Reliability of Conservation Actions Based on Elasticity Analysis of Matrix Models

Abstract: Matrix population models have entered the mainstream of conservation biology, with analysis of proportional sensitivities (elasticity analysis) of demographic rates becoming important components of conservation decision making. We identify areas where management applications using elasticity analysis potentially conflict with the mathematical basis of the technique, and we use a hypothetical example and three real data sets (Prairie Chicken [   Tympanuchus cupido ], desert tortoise [ Gopherus agassizii ], and killer whale [ Orcinus orca ]) to evaluate the extent to which conservation recommendations based on elasticities might be misleading. First, changes in one demographic rate can change the qualitative ranking of the elasticity values calculated from a population matrix, a result that dampens enthusiasm for ranking conservation actions based solely on which rates have the highest elasticity values. Second, although elasticities often provide accurate predictions of future changes in population growth rate under management perturbations that are large or that affect more than one rate concurrently, concordance frequently fails when different rates vary by different amounts. In particular, when vital rates change to their high or low values observed in nature, predictions of future growth rate based on elasticities of a mean matrix can be misleading, even predicting population increase when the population growth rate actually declines following a perturbation. Elasticity measures will continue to be useful tools for applied ecologists, but they should be interpreted with considerable care. We suggest that studies using analytical elasticity analysis explicitly consider the range of variation possible for different rates and that simulation methods are a useful tool to this end.     

Importance of individual and environmental variation for invasive species spread: a spatial integral projection model

Plant survival, growth, and flowering are size dependent in many plant populations but also vary among individuals of the same size. This individual variation, along with variation in dispersal caused by differences in, e.g., seed release height, seed characteristics, and wind speed, is a key determinant of the spread rate of species through homogeneous landscapes. Here we develop spatial integral projection models (SIPMs) that include both demography and dispersal with continuous state variables. The advantage of this novel approach over discrete-stage spread models is that the effect of variation in plant size and size-dependent vital rates can be studied at much higher resolution. Comparing Neubert-Caswell matrix models to SIPMs allowed us to assess the importance of including individual variation in the models. As a test case we parameterized a SIPM with previously published data on the invasive monocarpic thistle Carduus nutans in New Zealand. Spread rate (c*) estimates were 34% lower than for standard spatial matrix models and stabilized with as few as seven evenly distributed size classes. The SIPM allowed us to calculate spread rate elasticities over the range of plant sizes, showing the size range of seedlings that contributed most to c* through their survival, growth and reproduction. The annual transitions of these seedlings were also the most important ones for local population growth (lambda). However, seedlings that reproduced within a year contributed relatively more to c* than to lambda. In contrast, plants that grow over several years to reach a large size and produce many more seeds, contributed relatively more to lambda than to c*. We show that matrix models pick up some of these details, while other details disappear within wide size classes. Our results show that SIPMs integrate various sources of variation much better than discrete-stage matrix models. Simpler, heuristic models, however, remain very valuable in studies where the main goal is to investigate the general impact of a life history stage on population dynamics. We conclude with a discussion of future extensions of SIPMs, including incorporation of continuous time and environmental drivers.     

Effect of stage-specific vital rates on population growth rates and effective population sizes in an endangered iteroparous plant

Effective population size (N(e)) determines the strength of genetic drift and can influence the level of genetic diversity a population can maintain. Assessing how changes in demographic rates associated with environmental variables and management actions affect N(e) thus can be crucial to the conservation of endangered species. Calculation of N(e) through demographic models makes it possible to use elasticity analyses to study this issue. The elasticity of N(e) to a given vital rate is the proportional change in N(e) associated with a proportional increase in that vital rate. In addition, demographic models can be used to study N(e) and population growth rate (λ) simultaneously. Simultaneous examination is important because some vital rates differ diametrically in their associations with λ and N(e). For example, in some cases increasing these vital rates increases λ and decreases N(e). We used elasticity analysis to study the effect of stage-specific survival and flowering rates on N(e), annual effective population size (N(a)), and λ in seven populations of the endangered plant Austrian dragonhead (Dracocephalum austriacum). In populations with λ ≥ 1, the elasticities of N(e) and N(a) were similar to those of λ. Survival rates of adults were associated with greater elasticities than survival rates of juveniles, flowering rates, or fecundity. In populations with λ < 1, N(e) and N(a) exhibited greater elasticities to juvenile than to adult vital rates. These patterns are similar to those observed in other species with similar life histories. We did not observe contrasting effects of any vital rate on λ and N(e); thus, management actions that increase the λ of populations of Austrian dragonhead will not increase genetic drift. Our results show that elasticity analyses of N(e) and N(a) can complement elasticity analysis of λ. Moreover, such analyses do not require more data than standard matrix models of population dynamics.     

