Population Projection Matrix Model Research Articles

Genetic predictors of population resilience: A case study of native Brook Trout in headwater streams

AbstractObjectivePopulations of eastern Brook Trout Salvelinus fontinalis face threats from several sources, such as habitat fragmentation, climate change, and competition with introduced salmonids. As a native species, understanding how these populations will respond to disturbances is paramount to their management and effective conservation. A population's ability to respond to disturbance, its resilience, is influenced by several factors. One such group of factors is population genetics.MethodsWe calculated population resilience metrics based on transient dynamics using population projection matrix models. Long‐term demographic data from 23 headwater stream Brook Trout populations were used to parameterize models. Genetic data were collected, and genetic indices were calculated. Partial redundancy analysis was then used to evaluate relationships between resilience metrics and genetic indices.ResultInbreeding coefficient, rarefied allelic richness, pairwise genetic differentiation (FST), and effective population size were all found to be important variables in predicting resilience.ConclusionOur results suggest that genetic isolation may increase the demographic resilience in Brook Trout through faster generation times and higher juvenile survival, but this likely comes at the cost of increased extinction risk and truncated size structures. Genetic indices can provide insight into gene flow between populations, thus the relationship between population connectivity and resilience. Given the importance of connectivity to population resilience, restoring and maintaining movement corridors could affect resilience in headwater Brook Trout populations.

Demystifying individual heterogeneity.

Among-individual variation in vital rates, such as mortality and birth rates, exists in nearly all populations. Recent studies suggest that this individual heterogeneity produces substantial life-history and fitness differences among individuals, which in turn scale up to influence population dynamics. However, our ability to understand the consequences of individual heterogeneity is limited by inconsistencies across conceptual frameworks in the field. Studies of individual heterogeneity remain filled with contradicting and ambiguous terminology that introduces risks of misunderstandings, conflicting models and unreliable conclusions. Here, we synthesise the existing literature into a single and comparatively straightforward framework with explicit terminology and definitions. This work introduces a distinction between potential vital rates and realised vital rates to develop a coherent framework that maps directly onto mathematical models of individual heterogeneity. We suggest the terms "fixed condition" and "dynamic condition" be used to distinguish potential vital rates that are permanent from those that can change throughout an individual's life. To illustrate, we connect the framework to quantitative genetics models and to common classes of statistical models used to infer individual heterogeneity. We also develop a population projection matrix model that provides an example of how our definitions are translated into precise quantitative terms.

An analysis of elasticity vector distribution specific to semelparous species using randomly generated population projection matrices and the COMPADRE Plant Matrix Database

The elasticity of population growth rate λ is a notable statistic in population projection matrix models and has been used by many empirical researchers. Silvertown and his collaborators published a paper in 1996 in which they mapped elasticity vectors of stasis, growth and fecundity for 84 plant species in a ternary plot and reported that the elasticity vector distribution of semelparous species (10 species) was located in the upper-left region of the ternary plot. To understand and clarify why the elasticity vectors of semelparous species were distributed in the upper-left region, we conducted three analyses. First, we used 68 matrices of semelparous species populations in the COMPADRE Plant Matrix Database and plotted their elasticity vectors in the ternary plot. Second, we constructed randomly generated population projection matrices for three archetypes, where several biological assumptions were made, and obtained their elasticity vector distributions. We showed that the high fecundity and/or immediate death after reproduction shifted the elasticity vector distribution to the upper-left region of the plot. Third, we examined the evolutionary change in the distribution using a population projection matrix model in which the trade-off between fecundity and adult survival was incorporated. The evolutionary trajectory derived from the trade-off model showed that the evolutionary consequence of semelparity was located in the upper-left more than was that of iteroparity in the ternary plot. Synthesizing the results derived from the three analyses, we concluded that the distinctive distribution of the elasticity vectors of semelparous species is the outcome of natural selection under the trade-off with low convexity, and the outcome of the resultant big-bang reproduction and immediate death after reproduction.

