Stochastic life history modeling for managing regional-scale freshwater fisheries: an experimental study of brook trout

Environmental heterogeneity can combine with evolutionary responses to create very dynamic and often locally independent populations across a landscape. Such complexity creates difficulties for managers trying to conserve populations across large areas. This study develops, applies, and tests the use of stochastic life history modeling and Monte Carlo simulation to assess management scenarios related to the realities of regional fisheries management and conservation. We apply this approach to the management of recreational brook trout (Salvelinus fontinalis) fishing; an activity that can severely impact species balance, abundance, and the size structure of fish communities. Specifically, the model incorporates population-specific life-history information (e.g., growth rate, reproductive effort, and survival) to allow forecasts of the impact of various management strategies and/or changes to environmental conditions on a population's ecological characteristics (e.g., size structure, abundance, and probability of persistence). Sampling was carried out in 16 water bodies spread across four sites in Atlantic Canada. Each water body was sampled in 2005 and reassessed in 2008. This sampling had two primary objectives: (1) define a significant proportion of life-history variation of brook trout in Atlantic Canada, and (2) to test the precision and accuracy of model predictions of population responses to experimental exploitation and management changes. The model successfully predicted population responses to changes in adult survival in 12 of 13 populations having sufficient data for validation testing, while also proving to be a useful tool when engaging stakeholders regarding management options and their associated risk. We suggest that such models are cost-effective and have great potential for informing proactive management of jurisdictions with numerous and diverse populations.

Population viability of Pediocactus bradyi (Cactaceae) in a changing climate

PREMISE OF THE STUDY

A key question concerns the vulnerability of desert species adapted to harsh, variable climates to future climate change. Evaluating this requires coupling long-term demographic models with information on past and projected future climates. We investigated climatic drivers of population growth using a 22-yr demographic model for Pediocactus bradyi, an endangered cactus in northern Arizona.

METHODS

We used a matrix model to calculate stochastic population growth rates (λs) and the relative influences of life-cycle transitions on population growth. Regression models linked population growth with climatic variability, while stochastic simulations were used to (1) understand how predicted increases in drought frequency and extreme precipitation would affect λs, and (2) quantify variability in λs based on temporal replication of data.

KEY RESULTS

Overall λs was below unity (0.961). Population growth was equally influenced by fecundity and survival and significantly correlated with increased annual precipitation and higher winter temperatures. Stochastic simulations increasing the probability of drought and extreme precipitation reduced λs, but less than simulations increasing the probability of drought alone. Simulations varying the temporal replication of data suggested 14 yr were required for accurate λs estimates.

CONCLUSIONS

Pediocactus bradyi may be vulnerable to increases in the frequency and intensity of extreme climatic events, particularly drought. Biotic interactions resulting in low survival during drought years outweighed increased seedling establishment following heavy precipitation. Climatic extremes beyond historical ranges of variability may threaten rare desert species with low population growth rates and therefore high susceptibility to stochastic events.

Ability of matrix models to explain the past and predict the future of plant populations

Uncertainty associated with ecological forecasts has long been recognized, but forecast accuracy is rarely quantified. We evaluated how well data on 82 populations of 20 species of plants spanning 3 continents explained and predicted plant population dynamics. We parameterized stage-based matrix models with demographic data from individually marked plants and determined how well these models forecast population sizes observed at least 5 years into the future. Simple demographic models forecasted population dynamics poorly; only 40% of observed population sizes fell within our forecasts' 95% confidence limits. However, these models explained population dynamics during the years in which data were collected; observed changes in population size during the data-collection period were strongly positively correlated with population growth rate. Thus, these models are at least a sound way to quantify population status. Poor forecasts were not associated with the number of individual plants or years of data. We tested whether vital rates were density dependent and found both positive and negative density dependence. However, density dependence was not associated with forecast error. Forecast error was significantly associated with environmental differences between the data collection and forecast periods. To forecast population fates, more detailed models, such as those that project how environments are likely to change and how these changes will affect population dynamics, may be needed. Such detailed models are not always feasible. Thus, it may be wiser to make risk-averse decisions than to expect precise forecasts from models.

Beyond R0: demographic models for variability of lifetime reproductive output

The net reproductive rate R0 measures the expected lifetime reproductive output of an individual, and plays an important role in demography, ecology, evolution, and epidemiology. Well-established methods exist to calculate it from age- or stage-classified demographic data. As an expectation, R0 provides no information on variability; empirical measurements of lifetime reproduction universally show high levels of variability, and often positive skewness among individuals. This is often interpreted as evidence of heterogeneity, and thus of an opportunity for natural selection. However, variability provides evidence of heterogeneity only if it exceeds the level of variability to be expected in a cohort of identical individuals all experiencing the same vital rates. Such comparisons require a way to calculate the statistics of lifetime reproduction from demographic data. Here, a new approach is presented, using the theory of Markov chains with rewards, obtaining all the moments of the distribution of lifetime reproduction. The approach applies to age- or stage-classified models, to constant, periodic, or stochastic environments, and to any kind of reproductive schedule. As examples, I analyze data from six empirical studies, of a variety of animal and plant taxa (nematodes, polychaetes, humans, and several species of perennial plants).

Finding confidence limits on population growth rates: bootstrap and analytic methods

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model's predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.

