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The state of the art of population projection models: from the Leslie matrix to evolutionary demography

Authors:
Maja Kajin at Rio de Janeiro State University
Maja Kajin
Paulo Almeida at Laboratório Nacional de Computação Científica
Paulo Almeida
Marcus Vinícius Vieira at Federal University of Rio de Janeiro
Marcus Vinícius Vieira
Rui Cerqueira at Federal University of Rio de Janeiro
Rui Cerqueira
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Abstract and Figures

Population projection models have received much attention in ecology and made important theoretical advances in the last 50 years. They represent vital tools for improving conservation strategies and management actions. Here we attempt to join some of theoretical advances made in the field of population projection modelling, briefly revise the history and present some applications derived from population matrix models in ecological and evolutionary studies.
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Oecol.'Aust.,#16(1):#13*22,#2012
Oecologia)Australis
16(1):%13'22,%Março%2012
http://dx.doi.org/10.4257/oeco.2012.1601.02
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... For example, the works of Daskalopoulos et al. (1998), Christiansen and Fischer (1999) and Skovgaard et al. (2005) Despite the various attempts at modelling solid waste, nowhere in literature, the Leslie (1945) or Lefkovitch (1965) matrix models have been used to model solid waste generation. The models are normally used in population ecology and demographic studies (Leslie, 1945;Lefkovitch, 1965;Crouse et al. 1987;Lo et al. 1995 Kajin et al. 2012). In this work, the author uses the already established strength of the Leslie/Lefkovitch matrix models in modelling population growth to predict solid waste generation as there is a direct link between population growth and solid waste generation rates (Senzige et al. 2014). ...
... where A1 represents infant (age 0-4) group, A2 represents children (age 5-14) group, A3 represents adolescents (15)(16)(17)(18)(19), A4 represents young adults (age 20-34), A5 represents adults (age 35-49) A6 represents elderly (age 50 and above). The s1, s2, s3,s4 and s5 are ...
MODELLING SOLID WASTE GENERATION USING LEFKOVITCH MATRIX
Article
Full-text available
  • Jul 2016
Population projection models provide a vital tool for planning, not only in the provision of social services but also in various other fields such as conservation strategies for endangered plants and animal species. As the name implies, these models have traditionally been used in ecological and population studies. In this article, the Lefkovitch matrix is used to predict solid waste generation by linking population growth with solid waste generation. The results indicate gradual population growth results in gradual increase in solid waste generation rate. Variability analysis shows that survival rates of the first three groups have significant effect on the population growth rate and so solid waste generation rate. The results are based on the solid waste characterisation and quantification study carried out in Dar es Salaam, Tanzania.
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... In a successful management plan for control or removal of invasive species, the rate of removal must exceed the growth of the population (Bomford and O'Brien 1995). Therefore, the use of models to estimate population trends is essential (Kajin et al. 2012). The predictive value of those demographic models can be enhanced by incorporating information on the mating system (Wildermuth et al. 2013). ...
Mating Strategies of Invasive Versus Indigenous Crayfish: Multiple Paternity as a Driver for Invasion Success?
Article
Full-text available
  • Dec 2021
The invasive spiny-cheek crayfish (Faxonius limosus) has been able to colonize many European waterbodies since its first introduction into Europe, threatening the indigenous crayfish fauna. The remarkable reproductive plasticity of this species has been suggested as an important factor contributing to the alarming invasiveness of this species. In this study, we compared the reproductive strategies of an invasive (F. limosus) and a sympatric indigenous crayfish (Pontastacus leptodactylus). We questioned if the reproductive abilities, namely parthenogenesis and multiple paternity, may contribute to an ongoing invasion process in the Lower Danube. Using microsatellites, we genotyped the mothers and their offspring from 11 clutches of F. limosus and 18 clutches of P. leptodactylus. While no parthenogenesis has been found in F. limosus populations, multiple paternity has been detected for the first time in both species, with comparable incidence. The results of the study indicate that multiple paternity does not play a dominant role in the successful colonization of F. limosus in the Danube. However, the presented results have to be regarded as a pilot study, with a limited number of samples and loci investigated. Given the relevance of mating system knowledge for management measures, future studies with larger sample number could provide valuable contributions to ongoing conservation actions.