Demography and population growth rate of the tree Prosopis flexuosa with contrasting grazing regimes in the Central Monte Desert

One of the most important current challenges for ecologists is to evaluate how human-induced changes in ecosystems would impact viability of populations. Demographic response to anthropogenic impact could help us to understand how to manage those impacts. Using demographic techniques and population projection models, here we assess if demography and population dynamics of the tree Prosopis flexuosa change in cattle grazed areas compared to ungrazed areas in the Central Monte desert, Mendoza, Argentina. To this end, we quantified vital rates and constructed a population projection matrix model to compare the deterministic population growth rate (λ) between grazed and ungrazed areas. We also estimated elasticities of vital rates to evaluate their potential importance for future changes in λ and performed a life table response experiment (LTRE) to identify the life cycle transitions that contribute the most to the observed differences in λ between the two treatments. Although we found differences in demographic processes, such as lower seed production and higher probability of reversion to smaller size classes in young individuals when cattle were present, our results indicate that cattle grazing had no significant effect on λ for this species. According to the elasticity analysis, survival of large trees is the main driver of the population growth rate (λ) of P. flexuosa, and the vital rates related to tree reproduction, such as seed production and germination, have a poor contribution to λ. Therefore, limitations of activities that can affect survival of large trees should be considered as part of the conservation strategy for this species. Our study provides a compilation of demographic information that can be useful to set policies connecting the conservation objectives for this woodlands with that of ranch managers of the area.

Use of an Inverse Method for Time Series to Estimate the Dynamics of and Management Strategies for the Box Jellyfish Carybdea marsupialis.

Frequently, population ecology of marine organisms uses a descriptive approach in which their sizes and densities are plotted over time. This approach has limited usefulness for design strategies in management or modelling different scenarios. Population projection matrix models are among the most widely used tools in ecology. Unfortunately, for the majority of pelagic marine organisms, it is difficult to mark individuals and follow them over time to determine their vital rates and built a population projection matrix model. Nevertheless, it is possible to get time-series data to calculate size structure and densities of each size, in order to determine the matrix parameters. This approach is known as a “demographic inverse problem” and it is based on quadratic programming methods, but it has rarely been used on aquatic organisms. We used unpublished field data of a population of cubomedusae Carybdea marsupialis to construct a population projection matrix model and compare two different management strategies to lower population to values before year 2008 when there was no significant interaction with bathers. Those strategies were by direct removal of medusae and by reducing prey. Our results showed that removal of jellyfish from all size classes was more effective than removing only juveniles or adults. When reducing prey, the highest efficiency to lower the C. marsupialis population occurred when prey depletion affected prey of all medusae sizes. Our model fit well with the field data and may serve to design an efficient management strategy or build hypothetical scenarios such as removal of individuals or reducing prey. TThis This sdfsdshis method is applicable to other marine or terrestrial species, for which density and population structure over time are available.

The Implications of Habitat Management on the Population Viability of the Endangered Ohlone Tiger Beetle (Cicindela ohlone) Metapopulation

Despite their role in providing ecosystem services, insects remain overlooked in conservation planning, and insect management approaches often lack a rigorous scientific basis. The endangered Ohlone tiger beetle (Cicindela ohlone) occurs in a 24-km2 area in Santa Cruz County, California. The once larger metapopulation now consists of subpopulations inhabiting five patches of coastal prairie where it depends on bare ground for mating, foraging, and oviposition. Human activities have eliminated natural disturbances and spread invasive grasses, reducing C. ohlone's bare-ground habitat. Management actions to restore critical beetle habitat consist of cattle and horse grazing, maintaining slow bicycle speeds on occupied public trails, and artificial creation of bare-ground plots. Recreational biking trails help maintain bare ground, but can cause beetle mortality if left unregulated. We tracked C. ohlone survivorship and estimated fecundity for three years. We then constructed a stage-structured population projection matrix model to estimate population viability among the five patches, and to evaluate the success of management interventions. We demonstrate that habitat creation, regulation of bicycle speed, and migration between patches increase C. ohlone survival and population viability. Our results can be directly applied to management actions for conservation outcomes that will reduce species extinction risk and promote recolonization of extirpated patches.

