Integral projection models perform better for small demographic data sets than matrix population models: a case study of two perennial herbs

Applying stochastic and Bayesian integral projection modeling to amphibian population viability analysis

Integral projection models (IPMs) can estimate the population dynamics of species for which both discrete life stages and continuous variables influence demographic rates. Stochastic IPMs for imperiled species, in turn, can facilitate population viability analyses (PVAs) to guide conservation decision-making. Biphasic amphibians are globally distributed, often highly imperiled, and ecologically well suited to the IPM approach. Herein, we present a stochastic size- and stage-structured IPM for a biphasic amphibian, the U.S. federally threatened California tiger salamander (CTS) (Ambystoma californiense). This Bayesian model reveals that CTS population dynamics show greatest elasticity to changes in juvenile and metamorph growth and that populations are likely to experience rapid growth at low density. We integrated this IPM with climatic drivers of CTS demography to develop a PVA and examined CTS extinction risk under the primary threats of habitat loss and climate change. The PVA indicated that long-term viability is possible with surprisingly high (20%-50%) terrestrial mortality but simultaneously identified likely minimum terrestrial buffer requirements of 600-1000 m while accounting for numerous parameter uncertainties through the Bayesian framework. These analyses underscore the value of stochastic and Bayesian IPMs for understanding both climate-dependent taxa and those with cryptic life histories (e.g., biphasic amphibians) in service of ecological discovery and biodiversity conservation. In addition to providing guidance for CTS recovery, the contributed IPM and PVA supply a framework for applying these tools to investigations of ecologically similar species.

Modelling the dynamics of Pinus sylvestris forests after a die-off event under climate change scenarios

In recent decades, die-off events in Pinus sylvestris populations have increased. The causes of these phenomena, which are usually related to local and regional extreme hot droughts, have been extensively investigated from a physiological viewpoint. However, the consequences of die-off process in terms of demography and vegetation dynamics have been less thoroughly addressed. Here, we projected P. sylvestris plot dynamics after a die-off event, under climate change scenarios, considering also their early demographic stages (i.e., seedlings, saplings and ingrowth from the sapling to adult class), to assess the resilience of P. sylvestris populations after such events. We used Integral Projection Models (IPMs) to project future plot structure under current climate, and under RCP4.5 and RCP8.0 climate scenarios, using climatic suitability - extracted from Species Distribution Models - as a covariable in the estimations of vital rates over time. Field data feeding IPMs were obtained from two successive surveys, at the end of the die-off event (2013) and four years later (2017), undertaken on populations situated across the P. sylvestris range of distribution in Catalonia (NE Spain). Plots affected by die-off experienced a loss of large trees, which causes that basal area, tree diameter and tree density will remain lower for decades relative to unaffected plots. After the event, this situation is partially counterbalanced in affected plots by a greater increase in basal area and seedling recruitment into tree stage, thus promoting resilience. However, resilience is delayed under the climate-change scenarios with warmer and drier conditions involving additional physiological stress, due to a reduced abundance of seedlings and a smaller plot basal area. The study shows lagged effect of drought-induced die-off events on forest structure, also revealing stabilizing mechanisms, such as recruitment and tree growth release, which enhance resilience. However, these mechanisms would be jeopardized by oncoming regional warming.

The role of demographic compensation in stabilising marginal tree populations in North America

Demographic compensation-the opposing responses of vital rates along environmental gradients-potentially delays anticipated species' range contraction under climate change, but no consensus exists on its actual contribution. We calculated population growth rate (λ) and demographic compensation across the distributional ranges of 81 North American tree species and examined their responses to simulated warming and tree competition. We found that 43% of species showed stable population size at both northern and southern edges. Demographic compensation was detected in 25 species, yet 15 of them still showed a potential retraction from southern edges, indicating that compensation alone cannot maintain range stability. Simulated climatic warming caused larger decreases in λ for most species and weakened the effectiveness of demographic compensation in stabilising ranges. These findings suggest that climate stress may surpass the limited capacity of demographic compensation and pose a threat to the viability of North American tree populations.