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... PVAs have been central to population conservation for decades (Soule 1987;Gerber and González-Suárez 2010;Chirakkal and Gerber 2010;Saunders et al., 2018). PVAs can be conducted aspatially based on demographic vital rates with the use of a standard Leslie matrix life-table analysis and simulation projection models (Kajin et al., 2012), using programs such as Vortex and RAMAS (Lindenmayer et al., 2000;LaRue and Nielsen 2016). PVAs also can be spatialised with individual-based simulation models (Watkins and Rose 2017) such as HexSim (Schumaker and Brookes 2018) or through spatially and temporally explicit population simulations (Visitin et al., 2020). ...
Population viability analysis using Bayesian networks
Article
  • Nov 2021
  • ENVIRON MODELL SOFTW
Traditional population viability analysis (PVA) does not address the degree of measurement error or spatial and temporal variability of vital rate parameters, potentially leading to inappropriate conservation decision-making. We provide a methodology of applying Bayesian network (BN) modeling to PVA addressing these considerations, particularly for species with complex stage-class structures. We provide examples of three species from eastern Australia - hip pocket frog (Assa darilingtoni), squirrel glider (Petaurus norfolcensis) and giant burrowing frog (Heleioporus australiacus), comparing traditional matrix-based PVA with BN model analyses of mean stage abundance, quasi-extinction probability, and interval threshold extinction risk. Both approaches project similar population sizes, but BN PVA gave more clearly identifiable thresholds of population changes and extinction levels. The PVA BN uniquely represents complex stage-class structures and in a single network, including variation and uncertainty propagation of vital rates, to better inform conservation management decisions.
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... Usually, this fertility is zero for an individual in a pre-or post-productive age and positive fertility is observed for individuals of a reproductive age. These structured systems are employed to model the changes in a population of organisms and are widely used in studies of biology, ecology and demography to determine the growth of a population, (see, e.g., Caswell, 2001;Kajin et al., 2012;Leslie, 1948;Lefkovitch, 1965). Note that this kind of dynamic process is represented by a discrete linear system with nonnegative states and nonnegative inputs. ...
Parameter Identification and Estimation For Stage–Structured Population Models
Article
Full-text available
  • Jun 2019
  • INT J AP MAT COM-POL
A stage-structured population model with unknown parameters is considered. Our purpose is to study the identifiability of the model and to develop a parameter estimation procedure. First, we analyze whether the parameter vector can or cannot uniquely be determined with the knowledge of the input-output behavior of the model. Second, we analyze how the information in the experimental data is translated into the parameters of the model. Furthermore, we propose a process to improve the recursive values of the parameters when successive observation data are considered. The structure of the state matrix leads to an analysis of the inverse of a sum of rank-one matrices.
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... The use of such matrix has been found predominantly in the context of various living organisms and animals as discussed in Jensen (1974), Werner and Caswell (1977), Cheke (1978), Van Groenendael et al. (1988), and Gauthier et al. (2007). It has been used in demography and health by Gross et al. (2006), Kajin et al. (2012), Thomas and Clark (2008), and few others. Changes in the policy parameters over time could be incorporated in a model based on Leslie matrix by modifying the matrix elements appropriately. ...
Projection of Indian Population by Using Leslie Matrix with Changing Age Specific Mortality Rate, Age Specific Fertility Rate and Age Specific Marital Fertility Rate
Article
  • Jun 2013
The importance of projection in national and state level planning and policy formulation is quite well recognized in any country which attempts at achieving sustainability in human development and improving the quality of life. Population projection exercises are basically a part of forecasting growth of human population in the future years over a time horizon. There are various means of extrapolating past trend of change in human population over the future years. These means are determined by the assumptions that are made on the determinants of population change such as time, pattern of changes in fertility, mortality, and migration and other associated factors. The success of population projection depends not only on the technique of projection but also on the proximity of the assumptions to reality so that changes in the future years get estimated with least possible errors. Projections, however, might not suffice when there are significant deviations from the assumption that prevailing conditions would continue unchanged in the future. Also, the projection might not be satisfactory due to failure to incorporate adequately the changes in the policy parameters, technological changes, changes in the migration pattern, etc. Forecasting attempts at overcoming these drawbacks by incorporating the elements of judgment in the projection exercise. Forecasting enjoys the advantage of being based upon one or more assumptions that are likely to be realized in the future years. Thus, forecasts give more realistic picture of the future.
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... Esta condição permitiu, por exemplo, comparar o desempenho de diferentes métodos de estimativas de parâmetros populacionais, comparando abordagens (probabilística ou determinista), localidades de características diferentes, desenhos amostrais variados e espécies diferentes. No mesmo contexto, avançando cada vez mais em termos de sofisticação das análises empregadas, o conhecimento teórico avançou e as teorias ecológico--evolutivas puderam ser cada vez mais compreendidas e abordadas não apenas empiricamente (Kajin et al. 2012. Além disso, outras abordagens de populações foram realizadas dentro deste projeto, sob a coordenação do Prof. ...