Using a deterministic population model to evaluate population stability and the effects of fruit harvesting and livestock on baobab (Adansonia digitata L.) populations in five land-use types

The subsistence and commercial use of baobab (Adansonia digitata) fruit is important to many thousands of marginalized people in the arid tropics of Africa, yet sustainable harvest levels have not previously been studied. Size-class distributions of baobab populations tend to be stable, suggesting high tolerance to fruit harvesting. However, environmental conditions have changed substantially over the last 100years. Increasing livestock numbers, land modification and climate change are new threats which may affect tolerance to fruit harvesting. To investigate this, a deterministic stage-based population projection matrix model was developed using (a) long term baobab monitoring data from 2 sites, (b) radiocarbon age calculations, (c) extensive field surveys of population structure and fruit (and seed) production, and (d) experimental field trials on seed banks and seedling and sapling survival in relation to the presence of livestock. Projected population growth (λ) was then evaluated for five land-use types (nature reserves, rocky outcrops, plains, fields, and villages) under three levels of livestock (none, moderate and high stocking rates). Response to fruit harvest intensity was tested for each scenario by decreasing seed availability by 10% from 100%. High livestock numbers resulted in baobab population declines, with λ<1 in all land-use types. Under moderate and zero livestock numbers, baobab populations in plains, rocky outcrops, villages and fields were able to tolerate between 33% and 90% fruit harvest rates. In nature reserves there was already high predation on immature fruit by baboons, another cause of population decline, with the model showing no tolerance whatsoever to fruit harvesting. These results show that fruit harvesting can be sustainable in production landscapes under moderate livestock levels. However the future is uncertain, as a predicted lowering of rainfall due to climate change is a further concern, with likely negative impacts on fruit yields and recruitment and consequently population projections. Thus active planting and protection of seedlings should take place to mitigate current and future negative impacts facing baobab populations.

Popdemo: an R package for population demography using projection matrix analysis

Summary1. Effective population management requires accurate predictions of future population dynamics and how they may be manipulated to achieve management goals.2. The R package popdemo provides software tools for novel analytical methods that aim to enhance the predictive power of basic population projection matrix models. These include indices of transient population dynamics and transfer function analyses.3. We use a case study to demonstrate the use and importance of these methods for population management and briefly discuss their potential application outside population ecology.

The population dynamics and productivity of Acacia pennatula in the pasturelands of the Nature Reserve Mesas de Moropotente, Estelí, Nicaragua

Acacia pennatula trees are the most conspicuous woody species in the pasturelands of the Nature Reserve Mesas de Moropotente, Esteli, Nicaragua. Cattle ranchers keep A. pennatula because it produces fence posts, forage (pods) and firewood. A population projection matrix model was developed to: (1) estimate the sustainable harvest (H) of fence posts at different tree population densities, (2) explore the range of recruitment (R), and survival and growth of both saplings and small poles compatible with current population density, and (3) determine how much carbon is stored in the soil-pasture-tree system. Acacia pennatula trees take 40 years to reach H size (D30 ≥ 30 cm). Estimated sustainable H from current tree population density is 1.8l7 trees ha−1 year−1, yielding 2.8 large and 11.2 regular size fence posts ha−1 year−1. This annual output easily satisfies the needs of a typical 100 ha cattle ranch in the study area. Current population density is congruent with very low R (<100 saplings ha−1 year−1), very low survival rates (<0.30%) and/or retarded D30 growth of saplings and small poles. Total carbon in tree biomass was only 37 Mg ha−1. Cattle ranchers have learned to harness the invasive nature of the species to obtain valuable tree products for farm use or sale.

The Growth Rates of Population Projection Matrix Models in Random Environments

Population projection matrix models in random environments are random walk models. The growth rate of the mean population size, which is equal to the maximum eigenvalue of the mean matrix, is better than the average of the intrinsic rates of natural increase calculated by computer simulations, because the population size is more important than the growth rate. The arithmetic mean of the maximum eigenvalues of matrices for all permutations converges to the maximum eigenvalue of the mean matrix. The periodicity of environments is more important than the correlation between environments. Simple matrices and three numerical models are used as examples.