Prescriptions for the Control of a Clonal Invasive Species Using Demographic Models

Until recently, little research has focused on determination of the population dynamics of invasive species and evaluating their genetic variation. Consequently, not much is known of what drives clonal invasive species and their demography. Here, we describe the population dynamics of Kalanchoe delagoensis (Crassulaceae), considered invasive to several countries. We quantified the demography of a population in central Mexico using integral projection models (IPM) in a population that reproduced asexually exclusively through plantlets. The effect of clonal recruitment on population growth rate (λ) was evaluated by changing plantlet survival and simulating management scenarios that used previous data of watering and seven experimental herbicide treatments. The finite rate of population increase indicated that this Kalanchoe delagoensis population is growing (above one) and with water availability, growth rates will only accelerate. The IPM showed that plantlet survival and recruitment were the most critical steps in the cycle for the population, and simulations of different management scenarios showed that reducing plantlet survival significantly decreased λ only in two out of the seven herbicides used.

Integral Projection Models and Sustainable Forest Management of Agave inaequidens in Western Mexico

In México, at least 37 Agave species are extracted from wild populations for producing distilled spirits. This activity involves harvesting mature agaves just before producing their inflorescences, which cancels sexual reproduction of plants used. The increasing demand of agaves spirits in national and international markets is determining a strong pressure on wild populations, most of them lacking adequate management. In addition, the dynamics of agave populations may be affected by natural phenomena like oscillation of rainfall regimes, which affects the recruitment of agave seedlings, or the scarcity of pollinators that may affect seed production and general population dynamics. We studied the demography of wild populations of Agave inaequidens to analyze critical conditions for populations recovery, modelling the effects of rainfall trends on the demographic performance of this species, and exploring response of populations to hypothetical extraction regimes and reforestation efforts. Our study was performed in four well-conserved wild populations in Central Western Mexico, each population was sampled in a plot of about one hectare composed by 10 subplots 50 x 5 m (2500 m²). Populations were monitored yearly between 2011 and 2013, measuring plant size, reproductive individuals, and fecundity. Data were analyzed through integral projection models by using the IPMpack for R, to perform prospective analyses. We in addition constructed stochastic models to explore the possible influence of rainfall variation on species demography, using data for the drier and wetter years of the study period. Population growth varied from λ=1.003 to λ=0.899 among populations and years, and exceptionally λ=0.559 after a fire event. Low rainfall decreases λ values, indicating especial limitations to harvesting agaves during dry years whose frequency most probably will increase. In general, extraction rates from 10% to 30% of mature individuals are viable to maintain λ above 1, and these rates may be higher if new plants are introduced in populations. Depending on levels of extraction, our models suggest that it is necessary to carry out actions of reforestation, and in situ management according to the trends found in each site. This is one indispensable condition to maintain λ close to or greater than 1. Sustainable extraction of wild agaves is possible, some communities are already carrying out a repertoire of goods practices in this direction, but together with ecological criteria and good management techniques, strict regulations and social organization are needed to achieve it.

Microclimate and demography interact to shape stable population dynamics across the range of an alpine plant

Heterogeneous terrain in montane systems results in a decoupling of climatic gradients. Population dynamics across species' ranges in these heterogeneous landscapes are shaped by relationships between demographic rates and these interwoven climate gradients. Linking demography and climate variables across species' ranges refines our understanding of the underlying mechanisms of species' current and future ranges. We explored the importance of multiple microclimatic gradients in shaping individual demographic rates and population growth rates in 16 populations across the elevational distribution of an alpine plant (Ivesia lycopodioides var. scandularis). Using integral projection modeling, we ask how each rate varies across three microclimate gradients: accumulated degree-days, growing-season soil moisture, and days of snow cover. Range-wide variation in demographic rates was best explained by the combined influence of multiple microclimatic variables. Different pairs of demographic rates exhibited both similar and inverse responses to the same microclimatic gradient, and the microclimatic effects often varied with plant size. These responses resulted in range-wide projected population persistence, with no declining populations at either elevational range edge or at the extremes of the microclimate gradients. The complex relationships between topography, microclimate and demography suggest that populations across a species' range may have unique demographic pathways to stable population dynamics.

