Matrix Projection Models Research Articles

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.

Population recovery of black rhinoceros in north‐west Namibia following poaching

AbstractCurtailing overharvest, whether illegal or legal, is often a critical conservation objective. Yet even if overexploitation can be stopped, subsequent rates of population recovery can be highly variable due to Allee effects, alterations to age and sex structure and disruptions of animal social systems. Moreover, understanding the influence of density dependence can be difficult but important for long‐term management. Here, we investigate the dynamics of black rhinoceros Diceros bicornis in the Kunene region of Namibia as they recover from illegal hunting. We use multi‐strata mark–recapture models to examine survival and stage‐transition rates from 1992 to 2005. Survivorship estimates ranged from 0.793 for calves to 0.910 for adult males and 0.944 for adult females. The annual reproductive rate in adult females was estimated at 0.315. Model selection showed that these vital rates were time invariant, suggesting that Allee effects and transient dynamics did not have an important effect upon population dynamics, even in the early stages of recovery. Relative population density increased significantly from 1992 to 2005 once illegal hunting had ceased in Kunene. However, the best‐fit models did not include relative density in the estimation of survival or stage‐transition rates. We then used the vital rates generated from our mark–recapture analysis to build matrix projection models that assessed overall population dynamics. The female‐only model gave a population growth rate estimate of λ=1.011. Two‐sex models suggest that the growth rate of the population could range from 0.990 to 1.012. The relatively slow growth rate of this population, even without hunting or density dependence, could stem from the low productivity of the region. Adult females had the highest reproductive value and their survival had the highest elasticity among vital rates. Translocating adult females would lead to the fastest initial population growth rate in founder populations but would have the most impact on the source population.

How do plant ecologists use matrix population models?

Matrix projection models are among the most widely used tools in plant ecology. However, the way in which plant ecologists use and interpret these models differs from the way in which they are presented in the broader academic literature. In contrast to calls from earlier reviews, most studies of plant populations are based on < 5 matrices and present simple metrics such as deterministic population growth rates. However, plant ecologists also cautioned against literal interpretation of model predictions. Although academic studies have emphasized testing quantitative model predictions, such forecasts are not the way in which plant ecologists find matrix models to be most useful. Improving forecasting ability would necessitate increased model complexity and longer studies. Therefore, in addition to longer term studies with better links to environmental drivers, priorities for research include critically evaluating relative/comparative uses of matrix models and asking how we can use many short-term studies to understand long-term population dynamics.

Predicting Yellow Perch Population Responses Using a Density‐Dependent Age‐Structured Matrix Projection Model: How Many Annual Data Points Are Needed?

AbstractDensity‐dependent matrix projection models are commonly used to simulate fish population dynamics. Much of the data needed to specify the density‐dependent relationships are annual; thus, determining how many years of data are needed to accurately specify these relationships is critical. We used 200 years of simulated output from an individual‐based model (IBM) as “data,” and we estimated density‐dependent age‐0 survival, yearling survival, and adult growth (affected maturity and fecundity) for an age‐structured matrix projection model. We divided the 200‐year baseline simulation into 34 data sequences: three 60‐year sequences, four 40‐year sequences, nine 20‐year sequences, and eighteen 10‐year sequences. We refitted the density‐dependent survival and growth functions to each reduced data sequence and compared their shapes; we then substituted the refitted functions into a matrix model specific to each data sequence. We compared key output variables for the baseline simulations and the responses to decreased egg survival among the IBM, the matrix model based on 200 years, and the 34 matrix models based on the different data sequences. The variation in shape and the number of sequences that resulted in density‐independent survival and depensatory growth increased greatly for the 20‐ and 10‐year sequences. Predicted population dynamics under baseline and predicted responses to reduced egg survival were reasonably similar to those under the IBM and matrix model based on 200 years for the 60‐ and 40‐year sequences but showed increasing divergence for the 20‐ and 10‐year sequences. We suggest that 40 or more years of annual data will allow for reasonable estimation of density‐dependent relationships in age‐structured matrix projection models. Many applications of similar models used in management are based on fewer than 40 years of data, and yet their use is intended to generate predictions with sufficient precision and accuracy to resolve differences between relatively small changes in survival rates.