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.

Higher survival at low density counteracts lower fecundity to obviate Allee effects in a perennial plant

Summary 1. At low densities, plants may produce fewer seeds than at high densities, due to reduced pollinator visits or reduced receipt of compatible pollen. In principle, lower seed production could lead to an Allee effect (a decline in the population growth rate with declining density), which, if severe, could cause the population to decline to extinction when recruitment is insufficient to replace dying individuals. 2. However, plants at low density may also grow or survive better than those at high densities, due to release from intraspecific competition. Few studies have examined the combined effects on plant population dynamics of positive density dependence in fecundity and negative density dependence in survival or growth. 3. To do so, we parameterized a density-dependent population projection matrix model for Piriqueta caroliniana, a widespread sandhill plant that often occurs at low density, using data from a demographic study of natural populations and from field experiments testing for effects of density on fecundity, growth and survival. We then tested for an Allee effect when various combinations of positive and negative density dependence are included in the model. 4. The model exhibits an Allee effect when only positive density dependence in fecundity is included, but when negative density dependence in survival is added, the Allee effect disappears. Thus the increase in survival as density declines overwhelms the effect of declining fecundity on the population growth rate. This occurs because survival has a greater influence on the population growth rate than fecundity (as measured by vital rate elasticities), increasingly so as density declines. 5. Synthesis. Our results caution against considering only declining fecundity at low density. We show that survival may increase at low densities, and will often have stronger effects on population growth than does fecundity, especially for perennial plants. Of course, reproductive failure and population collapse must eventually occur as the density of any non-selfing species approaches zero, or if pollinator abundance falls to low levels.

On reducibility and ergodicity of population projection matrix models

Summary 1. Population projection matrices (PPMs) are probably the most commonly used empirical population models. To be useful for predictive or prospective analyses, PPM models should generally be irreducible (the associated life cycle graph contains the necessary transition rates to facilitate pathways from all stages to all other stages) and therefore ergodic (whatever initial stage structure is used in the population projection, it will always exhibit the same stable asymptotic growth rate). 2. Evaluation of 652 PPM models for 171 species from the literature suggests that 24·7% of PPM models are reducible (parameterized transition rates do not facilitate pathways from all stages to all other stages). Reducible models are sometimes ergodic but may be non‐ergodic (the model exhibits two or more stable asymptotic states with different asymptotic stable growth rates, which depend on the initial stage structure used in the population projection). In our sample of published PPMs, 15·6% are non‐ergodic. 3. This presents a problem: reducible–ergodic models often defy biological rationale in their description of the life cycle but may or may not prove problematic for analysis as they often behave similarly to irreducible models. Reducible–non‐ergodic models will usually defy biological rationale in their description of the both the life cycle and population dynamics, hence contravening most analytical methods. 4. We provide simple methods to evaluate reducibility and ergodicity of PPM models, present illustrative examples to elucidate the relationship between reducibility and ergodicity and provide empirical examples to evaluate the implications of these properties in PPM models. 5. As a prevailing tool for population ecologists, PPM models need to be as predictive as possible. However, there is a large incidence of reducibility in published PPMs, with significant implications for the predictive power of such models in many cases. We suggest that as a general rule, reducibility of PPM models should be avoided. However, we provide a guide to the pertinent analysis of reducible matrix models, largely based upon whether they are ergodic or not.

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.

On second order sensitivity for stage-based population projection matrix models

In this paper we present a simple method for identifying life-history perturbations in population projection matrices that yield an accelerating population growth rate. Accelerating growth means that the dependence of the growth rate on the perturbation is convex. Convexity, when the second sensitivity of the growth rate is positive, is calculated using a new formula derived from the transfer function of the perturbed system. This formula is used to explore the relationship between stasis and growth probabilities from stage-structured population projection matrices.