Long-term empirical studies of small mammals: the contribution of Dr. Rui Cerqueira to population biology
Article
Full-text available
  • Oct 2015
This article is a review of the developmental trajectory of the scientific reasoning on population biology in small mammal studies led by Prof. Rui Cerqueira throughout his career. The multidisciplinarity joining ecology, demography, genetics, epidemiology, biogeography, taxonomy and systematics, and the different approaches guided by relevant ecological questions have always been present in his research projects which led to the progress of the knowledge described here. We report three empirical studies of long-term population biology: the study in Restinga de Barra de Marica, RJ, a pioneer study in many aspects at the time; the study in Sumidouro, RJ, developed in collaboration with the Fundação Oswaldo Cruz, RJ, related to the role of wild rodents in the transmission dynamics of schistosomiasis; and the study in Garrafão, municipality of Guapimirim, RJ. The latter arose from ideas generated in the study of Restinga de Barra de Marica on movements and population structure of some marsupial species and is still in progress, being one of the longest population dynamics studies of small mammals in Brazil. New and important results were obtained in these three projects using the capture-mark-recapture technique that permitted to obtain high quality data. The contribution of Prof. Cerqueira raised the field of mammalogy and in particular the population biology of small mammals in Brazil to a level of international excellence, both in terms of generating new knowledge and training human resources, as well as in terms of progress and sophistication of methodologies and analyses.
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Analysis of the population growth model to maintain environmental stability in the Province of The Bangka Belitung Islands
Article
Full-text available
  • Nov 2022
The province of the Bangka Belitung archipelago is one of the relatively young provinces in Indonesia. Based on BPS data in 2021, the population of Bangka Belitung is around 1.473.165. This population data is very useful for formal demography and will influence local government policies, economic stability and environmental balance. In this study, the population growth of the Bangka Belitung Islands Province was modeled using the Leslie model. The model show that the population of the Bangka Belitung Islands Province will decrease slowly.
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Epidemic spreading by indirect transmission in a compartmental farm
Article
  • Dec 2020
  • APPL MATH COMPUT
In this paper, we present a discrete dynamic system which describes an epidemic spreading within a single farm, where animals are separated into batches. In this model, we consider an indirect transmission of the disease coming from the bacteria remaining in the reservoir and taking into account the transfer of bacteria between adjacent compartments. In our model, tridiagonal matrices of non-negative blocks are involved. The development of the matrix spectral properties allows us to improve our understanding of the epidemic spreading within a farm with the above mentioned characteristics. Based on the results obtained, we have determined some bounds to obtain the maximum number of batches and the maximum population in each batch to ensure that the disease dies out.
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A multilevel functional data method for forecasting population, with an application to the United Kingdom
Article
Full-text available
  • Jul 2016
  • INT J FORECASTING
Cohort component models are often used to model the evolution of an age-specific population, and are particularly useful for highlighting the demographic component that contributes the most to the population change. Recently, most of the attention has been devoted to the estimation of four specific demographic components, namely mortality, fertility, emigration and immigration. Many methods take a deterministic viewpoint, which can be quite restrictive in practice. The statistical method that we propose is a multilevel functional data method, where both mortality and migration are modelled and forecast jointly for females and males. The forecast uncertainty associated with each component is incorporated through parametric bootstrapping. Using the historical data for the United Kingdom from 1975 to 2009, we found that the proposed method shows a good in-sample forecast accuracy for the holdout data between 2001 and 2009. Moreover, we produce out-of-sample population forecasts from 2010 to 2030, and compare our forecasts with those produced by the Office for National Statistics.
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Bayesian Inference and Data Cloning in the Calibration of Population Projection Matrices
Article
  • Dec 2015
Discrete time models are used in Ecology for describing the dynamics of an age-structured population. They can be introduced from a deterministic or from a stochastic viewpoint. We analyze a stochastic model for the case in which the dynamics of the population is described by means of a projection matrix. In this statistical model, fertility rates and survival rates are unknown parameters which are estimated by using a Bayesian approach and also data cloning, which is a simulation-based method especially useful with complex hierarchical models. Both methodologies are applied to real data from the population of Steller sea lions located in the Alaska coast since 1978–2004. The estimates obtained from these methods show a good behavior when they are compared to the nonmissing actual values.