Structured Population Dynamics: An Introduction to Integral Modeling

SummaryA single species is often modeled as a structured population. In a matrix projection model, individuals in the population are partitioned into a finite number of stage classes. For example, an insect population can be partitioned into egg, larva, pupa and adult stages. For some populations the stages are better described by a continuous variable, such as the stem diameter of a plant. For such populations an integral projection model can be used to describe the population dynamics, and might be easier to use or more accurate than a matrix model. In this article we discuss the similarities and differences between matrix projection models and integral projection models. We illustrate integral projection modeling by a Platte thistle population, showing how the model is determined by basic life history functions.

Incorporating ‘recruitment’ in matrix projection models: estimation, parameters, and the influence of model structure

Advances in the estimation of population parameters using encounter data from marked individuals have made it possible to include estimates of the probability of recruitment in population projection models. However, the projected growth rate of the population, and the sensitivity of projected growth to changes in recruitment, can vary significantly depending upon both the structural form of the model and how recruitment is parameterized. We show that the common practices of (1) collapsing some age classes into a single, terminal ‘aggregated’ age-class, and (2) parameterizing recruitment using the proportion of recruited individuals (breeders) in a given age-class may confound analysis of age-based (Leslie) matrix projection models in some instances, relative to state-based projection models where recruited and pre-recruited individuals are treated as separate states. Failing to account for these differences can lead to misinterpretation of the relative role of recruitment in the dynamics of an age-structured population. We show that such problems can be avoided, either by structural changes to the terminal aggregated age-class in age-based models, or by using using a state-based model instead. Since all the metrics of general interest from a classical age-based matrix models are readily derived from a state-based model equivalent, this suggests there may be little reason to use the classical age-based approach in situations where recruitment is a parameter of interest.

Contributions of harvest slash to maintaining downed woody debris in selection-managed forests

Harvest slash can represent a major source of downed woody debris (DWD) in selection-managed forests. In this study, we analyze the volume, cover, size distribution, and decay-class distribution of DWD input by selection harvesting in central Ontario, Canada. Selection harvesting input 23.9 m3 DWD·ha–1 (0.013 m2 DWD·m–2), with cut basal area explaining 46% and 30% of the respective within-stand variation in cover and volume, respectively. The size distribution of the slash was similar to that of DWD in permanent sample plots (including old-growth stands and stands that have not been recently harvested), countering a common assumption that harvesting inputs only small-sized material. Harvest-origin DWD was bimodally distributed across decay classes, with the first peak (decay class 1) associated with fresh harvest slash and a second smaller peak (decay class 3) likely representing dead trees and branches that were felled or broken during harvest operations. A matrix projection model showed that slash can maintain DWD levels in managed, uneven-aged stands comparable with those in unmanaged stands, but the mean decay class increases steadily over a 20-year period after harvest. Our results underline the importance of harvest inputs for maintaining DWD pools in selection-managed forests and provide baseline information against which to compare forests managed with higher utilization standards.

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.

Choosing classes for size projection matrix models

Projection matrix models are intensely used in ecology to model the dynamics of structured populations. When dealing with size-structured populations, there is no satisfactory algorithm to partition size into discrete classes. We show that the Vandermeer–Moloney algorithm for choosing classes is inconsistent with the Usher model, and systematically selects the finest classes. Considering that the matrix model is a discrete approximation of a continuous model, we define an approximation error as the sum of a distribution error (the difference between the discrete distribution and its continuous counterpart), and a sample error. The optimal partition of size into classes is the one that minimizes the approximation error. This method for choosing classes also shows that the choice of the class width cannot be disconnected from the choice of the time step. When applied to 520 trees of Dicorynia guianensis in French Guiana, this algorithm identified 8 classes of 11.4 cm in width, which is in agreement with the empirical choice of foresters.