Modelling the effect of urbanization on the transmission of an infectious disease

This paper models the impact of urbanization on infectious disease transmission by integrating a CA land use development model, population projection matrix model and CA epidemic model in S-Plus. The innovative feature of this model lies in both its explicit treatment of spatial land use development, demographic changes, infectious disease transmission and their combination in a dynamic, stochastic model. Heuristically-defined transition rules in cellular automata (CA) were used to capture the processes of both land use development with urban sprawl and infectious disease transmission. A population surface model and dwelling distribution surface were used to bridge the gap between urbanization and infectious disease transmission. A case study is presented involving modelling influenza transmission in Southampton, a dynamically evolving city in the UK. The simulation results for Southampton over a 30-year period show that the pattern of the average number of infection cases per day can depend on land use and demographic changes. The modelling framework presents a useful tool that may be of use in planning applications.

Conservation biology for suites of species: Demographic modeling for Pacific island kingfishers

Conservation practitioners frequently extrapolate data from single-species investigations when managing critically endangered populations. However, few researchers initiate work with the intent of making findings useful to conservation efforts for other species. We presented and explored the concept of conducting conservation-oriented research for suites of geographically separated populations with similar natural histories, resource needs, and extinction threats. An example was provided in the form of an investigation into the population demography of endangered Micronesian kingfishers ( Todiramphus cinnamominus). We provided the first demographic parameter estimates for any of the 12 endangered Pacific Todiramphus species, and used results to develop a population projection matrix model for management throughout the insular Pacific. Further, we used the model for elasticity and simulation analyses with demographic values that randomly varied across ranges that might characterize congener populations. Results from elasticity and simulation analyses indicated that changes in breeding adult survival exerted the greatest magnitude of influence on population dynamics. However, changes in nestling survival were more consistently correlated with population dynamics as demographic rates were randomly altered. We concluded that conservation practitioners working with endangered Pacific kingfishers should primarily focus efforts on factors affecting nestling and breeder survival, and secondarily address fledgling juveniles and helpers. Further, we described how the generalized base model might be changed to focus on individual populations and discussed the potential application of multi-species models to other conservation situations.

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

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

ESTIMATING POSTNATAL DISPERSAL: TRACKING THE UNSEEN DISPERSERS

Postnatal dispersal, defined as dispersal from the natal patch soon after reaching maturity, is common in a variety of insect species. Postnatal dispersal may increase connections among subpopulations and, thus, significantly affect local population dynamics in metapopulation and source–sink contexts. Yet most mark–recapture studies of insects miss the postnatal dispersal phase because it comes soon after eclosion. Cephaloleia fenestrata (Chrysomelidae) is a specialist herbivore of the rolled leaves of Pleiostachya pruinosa (Marantaceae, common name “prayer plant”). Some P. pruinosa patches were flooded prior to the onset of this study, extirpating the majority of C. fenestrata in the flood zone and providing a natural experiment and an opportunity to observe recolonization dynamics. We built a spatially explicit population projection matrix model to simulate spatial population dynamics of C. fenestrata. The model was a poor fit to the empirical data. Adding postnatal dispersal to the model resulted in a much better model fit. This study suggests that 81% of C. fenestrata disperse from the natal site soon after eclosion. Model simulations indicated that postnatal dispersal increases the mean population size in the flood zone more than six-fold and the standard deviation more than three-fold. Herein we present an implicit method of estimating postnatal dispersal when marking animals before the postnatal dispersal phase is problematic.

Demographic variation in the Australian desert cassia under grazing pressure.

Demographic variation was examined in 3 populations of the Australian desert shrub Cassia nemophila which vary in their grazing histories. Age-specific life tables were constructed from 50 years of observations on mortality and recruitment at the Koonamore Vegetation Reserve in South Australia. Population projection matrix models were used to examine population responses to grazing pressure. The predicted population growth rates, reproductive values and stable age distributions are evaluated and compared with observed results. Grazing by sheep or rabbits, in high populations, prevents shrub recruitment and causes local population extinction. However, where protected from sheep and with low rabbit pressure, Cassia populations have increased. Current sheep grazing practices and rabbit population levels if continued will have a drastic affect on Cassia populations and other shrub species, and on the structure and composition of the Australian arid shrublands in general.