Landscape heterogeneity is key to forecasting outcomes of plant reintroduction

Conservation and restoration projects often involve starting new populations by introducing individuals into portions of their native or projected range. Such efforts can help meet many related goals, including habitat creation, ecosystem service provisioning, assisted migration, and the reintroduction of imperiled species following local extirpation. The outcomes of reintroduction efforts, however, are highly variable, with results ranging from local extinction to dramatic population growth; reasons for this variation remain unclear. Here, we ask whether population growth following plant reintroductions is governed by variation at two scales: the scale of individual habitat patches to which individuals are reintroduced, and larger among-landscape scales in which similar patches may be situated in landscapes that differ in matrix type, soil conditions, and other factors. Quantifying demographic variation at these two scales will help prioritize locations for introduction and, once introductions take place, forecast population growth. This work took place within a large-scale habitat fragmentation experiment, where individuals of two perennial forb species were reintroduced into eight replicate ~50-ha landscapes, each containing a set of five ~1-ha patches that varied in their degree of isolation (connected by habitat corridors or unconnected) and edge-to-area ratio. Using data on individual growth, survival, reproductive output, and recruitment collected one to two years after reintroduction, we developed models to forecast population growth, then compared forecasts to observed population sizes, three and six years later. Both the type of patch (connected and unconnected) and identity of the landscape to which individuals were reintroduced had effects on forecasted population growth rates, but only variation associated with landscape identity was an accurate predictor of subsequently observed population growth rates. Models that did not include landscape identity had minimal forecasting ability, revealing the key importance of variation at this scale for accurate prediction. Of the five demographic rates used to model population dynamics, seed production was the most important source of forecast error in population growth rates. Our results point to the importance of accounting for landscape-scale variation in demographic models and demonstrate how such models might assist with prioritizing particular landscapes for species reintroduction projects.

Delivering the promises of trait-based approaches to the needs of demographic approaches, and vice versa

Few facets of biology vary more than functional traits and life-history traits. To explore this vast variation, functional ecologists and population ecologists have developed independent approaches that identify the mechanisms behind and consequences of trait variation.Collaborative research between researchers using trait-based and demographic approaches remains scarce. We argue that this is a missed opportunity, as the strengths of both approaches could help boost the research agendas of functional ecology and population ecology.This special feature, which spans three journals of the British Ecological Society due to its interdisciplinary nature, showcases state-of-the-art research applying trait-based and demographic approaches to examine relationships between organismal function, life history strategies and population performance across multiple kingdoms. Examples include the exploration of how functional trait × environment interactions affect vital rates and thus explain population trends and species occurrence; the coordination of seed traits and dispersal ability with the pace of life in plants; the incorporation of functional traits in dynamic energy budget models; or the discovery of linkages between microbial functional traits and the fast-slow continuum.Despite their historical isolation, collaborative work between functional ecologists and population ecologists could unlock novel research pathways. We call for an integrative research agenda to evaluate which and when traits are functional, as well as their ability to describe and predict life history strategies and population dynamics. We highlight promising, complementary research avenues to overcome current limitations. These include a more explicit linkage of selection gradients in the context of functional trait-vital rate relationships, and the implementation of standardised protocols to track changes in traits and vital rates over time at the same location and individuals, thus allowing for the explicit incorporation of trade-offs in analyses of covariation of functional traits and life-history traits.

Effects of harvest on the sustainability and leaf productivity of populations of two palm species in Maya homegardens

Traditional management practices are usually thought to be sustainable. The Maya manage Sabal (Arecaceae) palms in homegardens, using their leaves for thatching. The sustainability of such production systems depends on the long-term persistence of palm populations, whereas resource availability also depends on the number of leaves on individual palms. We examined how leaf harvest affects Sabal yapa and S. mexicana population growth rates (λ) and leaf production, comparing traditional and alternative harvest regimes in terms of sustainability and productivity. Demographic, harvest and leaf production data were recorded for three years in two homegardens. We used general integral projection models linked to leaf-production models to describe population dynamics and productivity. Harvest had no effect on S. yapa’s vital rates or on λ, but it changed the growth rate of individuals of S. mexicana, with a negligible impact on λ. Homegardens affected λ values, reflecting the species’ ecological affinities. S. mexicana, introduced from mesic forests, required watering and shade; therefore, its population declined rapidly in the homegarden that lacked both water and shade. The λ of the xerophilic S. yapa was slightly larger without watering than with watering. Palms usually compensated for leaf extraction, causing the number of leaves harvested per individual to increase with harvest intensity. Nevertheless, traditional management is relatively mild, allowing standing leaves to accumulate but reducing the homegarden’s yield. Apparently, the Maya do not seek to maximize annual production but to ensure the availability of large numbers of leaves in homegardens. These leaves may then be used when the entire roof of a hut needs to be replaced every few years.