Matrix models for a changeable world: the importance of transient dynamics in population management

1. Matrix population models are tools for elucidating the association between demographic processes and population dynamics. A large amount of useful theory pivots on the assumption of equilibrium dynamics. The preceding transient is, however, of genuine conservation concern as it encompasses the short-term impact of natural or anthropogenic disturbance on the population. 2. We review recent theoretical advances in deterministic transient analysis of matrix projection models, considering how disturbance can alter population dynamics by provoking a new population trajectory. 3. We illustrate these impacts using plant and vertebrate systems across contiguous and fragmented landscapes. 4. Short-term responses are of fundamental relevance for applied ecology, because the time-scale of transient effects is often similar to the length of many conservation projects. Investigation of the immediate, post-disturbance phase is vital for understanding how population processes respond to widespread disturbance in the short- and into the long term. 5. Synthesis and applications. Transient analysis is critical for understanding and predicting the consequences of management activities. By considering short-term population responses to perturbations, especially in long-lived species, managers can develop more informed strategies for species harvesting or controlling of invasive species.

The Ecological and Evolutionary Drivers of Female‐Biased Sex Ratios: Two‐Sex Models of Perennial Seagrasses

Among sexually reproducing species, differences between the sexes within species are ubiquitous. Despite the clear effect of sex differences on sex ratios and population growth rates, demographic models rarely consider both sexes explicitly. Here I explore the causes of extreme female-biased sex ratios in two marine angiosperms (Phyllospadix spp.). Using demographic data, I develop two-sex matrix projection models to assess the magnitude of demographic differences necessary to generate observed sex ratios and the consequences of sex differences for population growth rates. I demonstrate that small sex differences in survival can generate biased sex ratios, but the importance of sexual reproduction differs markedly between species. Even in the absence of a direct trade-off between sexual and asexual reproduction, the presence of two reproductive modes affects both the importance of sex and the sex-ratio bias. Using sensitivity analyses, I quantify the contribution of shared and sex-specific vital rates and show that until males become rare, the sensitivity of sex-specific vital rates is small relative to that of shared vital rates. I demonstrate that placing sex differences in the context of a demographic model that includes biologically motivated life-history trade-offs can explain the maintenance of sex-specific life histories and the persistence of skewed sex ratios.

Matrix projection models meet variation in the real world

Summary 1. Projection matrices have become the dominant modelling approach in plant demography because they (i) are relatively easy to formulate, (ii) compile complex data in a structured and analytically tractable manner, (iii) provide numerous parameters with direct biological meaning, (iv) allow the investigator to address broad or specific, experimental and/or theoretical, ecological and evolutionary questions, and (v) produce uniform outputs, enabling direct comparisons between the results of different studies. 2. The last decade has witnessed major advancements in this field that have brought demographic models much closer to the real world, in particular in the analysis of effects of spatial and temporal environmental variation on populations. The present Special Feature contributes to that progress with novel methodologies and applications on Integral Projection Models, stochastic Life Table Response Experiment analyses, stochastic elasticities, transient dynamics and phylogenetic analyses. 3. Synthesis. Environmental stochasticity is an integral part of ecosystems, and plant populations exhibit a tremendous array of demographic strategies to deal with its effects. The analytical challenge of understanding how populations avoid, tolerate or depend on stochasticity is finally overcome with the new matrix approaches. The tools are now available to interpret the effects of changes in temporal and spatial variation on plant populations.

The stock recovery rate in a Central African rain forest: an index of sustainability based on projection matrix models

The stock recovery rate is used in most natural forests of the Congo Basin to assess logging sustainability. This rate is computed using the so-called Dimako formula. Although this formula has been used for many years now in management plans, its mathematical properties have not been closely reviewed. We show that the Dimako formula corresponds to a Leslie matrix model, and then we propose an extension of it as a Usher matrix model. The stock recovery rate at the end of the first felling cycle for six commercial species in the Central African Republic varied between 21.7% and 99.9%. As felling cycles follow each other, the stock recovery rate converged towards a limit that is the asymptotic stock recovery rate. This limit varies between 27.2% and 158.4% for the same six species. Comparing felling scenarios reveals that increasing the minimum harvest diameter was as efficient at increasing the stock recovery rate at the end of the first felling cycle as decreasing the logging intensity. The results for the other parameters of the felling scenarios varied among species, with changes in the stock recovery rate ranging from 0% to 180% at the end of the first felling cycle, and changes in the asymptotic rate ranging from 0% to 685%.