Asymptotic distribution of density-dependent stage-grouped population dynamics models

Matrix models are widely used in biology to predict the temporal evolution of stage-structured populations. One issue related to matrix models that is often disregarded is the sampling variability. As the sample used to estimate the vital rates of the models are of finite size, a sampling error is attached to parameter estimation, which has in turn repercussions on all the predictions of the model. In this study, we address the question of building confidence bounds around the predictions of matrix models due to sampling variability. We focus on a density-dependent Usher model, the maximum likelihood estimator of parameters, and the predicted stationary stage vector. The asymptotic distribution of the stationary stage vector is specified, assuming that the parameters of the model remain in a set of the parameter space where the model admits one unique equilibrium point. Tests for density-dependence are also incidentally provided. The model is applied to a tropical rain forest in French Guiana.

Life span correlates with population dynamics in perennial herbaceous plants

Survival and fecundity are basic components of demography and therefore have a strong influence on population dynamics. These two key parameters and their relationship are crucial to understand the evolution of life histories. It remains, however, to be empirically established how life span, fecundity, and population dynamics are linked in different organism groups. We conducted a comparative study based on demographic data sets of 55 populations of 23 perennial herbs for which structured demographic models and among-year natural variation in demographic attributes were available. Life span (from 4 to 128 yr old), estimated by using an algorithm, was inversely correlated with the deviance of the population growth rate from equilibrium as well as with among-year population fluctuations. Temporal variability was greater for short-lived species than for the long-lived ones because fecundity was more variable than survival and relatively more important for population dynamics for the short-lived species. The relationship between life span and population stability suggests that selection for longevity may have played an important role in the life history evolution of plants because of its ability to buffer temporal fluctuations in population size.

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.

Environmental stochasticity: empirical estimates of prairie vole survival with implications for demographic models

A rich theory exists regarding the potential impact of correlations among vital rates on population projections derived from demographic models. However, relatively little is known about the magnitude of correlations among vital rates in natural populations, particularly in mammals. We used 30 years of mark–recapture data from a population of prairie voles ( Microtus ochrogaster (Wagner, 1842)) to test for differences in survival among mass classes and sexes, in correlations among vital rates, in correlations between vital rates and environmental factors, and in autocorrelation in vital rates. Estimated monthly survival rates did not differ significantly among mass classes and there were no significant cross-correlations among mass classes. Survival of large prairie voles increased in mild winters (i.e., warm temperatures and low snowfall). Survival rates of medium and large voles were negatively autocorrelated at time lags of 9–12 months, and survivals of large voles were positively autocorrelated for time lags of <3 months. These autocorrelations were not explained by patterns of temperature or precipitation. The observed degree of autocorrelation in vital rates is sufficient to affect projections from demographic models, particularly in short-lived taxa that require seasonal or monthly estimation of vital rates.

Using population viability analysis to predict the effects of climate change on the extinction risk of an endangered limestone endemic shrub, Arizona cliffrose

The threat of global warming to rare species is a growing concern, yet few studies have predicted its effects on rare populations. Using demographic data gathered in both drought and nondrought years between 1996-2003 in central Arizona upper Sonoran Desert, we modeled population viability for the federally endangered Purshia subintegra (Kearney) Henrickson (Arizona cliffrose). We used deterministic matrix projection models and stochastic models simulating weather conditions during our study, given historical weather variation and under scenarios of increased aridity. Our models suggest that the P. subintegra population in Verde Valley is slowly declining and will be at greater risk of extinction with increased aridity. Across patches at a fine spatial scale, demographic performance was associated with environmental factors. Moist sites (patches with the highest soil moisture, lowest sand content, and most northern aspects) had the highest densities, highest seedling recruitment, and highest risk of extinction over the shortest time span. Extinction risk in moist sites was exacerbated by higher variance in recruitment and mortality. Dry sites had higher cumulative adult survival and lower extinction risk but negative growth rates. Steps necessary for the conservation of the species include introductions at more northern latitudes and in situ manipulations to enhance seedling recruitment and plant survival. We demonstrate that fine spatial-scale modeling is necessary to predict where patches with highest extinction risk or potential refugia for rare species may occur Because current climate projections for the 21st century imply range shifts at rates of 300 to 500 km/century, which are beyond even exceptional examples of shifts in the fossil record of 100-150 km, it is likely that preservation of many rare species will require human intervention and a long-term commitment. Global warming conditions are likely to reduce the carrying capacity of many rare species' habitats.

Forecasting extinction risk with nonstationary matrix models

Matrix population growth models are standard tools for forecasting population change and for managing rare species, but they are less useful for predicting extinction risk in the face of changing environmental conditions. Deterministic models provide point estimates of lambda, the finite rate of increase, as well as measures of matrix sensitivity and elasticity. Stationary matrix models can be used to estimate extinction risk in a variable environment, but they assume that the matrix elements are randomly sampled from a stationary (i.e., non-changing) distribution. Here we outline a method for using nonstationary matrix models to construct realistic forecasts of population fluctuation in changing environments. Our method requires three pieces of data: (1) field estimates of transition matrix elements, (2) experimental data on the demographic responses of populations to altered environmental conditions, and (3) forecasting data on environmental drivers. These three pieces of data are combined to generate a series of sequential transition matrices that emulate a pattern of long-term change in environmental drivers. Realistic estimates of population persistence and extinction risk can be derived from stochastic permutations of such a model. We illustrate the steps of this analysis with data from two populations of Sarracenia purpurea growing in northern New England. Sarracenia purpurea is a perennial carnivorous plant that is potentially at risk of local extinction because of increased nitrogen deposition. Long-term monitoring records or models of environmental change can be used to generate time series of driver variables under different scenarios of changing environments. Both manipulative and natural experiments can be used to construct a linking function that describes how matrix parameters change as a function of the environmental driver. This synthetic modeling approach provides quantitative estimates of extinction probability that have an explicit mechanistic basis.