Conserving Slow-Growing, Long-Lived Tree Species: Input from the Demography of a Rare Understory Conifer, Taxus floridana

Abstract:  Although land preservation and promotion of successful regeneration are important conservation actions, their ability to increase population growth rates of slow-growing, long-lived trees is limited. We investigated the demography of Taxus floridana Nutt., a rare understory conifer, in three populations in different ravine forests spanning its entire geographic range along the Apalachicola River Bluffs in northern Florida (U.S.A.). We examined spatial and temporal patterns in demographic parameters and projected population growth rates by using four years of data on the recruitment and survival of seedlings and established stems, and on diameter growth from cross-sections of dead stems. All populations experienced a roughly 10-fold increase in seedling recruitment in 1996 compared with other years. The fates of seedlings and stems between 8 and 16 mm differed among populations. The fates of stems in two other size classes (the 2- to 4-mm class and the 4- to 8-mm class) differed among both populations and years. Individual stems in all populations exhibited similarly slow growth rates. Stochastic matrix models projected declines in all populations. Stochastic matrix analysis revealed the high elasticity of a measure of stochastic population growth rate to perturbations in the stasis of large reproductive stems for all populations. Additional analyses also indicated that occasional episodes of high recruitment do not greatly affect population growth rates. Conservation efforts directed at long-lived, slow-growing rare plants like Taxus floridana should both protect established reproductive individuals and further enhance survival of individuals in other life-history stages, such as juveniles, that often do not appear to contribute greatly to population growth rates.     

Importance of Accounting for Detection Heterogeneity When Estimating Abundance: the Case of French Wolves

Abstract: Assessing conservation strategies requires reliable estimates of abundance. Because detecting all individuals is most often impossible in free‐ranging populations, estimation procedures have to account for a <1 detection probability. Capture–recapture methods allow biologists to cope with this issue of detectability. Nevertheless, capture–recapture models for open populations are built on the assumption that all individuals share the same detection probability, although detection heterogeneity among individuals has led to underestimating abundance of closed populations. We developed multievent capture–recapture models for an open population and proposed an associated estimator of population size that both account for individual detection heterogeneity (IDH). We considered a two‐class mixture model with weakly and highly detectable individuals to account for IDH. In a noninvasive capture–recapture study of wolves we based on genotypes identified in feces and hairs, we found a large underestimation of population size (27% on average) occurred when IDH was ignored.     

Life history and viability of a long-lived marine invertebrate: the octocoral Paramuricea clavata

The red gorgonian Paramuricea clavata is a long-lived, slow-growing sessile invertebrate of ecological and conservation importance in the northwestern Mediterranean Sea. We develop a series of size-based matrix models for two Paramuricea clavata populations. These models were used to estimate basic life history traits for this species and to evaluate the viability of the red gorgonian populations we studied. As for many other slow-growing species, sensitivity and elasticity analysis demonstrate that gorgonian population growth is far more sensitive to changes in survival rates than to growth, shrinkage, or reproductive rates. The slow growth and low mortality of red gorgonians results in low damping ratios, indicating slow convergence to stable size structures (at least 50 years). The stable distributions predicted by the model did not differ from the observed ones. However, our simulations point out the fragility of this species, showing both populations in decline and high risk of extinction over moderate time horizons. These declines appear to be related to a recent increase in anthropogenic disturbances. Relative to their life span, the values of recruitment elasticity for Paramuricea clavata are lower than those reported for other marine organisms but are similar to those reported for some long-lived plants. These values and the delayed age of sexual maturity, in combination with the longevity of the species, show a clear fecundity/mortality trade-off. Full demographic studies of sessile marine species are quite scarce but can provide insight into population dynamics and life history patterns for these difficult and under-studied species. While our work shows clear results for the red gorgonian, the variability in some of our estimates suggest that future work should include data collection over longer temporal and spatial scales to better understand the long-term effects of natural and anthropogenic disturbances on red gorgonian populations.     