Predicting the impact of stage‐specific harvesting on population dynamics

1. Perturbation analyses of population projection matrices predict the response of a population's growth rate to changes in lifestage-specific vital rates. Such predictions have been widely used in population management but their reliability remains hotly debated. 2. We grew replicate populations of the water flea Daphnia magna in controlled, density-independent conditions and subjected treatment populations to harvesting of the largest lifestage. We predicted the growth rate of treatment populations using sensitivity analysis (a linear approximation), and transfer function analysis (TFA; which captures nonlinear responses) applied to projection matrix models parameterized from the control populations. 3. When perturbation analyses considered only the direct effect of harvesting on adult survival, the growth rate of harvested populations (averaging 0.051) was significantly overestimated (average of 0.112) by TFA and non-significantly underestimated (average of 0.012) by sensitivity. 4. When the indirect effects of harvesting on other vital rates were accounted for in a structured perturbation, TFA gave accurate predictions (average growth rate of 0.068), while sensitivity gave significant underestimates (average of -0.043). 5. Our results demonstrate two crucial sources of error that may influence predictions of the impacts of demographic perturbations on population dynamics. First, impacts of stage-specific harvesting are inherently nonlinear, hence predictions based on sensitivity must be treated with caution. Second, stage-specific perturbations can change non-target demographic rates, even in the absence of adaptation. 6. Population managers should consider both nonlinear and indirect effects of perturbations when designing management interventions. We encourage the development of methods to assess the robustness of predictions to unforeseen perturbation structures and indirect harvesting impacts.

A slow life in hell or a fast life in heaven: demographic analyses of contrasting roe deer populations

1. Environmental conditions shape population growth through their impact on demographic parameters. While knowledge has accumulated concerning the effects of population density and climatic conditions, a topical question now concerns how predation and harvest influence demographic parameters and population growth (lambda). 2. We performed a comparative demographic analysis based on projection matrix models for female roe deer. Population-specific matrices were parameterized based on longitudinal data from five intensively monitored populations in Norway and France, spanning a large variability in environmental characteristics such as densities of large predators, hunter harvest and seasonality. 3. As expected for a large iteroparous vertebrate, temporal variation was invariably higher in recruitment than in adult survival, and the elasticity of adult survival was consistently higher than that of recruitment. However, the relative difference in elasticity of lambda to recruitment and adult survival varied strongly across populations, and was closely correlated with adult survival. 4. Different traits accounted for most of the variance in lambda in different ecological settings. Adult survival generally contributed more in populations with low mean adult survival and low mean growth rate during the study period. Hunters and predators (Eurasian lynx and red foxes) occurred in two of our study populations and contributed substantially to the variance in lambda, accounting for a total of 35% and 70% in the two populations respectively. 5. Across populations, we did not find any evidence that roe deer increased their reproductive output when faced with harsh conditions, resulting in some populations having negative growth rates. 6. Generation time, a measure of the speed of the life-history cycle, increased from less than 4 years in the most productive population ('roe deer heaven') to more than 6 years in declining populations facing predation from lynx, red fox and hunters ('roe deer hell'), and was tightly and inversely correlated with lambda. Such a deceleration of the life cycle in declining populations might be a general feature in large herbivores. 7. Our results shows that the plethora of environmental conditions faced by populations of large herbivores also induce high intraspecific variation in their ranking along the 'fast-slow' continuum of life-history tactics.

