The usefulness of sensitivity analysis for predicting the effects of cat predation on the population dynamics of their avian prey

Sensitivity analyses of population projection matrix (PPM) models are often used to identify life-history perturbations that will most influence a population's future dynamics. Sensitivities are linear extrapolations of the relationship between a population's growth rate and perturbations to its demographic parameters. Their effectiveness depends on the validity of the assumption of linearity. Here we assess whether sensitivity analysis is an appropriate tool to investigate the effect of predation by cats on the population growth rates of their avian prey. We assess whether predation by cats leads to non-linear effects on population growth and compare population growth rates predicted by sensitivity analysis with those predicted by a non-linear simulation. For a two-stage, age-classified House Sparrow Passer domesticus PPM slight non-linearity arose when PPM elements were perturbed, but perturbation to the vital rates underlying the matrix elements had a linear impact on population growth rate. We found a similar effect with a slightly larger three-stage, age-classified PPM for a Winter Wren Troglodytes troglodytes population perturbed by cat predation. For some avian species, predation by cats may cause linear or only slightly nonlinear impacts on population growth rates. For these species, sensitivity analysis appears to be a useful conservation tool. However, further work on multiple perturbations to avian prey species with more complicated life histories and higher-dimension PPM models is required.     

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.     

Elasticity analysis in epidemiology: an application to tick-borne infections

The application of projection matrices in population biology to plant and animal populations has a parallel in infectious disease ecology when next-generation matrices (NGMs) are used to characterize growth in numbers of infected hosts ( R ₀). The NGM is appropriate for multi-host pathogens, where each matrix element represents the number of cases of one type of host arising from a single infected individual of another type. For projection matrices, calculations of the sensitivity and elasticity of the population growth rate to changes in the matrix elements has generated insight into plant and animal populations. These same perturbation analyses can be used for infectious disease systems. To illustrate this in detail we parameterized an NGM for seven tick-borne zoonoses and compared them in terms of the contributions to R ₀ from three different routes of transmission between ticks, and between ticks and vertebrate hosts. The definition of host type may be the species of the host or the route of infection, or, as was the case for the set of tick-borne pathogens, a combination of species and the life stage at infection. This freedom means that there is a broad range of disease systems and questions for which the methodology is appropriate.     

Prediction vs. reality: Can a PVA model predict population persistence 13 years later?

The challenge of conservation biology is to make models that predict population dynamics and have a high probability of accurately tracking population change (increase, decrease, constancy). In this study we tested whether the transition model is accurate enough to predict population persistence and size 13 years down and whether after 13 years populations had achieved a stable stage distribution. We modeled 6 small populations of an epiphytic orchid using a Lefkovitch type analysis to predict population growth pattern based on monthly surveys for approximately 1.5 years. In addition, sensitivity and elasticity analyses were used to identify life stages with high sensitivity or elasticity that have the largest influence on population growth rate. We re-censused the populations 13 years after the first study and compared the structure of the populations to predictions based on the earlier census data. Effective population growth rates were similar to those expected except for one where the population went extinct. The prediction slightly (but not significantly) overestimated the actual population growth rates of some populations. Elasticity analysis revealed that the adult stage is critical in the life cycle. The observed stage distributions of the populations were not stable at the beginning of the survey and neither were they after 13 years. We suggest that this might be caused by external perturbations that result in unequal mortality between life stages and stochastic recruitment events. The ability of the matrices to predict population size approximately eight generations in the future is encouraging and warrants the continued use of these approaches for PVA.     

A framework for studying transient dynamics of population projection matrix models

Empirical models are central to effective conservation and population management, and should be predictive of real-world dynamics. Available modelling methods are diverse, but analysis usually focuses on long-term dynamics that are unable to describe the complicated short-term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density-independent, time-invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case-specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.     