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Sensitivity of willow grouse Lagopus lagopus population dynamics to variations in demographic parameters
Article
Full-text available
  • Mar 1996
In order to estimate the potential importance of each of the demographic traits egg, chick, and apparent winter survival on willow grouse Lagopus lagopus population dynamics, a sensitivity/elasticity analysis on a general model for the growth rate of breeding hen willow grouse numbers was performed. The demographic parameters used were taken from a 21- year study of willow grouse population dynamics on a Norwegian island. The growth rate of breeding hen numbers may be sensitive to changes in a parameter either because of the structure of the model or because of a substantial between year variation in the parameter. Therefore, three measures of parameter importance were used; sensitivity, elasticity and an actual elasticity coefficient (AE-coefficient). Sensitivity is an unsealed measure of the impact of a parameter, and thus cannot be used for comparisons between different parameters Elasticity is a scaled measure of sensitivity allowing direct comparison between effects of different parameters on the growth rate (λ). To interpret the true effect of a demographic parameter on λ, between year variations in the parameter must be accounted for. The actual elasticity coefficient of each parameter is the product of the elasticity coefficient and the coefficient of variation. CV, for the demographic parameter in question. Sensitivity, elasticity and AE-coefficients showed that apparent winter survival of juveniles had the largest potential impact on changes in λ, followed by egg survival, chick survival and adult apparent winter survival. Considering elasticity coefficients alone, apparent winter survival compared to survival from egg laying to four weeks alter hatching had 2.3 times the impact on λ. However, between year variation in apparent winter survival was far less than between year variation in survival from egg laying to four weeks after hatching, which leads to them both having a similar impact on λ. Management efforts on willow grouse have largely been devoted to enhancing survival from egg laying to four weeks after hatching which was believed to be the single most important factor determining population size. The results presented suggest that equal attention should be paid to winter survival. One cause of reduced winter survival may be hunting. A model incorporating the area specific survival and chick production resulted in λ = 1, when chick production was 1.8 chicks per adult and when there was no hunting. The model suggested that grouse could not be hunted in four of the 21 years with out reducing the population size on the island and in many years only a modest harvest could be recommended.
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Pod-Specific Demography of Killer Whales (Orcinus Orca)
Article
Full-text available
  • Jul 1993
Killer whales live in stable social groups, called @'pods.@' It has been suggested that the structure of such groups may influence the vital rates, and hence the fitness, of their members. To test this suggestion we used data from a long-term study of killer whales in the Pacific Northwest (Bigg et al. 1990). We constructed stage-classified matrix population models for the entire population, two sub-populations, and individuals pods. The population growth rate for the entire population is @l = 1.0254, with 90% bootstrap confidence interval from 1.0178 to 1.0322. The mean female population stage distribution is not significantly different from the predicted stable stage distribution. Population growth rate is most sensitive to changes in adult and juvenile survival, followed by fertility Factors that cause even small changes in survival will thus have a large impact on population growth. Pod-specific growth rates range from @l = 0.9949 to @l = 1.0498. Most of the interpod variance in growth rate is due to variance in adult reproductive output. Randomization tests show that this variance is not significantly greater than expected on the basis of variation in individual life histories within the population. We conclude that there is no evidence for an effect of social structure on pod-specific population growth rate. The restriction of population growth rates to such a narrow range suggests, but does not prove, a possible role of density-dependent processes.
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Models to Evaluate Headstarting as a Management Tool for Long-Lived Turtles
Article
Full-text available
  • May 1996
Most turtle species suffer high mortality in their first year, have a long juvenile period, and can live for decades once they reach adulthood. Conservationists have implemented a number of recovery plans for threatened turtle populations, including experimental ''headstart'' programs. Headstarting involves the captive rearing of hatchlings from eggs collected in the wild. The hatchlings are held for several months to help them avoid high mortality in their first year. It is hoped that these turtles survive and grow like wild turtles after release. The purpose of our study was to evaluate headstarting as a management tool for threatened turtle populations. We critically examined the population-level effects of headstarting with a series of deterministic matrix models for yellow mud turtles (Kinosternon flavescens), a ''non-threatened,'' well-studied species, and endangered Kemp's ridley sea turtles (Lepidochelys kempi). We show that management efforts focused exclusively on improving survival in the first year of life are unlikely to be effective for long-lived species such as turtles. Population projections for both turtles predict that headstarting can augment increasing populations when adult survival is returned to or maintained at high levels, provided that headstarted juveniles are as vigorous as wild turtles. However, when subadult and adult survival is reduced, headstarting cannot compensate for losses in later stages. Proportional sensitivity (elasticity) analyses of stage-based matrix models indicated that annual survival rates for subadult and adult turtles are most critical; small decreases in the survival of older turtles can quickly overcome any potential benefits of headstarting. In general, the biological benefits of headstarting programs may be overestimated for turtles, and a careful examination of stage-specific mortality sources, demography, and life history can guide us toward more effective management strategies.