Environment-specific elasticity and sensitivity analysis of the stochastic growth rate

For matrix population models, analyses of how sensitive the population growth rate is to changes in vital rates (i.e. perturbations) are important for studies of life history evolution as well as for management and conservation of threatened species. There are two types of sensitivity analyses corresponding to absolute (sensitivity) or relative (elasticity) changes in the vital rates and both types can be applied to both deterministic and stochastic matrix population models. To date, most empirical studies of elasticity and sensitivity of the stochastic growth rate have examined the response to perturbations in the vital rates in a complete set of possible environments. However, it is often of interest to examine the response to perturbations occurring in only a subset of the possible environments. This has been done for periodic time-varying models elsewhere, but here we describe a recently published method for calculating the environment-specific sensitivity and elasticity of the stochastic growth rate and apply this method to data. These environment-specific perturbation analyses provide a logical way of dividing the sensitivity and elasticity among the environments. They give important insight into the selection regime in different environments and can provide valuable information for making management decisions and management evaluations in stochastic environments.     

Invasion dynamics of the varnish clam (Nuttallia obscurata): a matrix demographic modeling approach

Marine invaders have become a significant threat to native biodiversity and ecosystem function. In this study, the invasion of the varnish clam (Nuttallia obscurata) in British Columbia, Canada, is investigated using a matrix modeling approach to identify the life history characteristics most crucial for population growth and to investigate population differences. Mark-recapture analyses and field collections from 2003 to 2004 were used to determine individual growth, survival rates, and fecundity for two sites. A multi-state matrix model was used to determine population growth rates and to conduct sensitivity and elasticity analyses. A life table response experiment was also used to determine what life history stage contributed most to observed differences in population growth rates. Population survey data were used in conjunction with the matrix model to determine plausible recruitment levels and to investigate recruitment scenarios. Both populations are currently declining but are likely sustainable because of the pulsed nature of large recruitment events. Survival of larger clams (>40 mm) is the most important for population growth based on elasticity and sensitivity analyses. Adult survival also had the largest influence on observed differences between site-specific population growth rates. The two populations studied differed in recruitment dynamics; one experiencing annual recruitment with higher post-settlement mortality and the other, episodic recruitment and lower post-settlement mortality. The most influential factor for the successful invasion of the varnish clam appears to be survival of the larger size classes. Therefore, any process that decreases adult survival (e.g., predation, commercial harvest) will have the greatest impact on population growth.     

Indirect Benefits of Marine Protected Areas for Juvenile Abalone

Abstract: Marine protected areas ( MPAs) designed to provide harvest refugia for red sea urchins (  Strongylocentrotus franciscanus ) offer a unique opportunity to study the indirect effects of urchin fishing on subtidal communities. Sea urchins may provide important cryptic microhabitat for juvenile abalone sheltering beneath urchin spines in shallow habitats worldwide. We investigated the abundance of juvenile (3–90 mm) red abalone, (    Haliotis rufescens ) and the rare flat ( <90 mm) abalone (   H. walallensis ) on protected and fished rocky reefs in California. Abalone abundance surveys were conducted inside 24 × 30 m plots on three protected reefs with red sea urchins present and three fished reefs where red sea urchins were removed by commercial or experimental fishing. Significantly more juvenile abalone were found in 1996 and 1997 on protected reefs with urchins present than on fished reefs ( χ   ² = 188, df = 1, p < 0.001 ). Juvenile red abalone abundance was not correlated with local adult red abalone abundance or habitat rugosity. One-third of the juveniles inside the MPAs were found under the urchins' spine canopy, as were a suite of unfished marine organisms. In the laboratory, juvenile abalone survived better (  χ   ² = 7.31, df = 1, p < 0.01) in crab predation experiments in which red sea urchins were available as shelter. Fishing red urchins reduced structural complexity, potentially decreasing microhabitat available for juvenile abalone. This example demonstrates how MPAs designed for one fished species may help other species, illustrating their usefulness for ecosystem-based fishery management and marine conservation.     