Spatial Variation in Lesser Prairie-Chicken Demography: A Sensitivity Analysis of Population Dynamics and Management Alternatives

ABSTRACT The lesser prairie-chicken (Tympanuchus pallidicinctus) is currently considered a candidate for protection under the Endangered Species Act. To identify potential limiting factors for lesser prairie-chicken populations, we developed an age-based matrix model of lesser prairie-chicken population dynamics to compare the relative importance of components of reproduction and survival, and determine if various management alternatives stabilize or increase rates of population change. We based our analyses on an intensive 6-year population study from which demographic rates were estimated for each age class in Kansas. We used deterministic models and elasticity values to identify parameters predicted to have the greatest effect on the rate of population change (λ) at 2 study sites. Last, we used life-stage simulation analysis to simulate various management alternatives. Lambda was <1 for both populations (site 1: λ = 0.54, site 2: λ = 0.74). However, we found differences in sensitivity to nest success and chick survival between populations. The results of the simulated management scenarios complemented the lower-level elasticity analysis and indicated the relative importance of female survival during the breeding season compared with winter. If management practices are only capable of targeting a single demographic rate, changes to either nest success or chick survival had the greatest impact on λ at site 1 and 2, respectively. Management that simultaneously manipulated both nest success and chick survival was predicted to have a greater effect on λ than changes in survival of adult females. In practice, our demographic analyses indicate that effective management should be based on habitat conservation measures to increase components of fecundity.     

Population growth of a gorgonian coral: equilibrium and non-equilibrium sensitivity to changes in life history variables

Summary A size dependent model of population growth of the Caribbean gorgonian Plexaura A is developed based on observed rates of survival, growth and colony fragmentation at a site in the San Blas Islands, Panama. Sensitivity and elasticity analyses indicate that the fate of large colonies has the greatest effect on population growth. Variables which directly affect the generation of large colonies have the next greatest effect on population growth. These variables include the recruitment of large fragments, and the survivorship of colonies in the next smaller size class. Sexual reproduction has an extremely limited ability to affect population growth. Vegetative reproduction has a greater potential effect on growth rates. Environmental conditions regularly change the matrix of transition probabilities which predicts population growth. This keeps the population from approaching its stable size class distribution. Deviations from the stable size class distribution alter sensitivity and elasticity and in this case have the effect of increasing the importance of survivorship of the smallest colonies. Nonequilibrium conditions alter sensitivity analyses and it is important to assess whether populations are at equilibrium and to determine the effects of such deviations on the sensitivity analysis.     

Elasticity analysis of density-dependent matrix population models: the invasion exponent and its substitutes

In density-independent models, the population growth rate lambda measures population performance, and the perturbation analysis of lambda (its sensitivity and elasticity) plays an important role in demography. In density-dependent models, the invasion exponent lambdaI replaces lambda as a measure of population performance. The perturbation analysis of lambdaI reveals the effects of environmental changes and management actions, gives the direction and intensity of density-dependent natural selection on life history traits, and permits calculation of the sampling variance of the invasion exponent. Because density-dependent models require more data than density-independent models, it is tempting to look for substitutes for the invasion exponent, the sensitivity and elasticity of which can be calculated from a density-independent model. Here we examine the accuracy of two such substitutes: the dominant eigenvalue of the projection matrix evaluated at equilibrium (An) and the dominant eigenvalue of the matrix averaged over the attractor (A). Using a two-stage model that represents a wide range of life history types, we find that the elasticities of An or A often agree to within less than 5% error with those of the invasion exponent, even when population dynamics are chaotic. The exceptions are for semelparous life histories, especially when density-dependence affects fertility. This suggests that the elasticity analysis of density-independent models near equilibrium, or averaged over the attractor, provides useful information about the elasticity of the invasion exponent in density-dependent models.     

Sensitivity analysis of transient population dynamics

Short-term, transient population dynamics can differ in important ways from long-term asymptotic dynamics. Just as perturbation analysis (sensitivity and elasticity) of the asymptotic growth rate reveals the effects of the vital rates on long-term growth, the perturbation analysis of transient dynamics can reveal the determinants of short-term patterns. In this article, I present a completely new approach to transient sensitivity and elasticity analysis, using methods from matrix calculus. Unlike previous methods, this approach applies not only to linear time-invariant models but also to time-varying, subsidized, stochastic, nonlinear and spatial models. It is computationally simple, and does not require calculation of eigenvalues or eigenvectors. The method is presented along with applications to plant and animal populations.     