Matrix models for size-structured populations: unrealistic fast growth or simply diffusion?

Matrix population models are widely used to study population dynamics but have been criticized because their outputs are sensitive to the dimension of the matrix (or, equivalently, to the class width). This sensitivity is concerning for the population growth rate (λ) because this is an intrinsic characteristic of the population that should not depend on the model specification. It has been suggested that the sensitivity of λ to matrix dimension was linked to the existence of fast pathways (i.e. the fraction of individuals that systematically move up a class), whose proportion increases when class width increases. We showed that for matrix population models with growth transition only from class i to class i + 1, λ was independent of the class width when the mortality and the recruitment rates were constant, irrespective of the growth rate. We also showed that if there were indeed fast pathways, there were also in about the same proportion slow pathways (i.e. the fraction of individuals that systematically remained in the same class), and that they jointly act as a diffusion process (where diffusion here is the movement in size of an individual whose size increments are random according to a normal distribution with mean zero). For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic. Yet, the sensitivity of λ to matrix dimension for a class width in the range 1-10 cm was small, much smaller than the sampling uncertainty on the value of λ. Moreover, λ could either increase or decrease when class width increased depending on the species. Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

Developing, testing and application of rodent population dynamics and capture models based on an adjusted Leslie matrix-based population approach

Small rodents in general and the multimammate rat Apodemus agrarius in particular, damage crops and cause major economic losses in China. Therefore, accurate predictions of the population size of A. agrarius and an efficient control strategy are urgently needed. We developed a population dynamics model by applying a Leslie matrix method, and a capture model based on optimal harvesting theory for A. agrarius. Our models were parametrized using demographic estimates from a capture–mark–recapture (CMR) study conducted on the Qinshui Forest Farm in Northwestern China. The population dynamics model incorporated 12 equally balanced age groups and included immigration and emigration parameters. The model was evaluated by assessing the predictions for four years based on the known starting population in 2004 from the 2004–2007 CMR data. The capture model incorporated two functional age categories (juvenile and adult) and used density-dependent and density-independent factors. The models were used to assess the effect of rodent control measures between 2004 and 2023 on population dynamics and the resulting numbers of rats. Three control measures affecting survival rates were considered. We found that the predicted population dynamics of A. agrarius between 2004 and 2007 compared favorably with the observed population dynamics. The models predicted that the population sizes of A. agrarius in the period between 2004 and 2023 under the control measure applied in August 2004 were very similar to the optimal population sizes, and no significant difference was found between the two population sizes. We recommend using the population dynamics and capture models based on CMR-estimated demographic schedules for rodent, provided these data are available. The models that we have developed have the potential to play an important role in predicting the effects of rodent management and in evaluating different control strategies.

Building integral projection models: a user's guide

In order to understand how changes in individual performance (growth, survival or reproduction) influence population dynamics and evolution, ecologists are increasingly using parameterized mathematical models. For continuously structured populations, where some continuous measure of individual state influences growth, survival or reproduction, integral projection models (IPMs) are commonly used. We provide a detailed description of the steps involved in constructing an IPM, explaining how to: (i) translate your study system into an IPM; (ii) implement your IPM; and (iii) diagnose potential problems with your IPM. We emphasize how the study organism's life cycle, and the timing of censuses, together determine the structure of the IPM kernel and important aspects of the statistical analysis used to parameterize an IPM using data on marked individuals. An IPM based on population studies of Soay sheep is used to illustrate the complete process of constructing, implementing and evaluating an IPM fitted to sample data. We then look at very general approaches to parameterizing an IPM, using a wide range of statistical techniques (e.g. maximum likelihood methods, generalized additive models, nonparametric kernel density estimators). Methods for selecting models for parameterizing IPMs are briefly discussed. We conclude with key recommendations and a brief overview of applications that extend the basic model. The online Supporting Information provides commented R code for all our analyses.