Multi‐scale analysis of species introductions: combining landscape and demographic models to improve management decisions about non‐native species

Summary Non‐native, invasive species can affect biological patterns and processes at multiple ecological scales. The multi‐scalar effects of invasions can influence community structure, ecosystem processes and function, and the nature and intensity of ecological interactions. Consequently, efforts to assess the spread of invasive species may benefit from a multi‐scale analytic approach. We analysed results from landscape‐ and population‐scale models for Syzygium jambos, a non‐native tree in the Luquillo Mountains of Puerto Rico, to demonstrate a multi‐scale approach that can be used to inform management decisions about invasive plants. At the landscape‐level, we used an Ecological Niche Modelling approach to predict environmentally suitable habitats for the target plant. At the population‐level, we constructed matrix projection models to determine the finite rate of population increase (λ) for S. jambos. We then extrapolated λ values to the landscape‐scale to obtain a distribution map of λ values for the Luquillo forest. The landscape analyses suggested that the most environmentally suitable habitats were those most similar to where S. jambos had already been observed. The population‐level analyses showed that four of the seven populations had λ values less than 1, indicating that they were projected to be below replacement. The λ distribution map showed that S. jambos growth was highest in areas where it was most common and lowest in areas where it was most rare. Our analyses further suggested that the importance of different drivers of invasion and the environmental variables that mediate them appear to be strongly scale‐dependent. Past disturbances seemed most important for controlling invasions at fine‐spatial scales; while abiotic environmental variables modulated coarse‐scale invasion dynamics. Synthesis and applications. We have shown that a multi‐scale analytic approach can be used to manage invasive species by simultaneously targeting susceptible life stages and rapidly growing populations in a landscape. The utility of this approach stems from an ability to: (i) map the distribution of habitats that can potentially sustain λ values above replacement; (ii) identify populations to manage or monitor during selected stages of an invasion; (iii) forecast the probability for a target species to increase above a critical threshold abundance; and (iv) set priorities for control and monitoring actions.

Erratum et addendum: transient amplification and attenuation in stage‐structured population dynamics

Summary Not all members of natural populations contribute equally to population growth or decline. Populations that are disturbed away from stable stage structure will amplify (i.e. get bigger than expected) and/or attenuate (i.e. get smaller than expected) in the short term. We provide mathematical bounds for the magnitude of this amplification and attenuation, both in terms of absolute population change and population change relative to the long‐term rate of population increase. Our results correct an important error in an earlier analysis of transient population amplification, and provide new transient bounds for the analysis of population attenuation. Synthesis and applications. Bounds on transient amplification and attenuation help population managers to gauge ‘worst case’ and ‘best case’ scenarios for the response of stage‐structured populations to disturbance and management strategies. Such bounds help to create an envelope of possible future population scenarios around the mean, long‐term predictions made by eigenvalues and eigenvectors of projection matrix models. Transient amplification, caused by stage structures biased towards reactive life stages, may be exploited by conservation managers wishing to boost population densities in the short term and may be avoided in pest species by stage‐specific control strategies. Similarly, transient attenuation should be avoided by conservation managers and exploited by pest managers.

Choosing Fagus sylvatica L. matrix model dimension by sensitivity analysis of the population growth rate with respect to the width of the diameter classes

In this study we use a projection matrix model of order n of an uneven-aged beech stand to analyse how the width of the diameter (physical tree diameter) classes d influence the population growth rate λ 0, and to obtain the sensitivity and the elasticity of λ 0 with respect to d. Numerical computations using systematic variations in d and n (where n represents the number of diameter classes), three levels of diameter growth, three levels of stand basal area, G = 20, 22 and 24 m 2/ha, and three levels of recruitment, R = 200, 520 and 840 stems/ha, were conducted. The lowest values for λ 0, sensitivities (∂ λ 0/∂ d) and elasticities (∂ λ 0/ λ 0)/(∂ d/ d), were obtained at the lowest width of the diameter classes, d = 5 cm. The results have also shown that: (a) the estimation of λ 0 was not significantly affected by small or even moderate changes in n and/or d; (b) (∂ λ 0/∂ d) and (∂ λ 0/ λ 0)/(∂ d/ d) increased with increasing d; (c) within each quality level (diameter growth level), λ 0 increased with increasing R and decreased with increasing G, (∂ λ 0/∂ d) decreased with increasing R and increased with increasing G, and (∂ λ 0/ λ 0)/(∂ d/ d) decreased with increasing R and increased with increasing G. We suggest that narrow size categories should be used in this model to obtain conservative estimates of λ 0 and sustainable harvest rates, with minimum sensitivities and elasticities, although the results were not substantially affected by small or even moderate changes in the width and/or number of the diameter classes.