Projected population persistence of eastern hellbenders (Cryptobranchus alleganiensis alleganiensis) using a stage-structured life-history model and population viability analysis

The population of eastern hellbenders (Cryptobranchus alleganiensis alleganiensis) in the Blue River, Indiana has undergone a dramatic decline over the last decade. Recruitment in these declining populations has been negligible, and populations are now composed almost entirely of older age classes (upwards of 20 years old). Given this dramatic decline, it is imperative to assess the impacts of these demographic patterns on population growth and long-term stability. Therefore, we developed a stage-structured, life-history model to examine the effects of varying levels of egg, juvenile, and adult survivorship on abundance, recruitment, and long-term population projections. We performed a sensitivity analysis of the model and determine which life-history parameters have the greatest potential to increase/stabilise hellbender population growth. Finally, we conducted a population viability analysis to determine the probability of extinction associated with varying management strategies. For eastern hellbender populations in Indiana, adults (especially females) are the most important component of long-term population viability. Sensitivity and elasticity analyses of the Lefkovitch matrix revealed that survival of adult and egg/larvae life-history stages are the most important for focused management efforts. Indeed, adults had the highest elasticity and reproductive value in the matrix model. Increasing survival by as little as 20% corresponded to the turning point at which the population ceased to decline and increased abundance (28% survival of egg/larvae). The importance of the transition from subadult to adult (transitional matrix element) was identified as an additional factor in maintaining abundance based on the relatively long period spent in this life-history stage (seven years for females). A population viability analysis was conducted to assess the likelihood and projected time frame of extinction for this population under no management (～25 years to complete extirpation; probability of extinction = 1) and if management efforts such as captive rearing and headstarting are undertaken (probability of extinction <0.2 at 25–30% survival of egg/larvae). Adult females had the greatest effect in reducing growth rate and population abundance when removed in exploitation simulations (91.3% versus 51.8% reduction in population growth rate), indicating translocation efforts should be designed to maintain females in the breeding pool. These models indicated that conservation management strategies aimed at ensuring the presence of adult females while concomitantly ameliorating survival at early life stages (population augmentation, translocations, introduction of artificial nest structures) are needed to stabilise the Indiana population of eastern hellbenders. This stage-structured model is the first to model eastern hellbenders and has broad implications for use across the geographic range where populations of eastern hellbenders are monitored and vital rates can be estimated.     

Carnivora Population Dynamics Are as Slow and as Fast as Those of Other Mammals: Implications for Their Conservation

Of the 285 species of Carnivora 71 are threatened, while many of these species fulfill important ecological roles in their ecosystems as top or meso-predators. Population transition matrices make it possible to study how age-specific survival and fecundity affect population growth, extinction risks, and responses to management strategies. Here we review 38 matrix models from 35 studies on 27 Carnivora taxa, covering 11% of the threatened Carnivora species. We show that the elasticity patterns (i.e. distribution over fecundity, juvenile survival and adult survival) in Carnivora cover the same range in triangular elasticity plots as those of other mammal species, despite the specific place of Carnivora in the food chain. Furthermore, reproductive loop elasticity analysis shows that the studied species spread out evenly over a slow-fast continuum, but also quantifies the large variation in the duration of important life cycles and their contributions to population growth rate.These general elasticity patterns among species, and their correlation with simple life history characteristics like body mass, age of first reproduction and life span, enables the extrapolation of population dynamical properties to unstudied species. With several examples we discuss how this slow-fast continuum, and related patterns of variation in reproductive loop elasticity, can be used in the formulation of tentative management plans for threatened species that cannot wait for the results of thorough demographic studies. We argue, however, that such management programs should explicitly include a plan for learning about the key demographic rates and how these are affected by environmental drivers and threats.     

The influence of time since introduction on the population growth of introduced species and the consequences for management

Several processes likely act to change the demographic rates of introduced species over time, and this changing demography could influence the optimal management of invasive populations. Optimal management strategies should be derived based on the demography. However, we have a poor understanding of the degree to which the demography of introduced species changes following initial introduction. We used published matrix population models of introduced plant populations to test how population growth rate and elasticity change with time since introduction. We did not find a significant relationship between population growth rate and time since introduction. However, elasticity to stasis increased while elasticity to growth decreased with time since introduction. Broadly, as time since introduction progressed the elasticities of the introduced plant populations became more similar to those that have been reported for native species. These results suggest that the optimal management strategy should be derived incorporating elasticity through time, especially when the time scope of management is long or the available demographic data were obtained in the past.     