Linking vital rates to invasiveness of a perennial herb

Invaders generally show better individual performance than non-invaders and, therefore, vital rates (survival, growth, fecundity) could potentially be used to predict species invasiveness outside their native range. Comparative studies have usually correlated vital rates with the invasiveness status of species, while few studies have investigated them in relation to population growth rate. Here, I examined the influence of five vital rates (plant establishment, survival, growth, flowering probability, seed production) and their variability (across geographic regions, habitat types, population sizes and population densities) on population growth rate (λ) using data from 37 populations of an invasive, iteroparous herb (Lupinus polyphyllus) in a part of its invaded range in Finland. Variation in vital rates was often related to habitat type and population density. The performance of the populations varied from declining to rapidly increasing independently of habitat type, population size or population density, but differed between regions. The population growth rate increased linearly with plant establishment, and with the survival and growth of vegetative individuals, while the survival of flowering individuals and annual seed production were not related to λ. The vital rates responsible for rapid population growth varied among populations. These findings highlight the importance of both regional and local conditions to plant population dynamics, demonstrating that individual vital rates do not necessarily correlate with λ. Therefore, to understand the role of individual vital rates in a species ability to invade, it is necessary to quantify their effect on population growth rate.

Actual and potential use of population viability analyses in recovery of plant species listed under the US endangered species act

Use of population viability analyses (PVAs) in endangered species recovery planning has been met with both support and criticism. Previous reviews promote use of PVA for setting scientifically based, measurable, and objective recovery criteria and recommend improvements to increase the framework's utility. However, others have questioned the value of PVA models for setting recovery criteria and assert that PVAs are more appropriate for understanding relative trade-offs between alternative management actions. We reviewed 258 final recovery plans for 642 plants listed under the U.S. Endangered Species Act to determine the number of plans that used or recommended PVA in recovery planning. We also reviewed 223 publications that describe plant PVAs to assess how these models were designed and whether those designs reflected previous recommendations for improvement of PVAs. Twenty-four percent of listed species had recovery plans that used or recommended PVA. In publications, the typical model was a matrix population model parameterized with ≤5 years of demographic data that did not consider stochasticity, genetics, density dependence, seed banks, vegetative reproduction, dormancy, threats, or management strategies. Population growth rates for different populations of the same species or for the same population at different points in time were often statistically different or varied by >10%. Therefore, PVAs parameterized with underlying vital rates that vary to this degree may not accurately predict recovery objectives across a species' entire distribution or over longer time scales. We assert that PVA, although an important tool as part of an adaptive-management program, can help to determine quantitative recovery criteria only if more long-term data sets that capture spatiotemporal variability in vital rates become available. Lacking this, there is a strong need for viable and comprehensive methods for determining quantitative, science-based recovery criteria for endangered species with minimal data availability. Uso Actual y Potencial del Análisis de Viabilidad Poblacional para la Recuperación de Especies de Plantas Enlistadas en el Acta de Especies En Peligro de E.U.A.

Getting the biggest birch for the bang: restoring and expanding upland birchwoods in the Scottish Highlands by managing red deer

