seedlings, to change future population growth. Under herbivore pressure, survival and stasis of large vegetative ramets have the largest potential impact on future population growth. This study provides empirical evidence that white-tailed deer are an important ecological factor affecting long-term population dynamics of rare plant populations and offers management suggestions for remaining populations of P. vanbruntiae.     

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.     

Estimating ungulate recruitment and growth rates using age ratios

Trends in population growth can be monitored with data for key vital rates without knowledge of abundance. Although adult female survival has the highest elasticity for ungulate population dynamics, the more variable recruitment rates are commonly monitored to track local variation in growth rates. Specifically, recruitment is often measured using late winter young:adult age ratios, though these age ratios are difficult to reliably interpret given the contribution of multiple vital rates to annual ratios. We show that the supplementation of age ratio data with concurrent radio-telemetry monitoring of adult female survival allows both retrospective estimation of empirical population growth rates and the decomposition of recruitment-specific vital rates. We demonstrate the estimation of recruitment and population growth rates for 1 woodland caribou population using these methods, including elasticity and life-stage simulation analysis of the relative contribution of adult female survival and recruitment rates to variation in population growth. We show, for this woodland caribou population, that adult female survival and recruitment rates were nearly equivalent drivers of population growth. We recommend the concurrent monitoring of adult female survival to reliably interpret age ratios when managing caribou and other ungulates. © 2011 The Wildlife Society.     

Impact of Spatial and Temporal Variation in Calf Survival on the Growth of Elk Populations

Abstract: The realized impact of a vital rate on population growth (λ) is determined by both the relative influence of the vital rate on λ (elasticity) and its magnitude of variability. We estimated mean survival and reproductive rates in elk (Cervus elaphus) and spatial and temporal variation in these rates from 37 sources located primarily across the Rocky Mountain region and northwestern United States. We removed sampling variance from estimates of process variance both within and across vital-rate data sets using the variance discounting method developed by White (2000). Deterministic elasticities calculated from a population matrix model parameterized with these mean vital rates ranked adult female survival (e_Scow = 0.869) much higher than calf survival (e_Scalf = 0.131). However, process variance in calf survival was >11 times greater than process variance in female survival across data sets and 10 times greater on average within studies. We conducted Life-Stage Simulation Analysis to incorporate both vital-rate elasticity patterns and empirical estimates of variability to identify those vital rates most influential in elk population dynamics. The overwhelming magnitude of variation in calf survival explained 75% of the variation in the population growth rates generated from 1,000 matrix replicates, compared to just 16% of the variation in λ explained by variation in female survival. Variation in calf survival greatly impacts elk population growth and calls into question the utility of classical elasticity analysis alone for guiding elk management. These results also suggest that the majority of interannual variability that wildlife managers document in late-winter and spring elk surveys is attributable to variation in calf survival over the previous year and less influenced by variation in the harvest of females during the preceding autumn. To meet elk population size objectives, managers should consider the inherent variation in calf survival, and its apparent sensitivity to management, in addition to female harvest.     

Stochastic density effects on adult fish survival and implications for population fluctuations

The degree to which population fluctuations arise from variable adult survival relative to variable recruitment has been debated widely for marine organisms. Disentangling these effects remains challenging because data generally are not sufficient to evaluate if and how adult survival rates are regulated by stochasticity and/or population density. Using unique time series for a largely unexploited reef fish, we found both population density and stochastic food supply impacted adult survival. The estimated effect of variable survival on adult abundance (both mean and variability) rivalled that of variable recruitment. Moreover, we show density‐dependent adult survival can dampen impacts of stochastic recruitment. Thus, food variability may alter population fluctuations by simultaneously regulating recruitment and compensatory adult survival. These results provide an additional mechanism for why intensified density‐independent mortality (via harvest or other means) amplifies population fluctuations and emphasises need for research evaluating the causes and consequences of variability in adult survival.