Estimates of sex- and age-class-specific survival probabilities for a southern Great Barrier Reef green sea turtle population

Sex- and age-class-specific survival probabilities of a southern Great Barrier Reef green sea turtle population were estimated using a capture–mark–recapture (CMR) study and a Cormack–Jolly–Seber (CJS) modelling approach. The CMR history profiles for 954 individual turtles tagged over a 9-year period (1984–1992) were classified into three age classes (adult, subadult, juvenile) based on somatic growth and reproductive traits. Reduced-parameter CJS models, accounting for constant survival and time-specific recapture, fitted best for all age classes. There were no significant sex-specific differences in either survival or recapture probabilities for any age class. Mean annual adult survival was estimated at 0.9482 (95% CI: 0.92–0.98) and was significantly higher than survival for either subadults or juveniles. Mean annual subadult survival was 0.8474 (95% CI: 0.79–0.91), which was not significantly different from mean annual juvenile survival estimated at 0.8804 (95% CI: 0.84–0.93). The time-specific adult recapture probabilities were a function of sampling effort but this was not the case for either juveniles or subadults. The sampling effort effect was accounted for explicitly in the estimation of adult survival and recapture probabilities. These are the first comprehensive sex- and age-class-specific survival and recapture probability estimates for a green sea turtle population derived from a long-term CMR program.Communicated by M.S. Johnson, Crawley     

Optimization of capture–recapture monitoring of elusive species illustrated with a threatened grasshopper

Information on population sizes and trends of threatened species is essential for their conservation, but obtaining reliable estimates can be challenging. We devised a method to improve the precision of estimates of population size obtained from capture–recapture studies for species with low capture and recapture probabilities and short seasonal activity, illustrated with population data of an elusive grasshopper (Prionotropis rhodanica). We used data from 5 capture–recapture studies to identify methodological and environmental factors affecting capture and recapture probabilities and estimates of population size. In a simulation, we used the population size and capture and recapture probability estimates obtained from the field studies to identify the minimum number of sampling occasions needed to obtain unbiased and robust estimates of population size. Based on these results we optimized the capture–recapture design, implemented it in 2 additional studies, and compared their precision with those of the nonoptimized studies. Additionally, we simulated scenarios based on thresholds of population size in criteria C and D of the International Union for Conservation of Nature (IUCN) Red List to investigate whether estimates of population size for elusive species can reliably inform red-list assessments. Identifying parameters that affect capture and recapture probabilities (for the grasshopper time since emergence of first adults) and optimizing field protocols based on this information reduced study effort (−6% to −27% sampling occasions) and provided more precise estimates of population size (reduced coefficient of variation) compared with nonoptimized studies. Estimates of population size from the scenarios based on the IUCN thresholds were mostly unbiased and robust (only the combination of very small populations and little study effort produced unreliable estimates), suggesting capture–recapture can be considered reliable for informing red-list assessments. Although capture–recapture remains difficult and costly for elusive species, our optimization procedure can help determine efficient protocols to increase data quality and minimize monitoring effort.     

Linking irreplaceable landforms in a self‐organizing landscape to sensitivity of population vital rates for an ecological specialist

Irreplaceable, self‐organizing landforms and the endemic and ecologically specialized biodiversity they support are threatened globally by anthropogenic disturbances. Although the outcome of disrupting landforms is somewhat understood, little information exists that documents population consequences of landform disturbance on endemic biodiversity. Conservation strategies for species dependent upon landforms have been difficult to devise because they require understanding complex feedbacks that create and maintain landforms and the consequences of landform configuration on demography of species. We characterized and quantified links between landform configuration and demography of an ecological specialist, the dunes sagebrush lizard (Sceloporus arenicolus), which occurs only in blowouts (i.e., wind‐blown sandy depressions) of Shinnery oak (Quercus havardii) sand‐dune landforms. We used matrix models to estimate vital rates from a multisite mark‐recapture study of 6 populations occupying landforms with different spatial configurations. Sensitivity and elasticity analyses demonstrated demographic rates among populations varied in sensitivity to different landform configurations. Specifically, significant relationships between blowout shape complexity and vital rate elasticities suggested direct links between S. arenicolus demography and amount of edge in Shinnery oak sand‐dune landforms. These landforms are irreplaceable, based on permanent transition of disturbed areas to alternative grassland ecosystem states. Additionally, complex feedbacks between wind, sand, and Shinnery oak maintain this landform, indicating restoration through land management practices is unlikely. Our findings that S. arenicolus population dynamics depended on landform configuration suggest that failure to consider processes of landform organization and their effects on species’ population dynamics may lead to incorrect inferences about threats to endemic species and ineffective habitat management for threatened or endangered species. As such, successful conservation of these systems and the biodiversity they support must be informed by research linking process‐oriented studies of self‐organized landforms with studies of movement, behavior, and demography of species that dwell in them.     