High deer populations threaten the conservation value of woodlands and grasslands, but predicting the success of deer culling, in terms of allowing vegetation to recover, is difficult. Numerical simulation modeling is one approach to gain insight into the outcomes of management scenarios. We develop a spatially explicit model to predict the responses of Betula spp. to red deer (Cervus elaphus) and land management in the Scottish Highlands. Our model integrates a Bayesian stochastic stage-based matrix model within the framework of a widely used individual-based forest simulation model, using data collected along spatial and temporal gradients in deer browsing. By initializing our model with the historical spatial locations of trees, we find that densities of juvenile trees (<3 m tall) predicted after 9–13 years closely match counts observed in the field. This is among the first tests of the accuracy of a dynamical simulation model for predicting the responses of tree regeneration to herbivores. We then test the relative importance of deer browsing, ground cover vegetation, and seed availability in facilitating landscape-level birch regeneration using simulations in which we varied these three variables. We find that deer primarily control transitions of birch to taller (>3 m) height tiers over 30 years, but regeneration also requires suitable ground cover for seedling establishment. Densities of adult seed sources did not influence regeneration, nor did an active management scenario where we altered the spatial configuration of adults by creating “woodland islets”. Our results show that managers interested in maximizing tree regeneration cannot simply reduce deer densities but must also improve ground cover for seedling establishment, and the model we develop now enables managers to quantify explicitly how much both these factors need to be altered. More broadly, our findings emphasize the need for land managers to consider the impacts of large herbivores rather than their densities.

A demographic approach to study effects of climate change in desert plants

Desert species respond strongly to infrequent, intense pulses of precipitation. Consequently, indigenous flora has developed a rich repertoire of life-history strategies to deal with fluctuations in resource availability. Examinations of how future climate change will affect the biota often forecast negative impacts, but these-usually correlative-approaches overlook precipitation variation because they are based on averages. Here, we provide an overview of how variable precipitation affects perennial and annual desert plants, and then implement an innovative, mechanistic approach to examine the effects of precipitation on populations of two desert plant species. This approach couples robust climatic projections, including variable precipitation, with stochastic, stage-structured models constructed from long-term demographic datasets of the short-lived Cryptantha flava in the Colorado Plateau Desert (USA) and the annual Carrichtera annua in the Negev Desert (Israel). Our results highlight these populations' potential to buffer future stochastic precipitation. Population growth rates in both species increased under future conditions: wetter, longer growing seasons for Cryptantha and drier years for Carrichtera. We determined that such changes are primarily due to survival and size changes for Cryptantha and the role of seed bank for Carrichtera. Our work suggests that desert plants, and thus the resources they provide, might be more resilient to climate change than previously thought.

Avoiding unintentional eviction from integral projection models

Integral projection models (IPMs) are increasingly being applied to study size-structured populations. Here we call attention to a potential problem in their construction that can have important consequences for model results. IPMs are implemented using an approximating matrix and bounded size range. Individuals near the size limits can be unknowingly "evicted" from the model because their predicted future size is outside the range. We provide simple measures for the magnitude of eviction and the sensitivity of the population growth rate (lambda) to eviction, allowing modelers to assess the severity of the problem in their IPM. For IPMs of three plant species, we found that eviction occurred in all cases and caused underestimation of the population growth rate (lambda) relative to eviction-free models; it is likely that other models are similarly affected. Models with frequent eviction should be modified because eviction is only possible when size transitions are badly mis-specified. We offer several solutions to eviction problems, but we emphasize that the modeler must choose the most appropriate solution based on an understanding of why eviction occurs in the first place. We recommend testing IPMs for eviction problems and resolving them, so that population dynamics are modeled more accurately.

Identifying the demographic processes relevant for species conservation in human-impacted areas: does the model matter?

The identification of the demographic processes responsible for the decline in population growth rate (λ) in disturbed areas would allow conservation efforts to be efficiently directed. Integral projection models (IPMs) are used for this purpose, but it is unclear whether the conclusions drawn from their analysis are sensitive to how functional structures (the functions that describe how survival, growth and fecundity vary with individual size) are selected. We constructed 12 IPMs that differed in their functional structure by combining two reproduction models and three functional expressions (generalized linear, cubic and additive models), each with and without simplification. Models were parameterized with data from two populations of two endangered cacti subject to different disturbance intensities. For each model, we identified the demographic processes that most affected λ in the presence of disturbance. Simulations were performed on artificial data and analyzed as above to assess the generality of the results. In both empirical and simulated data, the same processes were identified as making the largest contribution to changes in λ regardless of the functional structure. The major differences in the results were due to misspecification of the fecundity functions, whilst functional expression and model simplification had lesser effects. Therefore, as long as the demographic attributes of the species are well known and incorporated into the model, IPMs will robustly identify the processes that most affect the growth of populations subject to disturbance, making them a reliable tool for developing conservation strategies.

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.