Robustness: predicting the effects of life history perturbations on stage-structured population dynamics

Matrix-based models lie at the core of many applications across the physical, engineering and life sciences. In ecology, matrix models arise naturally via population projection matrices (PPM). The eigendata of PPMs provide detailed quantitative and qualitative information on the dynamic behaviour of model populations, especially their asymptotic rates of growth or decline. A fundamental task in modern ecology is to assess the effect that perturbations to life-cycle transition rates of individuals have on such eigendata. The prevailing assessment tools in ecological applications of PPMs are direct matrix simulations of eigendata and linearised extrapolations to the typically non-linear relationship between perturbation magnitude and the resulting matrix eigenvalues. In recent years, mathematical systems theory has developed an analytical framework, called 'Robustness Analysis and Robust Control', encompassing also algorithms and numerical tools. This framework provides a systematic and precise approach to studying perturbations and uncertainty in systems represented by matrices. Here we lay down the foundations and concepts for a 'robustness' inspired approach to predictive analyses in population ecology. We treat a number of application-specific perturbation problems and show how they can be formulated and analysed using these robustness methodologies.     

Nonlinear Actomyosin Elasticity in Muscle?

Cyclic interactions between myosin II motor domains and actin filaments that are powered by turnover of ATP underlie muscle contraction and have key roles in motility of nonmuscle cells. The elastic characteristics of actin-myosin cross-bridges are central in the force-generating process, and disturbances in these properties may lead to disease. Although the prevailing paradigm is that the cross-bridge elasticity is linear (Hookean), recent single-molecule studies suggest otherwise. Despite convincing evidence for substantial nonlinearity of the cross-bridge elasticity in the single-molecule work, this finding has had limited influence on muscle physiology and physiology of other ordered cellular actin-myosin ensembles. Here, we use a biophysical modeling approach to close the gap between single molecules and physiology. The model is used for analysis of available experimental results in the light of possible nonlinearity of the cross-bridge elasticity. We consider results obtained both under rigor conditions (in the absence of ATP) and during active muscle contraction. Our results suggest that a wide range of experimental findings from mechanical experiments on muscle cells are consistent with nonlinear actin-myosin elasticity similar to that previously found in single molecules. Indeed, the introduction of nonlinear cross-bridge elasticity into the model improves the reproduction of key experimental results and eliminates the need for force dependence of the ATP-induced detachment rate, consistent with observations in other single-molecule studies. The findings have significant implications for the understanding of key features of actin-myosin-based production of force and motion in living cells, particularly in muscle, and for the interpretation of experimental results that rely on stiffness measurements on cells or myofibrils.     

Demography of squirrel monkeys (Saimiri sciureus) in captive environments and its effect on population growth

Understanding which life-history variables have the greatest influence on population growth rate has great ecological and conservation importance. Applying models of population regulation and demographic mechanisms can aid management and conservation of both wild and captive populations. By comparisons of sensitivity, elasticity, and life-table response analyses, we identified demographic processes that were most likely to produce changes in population size (via prospective analyses) and the traits that actually influenced population changes (via retrospective analyses) among sexes, zoological facilities, and generations of captive squirrel monkey populations (Saimiri sciureus). Variation in life-history traits occurs within each group analyzed. Those traits that vary the most include age at maturity, age at last reproduction, and fertility. Zoos with increasing population growth rates maintain earlier ages of maturity, later ages of last reproduction, high rates of juvenile and adult survival, and most importantly greater fertility, reflecting shorter inter-birth intervals. Using prospective analyses, juvenile and adult survivals were predicted to be demographic traits with the greatest effect on population growth. Surprisingly, and despite predictions, retrospective analyses revealed that fertility was the life-history characteristic trait that contributed the most to changes in population size.     

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

Matrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage-structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross-taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.     

Significant Impact on Muscle Mechanics of Small Nonlinearities in Myofilament Elasticity

Important mechanisms in muscle contraction have recently been reevaluated based on analyses that rely on the assumption of linear myofilament elasticity. However, the present theoretical study shows that nonlinearity of this elasticity, even when so minor that it may be difficult to detect in experimental data, could have great impact on the interpretation of muscle mechanical experiments. This is illustrated by using simulated stiffness and strain-versus-force data for muscle fibers shortening at different constant velocities. There is substantial quantitative agreement, for this condition, between models with distributed myofilament compliance and models where the compliance of the myofilaments and the actomyosin cross-bridges are lumped together into two separate elastic elements acting in series. The data thus support the usefulness of the latter, simpler, type of model in the analysis. However, most importantly, the data emphasize the importance of caution before reevaluating fundamental mechanisms of muscle contraction based on analyses relying on the assumption of linear myofilament elasticity.     