Growth of the scyllarid lobsters Ibacus peronii and I. chacei

The growth rates of the morphologically similar scyllarid lobsters Ibacus peronii (Leach, 1815) and I. chacei (Brown and Holthuis, 1998) are described using data from a tag/recapture study and from tagged lobsters kept in captivity. Within particular size classes, we found no differences in moult increments between male and female I. peronii nor between male and female I. chacei. Small individuals of both species always had larger moult increments than larger individuals. For I. peronii, females moulted more frequently than males, and smaller size classes moulted more frequently than larger size classes. Female I. peronii therefore grew more quickly than males and reached their estimated size at sexual maturity (51 mm carapace length) after ∼2 yr. Moulting of I. peronii was seasonal, with most lobsters (96.3%) moulting between October and January. We found no differences in growth rates of I. peronii at two locations along the east coast of Australia: Coffs Harbour in New South Wales (30°18′S; 153°08′E), and Lakes Entrance in Victoria (37°53′S; 148°00′E). For I. chacei, we found no differences in the frequency of moulting between males and females and, because we also found no differences in the moult increments between males and females, the growth rates of both sexes were the same. Received: 14 August 1999 / Accepted: 20 January 2000     

Demography of central and marginal populations of monkeyflowers (Mimulus cardinalis and M. lewisii)

Every species occupies a limited geographic area, but how spatiotemporal environmental variation affects individual and population fitness to create range limits is not well understood. Because range boundaries arise where, on average, populations are more likely to go extinct than to persist, range limits are an inherently population-level problem for which a demographic framework is useful. In this study, I compare demographic parameters and population dynamics between central and marginal populations of monkeyflowers, Mimulus cardinalis and M. lewisii, along an elevation gradient spanning both species' ranges. Central and marginal populations of both species differed in survival and fecundity. For M. lewisii, these components of fitness were higher in central than in marginal populations, but for M. cardinalis the converse was true. To assess spatiotemporal variation in population dynamics, I used transition matrix models to estimate asymptotic population growth rates (lambda) and found that population growth rates of M. lewisii were highest at the range center and reduced at the range margin. Population growth rates of M. cardinalis were highest at the range margin and greatly reduced at the range center. Life table response analysis decomposed spatiotemporal variation in lambda into contributions from each transition between life stages, finding that transitions from large nonreproductive and reproductive plants to the seed class and stasis in the reproductive class made the largest contributions to spatial differences in lambda. These transitions had only low to moderate sensitivities, indicating that differences in projected population growth rates resulted mainly from observed differences in transition matrix parameters and their underlying vital rates.     

Minimum area requirements for an at‐risk butterfly based on movement and demography

Determining the minimum area required to sustain populations has a long history in theoretical and conservation biology. Correlative approaches are often used to estimate minimum area requirements (MARs) based on relationships between area and the population size required for persistence or between species’ traits and distribution patterns across landscapes. Mechanistic approaches to estimating MAR facilitate prediction across space and time but are few. We used a mechanistic MAR model to determine the critical minimum patch size (CMP) for the Baltimore checkerspot butterfly (Euphydryas phaeton), a locally abundant species in decline along its southern range, and sister to several federally listed species. Our CMP is based on principles of diffusion, where individuals in smaller patches encounter edges and leave with higher probability than those in larger patches, potentially before reproducing. We estimated a CMP for the Baltimore checkerspot of 0.7–1.5 ha, in accordance with trait‐based MAR estimates. The diffusion rate on which we based this CMP was broadly similar when estimated at the landscape scale (comparing flight path vs. capture‐mark‐recapture data), and the estimated population growth rate was consistent with observed site trends. Our mechanistic approach to estimating MAR is appropriate for species whose movement follows a correlated random walk and may be useful where landscape‐scale distributions are difficult to assess, but demographic and movement data are obtainable from a single site or the literature. Just as simple estimates of lambda are often used to assess population viability, the principles of diffusion and CMP could provide a starting place for estimating MAR for conservation.     