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Visualizing Multivariate Selection
Article
Full-text available
  • Sep 1989
Recent developments in quantitative-genetic theory have shown that natural selection can be viewed as the multivariate relationship between fitness and phenotype. This relationship can be described by a multidimensional surface depicting fitness as a function of phenotypic traits. We examine the connection between this surface and the coefficients of phenotypic selection that can be estimated by multiple regression and show how the interpretation of multivariate selection can be facilitated through the use of the method of canonical analysis. The results from this analysis can be used to visualize the surface implied by a set of selection coefficients. Such a visualization provides a compact summary of selection coefficients, can aid in the comparison of selection surfaces, and can help generate testable hypotheses as to the adaptive significance of the traits under study. Further, we discuss traditional definitions of directional, stabilizing, and disruptive selection and conclude that selection may be more usefully classified into two general modes, directional and nonlinear selection, with stabilizing and disruptive selection as special cases of nonlinear selection.
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Monitoring Plant and Animal Populations
Article
  • Jan 2002
ELZINGA et al. have brought together a wealth of experience from their employment in private, governmental, educational and voluntary organizations to produce Monitoring Plant and Animal Populations. This knowledgeable book is intended to assist a range of audiences, from students to experienced wildlife biologists, encouraging them to produce high-quality population monitoring studies, with adaptations for community monitoring.
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How to Keep Fit in the Real World: Elasticity Analyses and Selection Pressures on Life Histories in a Variable Environment
Article
  • Jan 1996
Most life-history theory assumes the environment is invariant. For the first time, analytical and numerical techniques were employed to investigate the impact of environmental variability on selection pressures (elasticities = proportional sensitivities) on a range of life histories. We find that the impact of variability is influenced significantly by the amount of variability an organism experiences (more variability affects selection pressures more), the correlations between variations among the vital rates (negative correlations are more likely to relax selection on fecundities and increase it on survival rates), and the life history in question (shorter life histories are more affected). In addition, the impact of a variable environment on the elastici-ties of life histories is sensitive to the sampling distribution used to generate the variability, and it is particularly sensitive to extreme values, such as those caused by occasional catastrophic events. The elasticities of life histories in highly variable environments may bear little relationship to those in a constant environment. In detailed optimality or evolutionarily stable strategy (ESS) modeling, variability in vital rates as small as a standard deviation being 10%-15% of the mean may appreciably alter the conclusions. Thus, it may be very important to consider the possible impact of environmental stochasticity and not to assume that it has no effect.
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A Matrix Approach to the Management of Renewable Resources, with Special Reference to Selection Forests-Two Extensions
Article
  • Nov 1966
The management of a renewable resource, classified by size classes, is considered. A mathematical model which predicts the stable structure of such a resource is developed. Data on the percentage recruitment of organisms from one class to the class above and data on the regeneration of young organisms are used as elements of a matrix. The largest real latent root of this matrix gives the maximum exploitation, and the latent vector associated with this root gives the stable structure. The model is illustrated by reference to a Scots pine forest in Inverness-shire. An iterative process for finding the latent root and vector is described. This matrix method is compared with a geometric method.
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Second Derivatives of Population Growth Rate: Calculation and Applications
Article
  • Apr 1996
The rate of increase of a population depends on the age- or stage-specific vital rates (survival, reproduction, etc.). Sensitivity analysis explores this dependence by calculating the fir st derivatives of the rate of increase with respect to each of the vital rates, and has become a standard part of demographic analysis. This paper presents a formula for the second derivatives, for the case where the rate of increase is measured by the dominant eigenvalue lambda of a population projection matrix. Because the first derivatives of lambda are written in terms of the eigenvectors, the second derivatives of lambda can be written in terms of the first derivatives of the eigenvectors. I present examples of the second derivatives of lambda in age- and size-classified life cycles, and discuss several potential applications of the results: (1) the perturbation analysis of elasticity patterns with applications to conservation biology, (2) the sensitivity analysis of stochastic growth rates, (3) the inclusion of second-order terms in the decomposition analysis of life table response experiments, and (4) the analysis of directional, stabilizing, and correlational selection on life history traits.
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