Managing multiple vital rates to maximize greater sage-grouse population growth

Despite decades of field research on greater sage-grouse, range-wide demographic data have yet to be synthesized into a sensitivity analysis to guide management actions. We reviewed range-wide demographic rates for greater sage-grouse from 1938 to 2011 and used data from 50 studies to parameterize a 2-stage, female-based population matrix model. We conducted life-stage simulation analyses to determine the proportion of variation in population growth rate (λ) accounted for by each vital rate, and we calculated analytical sensitivity, elasticity, and variance-stabilized sensitivity to identify the contribution of each vital rate to λ. As expected for an upland game bird, greater sage-grouse showed marked annual and geographic variation in several vital rates. Three rates were demonstrably important for population growth: female survival, chick survival, and nest success. Female survival and chick survival, in that order, had the most influence on λ per unit change in vital rates. However, nest success explained more of the variation in λ than did the survival rates. In lieu of quantitative data on specific mortality factors driving local populations, we recommend that management efforts for greater sage-grouse first focus on increasing female survival by restoring large, intact sagebrush-steppe landscapes, reducing persistent sources of human-caused mortality, and eliminating anthropogenic habitat features that subsidize species that prey on juvenile, yearling, and adult females. Our analysis also supports efforts to increase chick survival and nest success by eliminating anthropogenic habitat features that subsidize chick and nest predators, and by managing shrub, forb, and grass cover, height, and composition to meet local brood-rearing and nesting habitat guidelines. We caution that habitat management to increase chick survival and nest success should not reduce the cover or height of sagebrush below that required for female survival in other seasons (e.g., fall, winter). The success or failure of management actions for sage-grouse should be assessed by measuring changes in vital rates over long time periods to avoid confounding with natural, annual variation. © 2011 The Wildlife Society.     

The contributions of interlocking loops and extensive nonlinearity to the properties of circadian clock models

Background

Sensitivity and robustness are essential properties of circadian clock systems, enabling them to respond to the environment but resist noisy variations. These properties should be recapitulated in computational models of the circadian clock. Highly nonlinear kinetics and multiple loops are often incorporated into models to match experimental time-series data, but these also impact on model properties for clock models.

Methodology/Principal Findings

Here, we study the consequences of complicated structure and nonlinearity using simple Goodwin-type oscillators and the complex Arabidopsis circadian clock models. Sensitivity analysis of the simple oscillators implies that an interlocked multi-loop structure reinforces sensitivity/robustness properties, enhancing the response to external and internal variations. Furthermore, we found that reducing the degree of nonlinearity could sometimes enhance the robustness of models, implying that ad hoc incorporation of nonlinearity could be detrimental to a model''s perceived credibility.

Conclusion

The correct multi-loop structure and degree of nonlinearity are therefore critical in contributing to the desired properties of a model as well as its capacity to match experimental data.     

Disentangling community patterns of nestedness and species co-occurrence

Two opposing patterns of meta‐community organization are nestedness and negative species co‐occurrence. Both patterns can be quantified with metrics that are applied to presence‐absence matrices and tested with null model analysis. Previous meta‐analyses have given conflicting results, with the same set of matrices apparently showing high nestedness (Wright et al. 1998) and negative species co‐occurrence (Gotelli and McCabe 2002). We clarified the relationship between nestedness and co‐occurrence by creating random matrices, altering them systematically to increase or decrease the degree of nestedness or co‐occurrence, and then testing the resulting patterns with null models. Species co‐occurrence is related to the degree of nestedness, but the sign of the relationship depends on how the test matrices were created. Low‐fill matrices created by simple, uniform sampling generate negative correlations between nestedness and co‐occurrence: negative species co‐occurrence is associated with disordered matrices. However, high‐fill matrices created by passive sampling generate the opposite pattern: negative species co‐occurrence is associated with highly nested matrices. The patterns depend on which index of species co‐occurrence is used, and they are not symmetric: systematic changes in the co‐occurrence structure of a matrix are only weakly associated with changes in the pattern of nestedness. In all analyses, the fixed‐fixed null model that preserves matrix row and column totals has lower type I and type II error probabilities than an equiprobable null model that relaxes row and column totals. The latter model is part of the popular nestedness temperature calculator, which detects nestedness too frequently in random matrices (type I statistical error). When compared to a valid null model, a matrix with negative species co‐occurrence may be either highly nested or disordered, depending on the biological processes that determine row totals (number of species occurrences) and column totals (number of species per site).