Measuring terrestrial movement behavior using passive integrated transponder (PIT) tags: effects of tag size on detection, movement, survival, and growth

Movement behaviors have broad ecological and evolutionary implications, affecting individual fitness, metapopulation dynamics, the distribution and abundance of species, as well as gene flow and thus adaptation and speciation. However, movement behaviors such as dispersal, station keeping, and ranging are poorly understood in many taxa due to the incompatibility of traditional tracking methods with long-term observations. This is particularly true for small-bodied life history stages and species. While the introduction of smaller passive integrated transponder (PIT) tags and the development of PIT telemetry have removed some barriers, the trade-offs between different tag sizes are unknown. Through a series of experiments, we tested for effects of PIT tag size on detection, movement, tag retention, growth, and survival of a juvenile amphibian. We found no effect of PIT tag size on initial movement distance, survival, or growth; and all individuals retained their tag for the course of the experiment. Detection and recapture rates, however, were increased with PIT tag size. The orientation of the tag relative to the vertical axis of the antenna also affected the size of the detection field, which was 15.78–43.90 % smaller when the antenna was moved perpendicular rather than parallel to the long axis of the tag. We conclude that PIT telemetry is a suitable technique for marking previously untraceable species or life history stages and may offer insight into the behaviors of these individuals. Investigations using multiple PIT tag sizes should include this in statistical analyses to account for tag size biased detection differences.     

Setting Realistic Recovery Targets for Two Interacting Endangered Species,Sea Otter and Northern Abalone

Failure to account for interactions between endangered species may lead to unexpected population dynamics, inefficient management strategies, waste of scarce resources, and, at worst, increased extinction risk. The importance of species interactions is undisputed, yet recovery targets generally do not account for such interactions. This shortcoming is a consequence of species‐centered legislation, but also of uncertainty surrounding the dynamics of species interactions and the complexity of modeling such interactions. The northern sea otter (Enhydra lutris kenyoni) and one of its preferred prey, northern abalone (Haliotis kamtschatkana), are endangered species for which recovery strategies have been developed without consideration of their strong predator–prey interactions. Using simulation‐based optimization procedures from artificial intelligence, namely reinforcement learning and stochastic dynamic programming, we combined sea otter and northern abalone population models with functional‐response models and examined how different management actions affect population dynamics and the likelihood of achieving recovery targets for each species through time. Recovery targets for these interacting species were difficult to achieve simultaneously in the absence of management. Although sea otters were predicted to recover, achieving abalone recovery targets failed even when threats to abalone such as predation and poaching were reduced. A management strategy entailing a 50% reduction in the poaching of northern abalone was a minimum requirement to reach short‐term recovery goals for northern abalone when sea otters were present. Removing sea otters had a marginally positive effect on the abalone population but only when we assumed a functional response with strong predation pressure. Our optimization method could be applied more generally to any interacting threatened or invasive species for which there are multiple conservation objectives. Definición de Metas de Recuperación Realistas para Dos Especies en Peligro Interactuantes, Enhydra lutris y Haliotis kamtschatkana     

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.     

Nonlinear relationships between vital rates and state variables in demographic models

To accurately estimate population dynamics and viability, structured population models account for among-individual differences in demographic parameters that are related to individual state. In the widely used matrix models, such differences are incorporated in terms of discrete state categories, whereas integral projection models (IPMs) use continuous state variables to avoid artificial classes. In IPMs, and sometimes also in matrix models, parameterization is based on regressions that do not always model nonlinear relationships between demographic parameters and state variables. We stress the importance of testing for nonlinearity and propose using restricted cubic splines in order to allow for a wide variety of relationships in regressions and demographic models. For the plant Borderea pyrenaica, we found that vital rate relationships with size and age were nonlinear and that the parameterization method had large effects on predicted population growth rates, X (linear IPM, 0.95; nonlinear IPMs, 1.00; matrix model, 0.96). Our results suggest that restricted cubic spline models are more reliable than linear or polynomial models. Because even weak nonlinearity in relationships between vital rates and state variables can have large effects on model predictions, we suggest that restricted cubic regression splines should be considered for parameterizing models of population dynamics whenever linearity cannot be assumed.