Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

A density-dependent Leslie matrix model

A density-dependent Leslie matrix model introduced in 1948 by Leslie is mathematically analyzed. It is shown that the behavior is similar to that of the constant Leslie matrix. In the primitive case, the density-dependent Leslie matrix model has an asymptotic distribution corresponding to the logistic equation. However, in the imprimitive case, the asymptotic distribution is periodic, with period depending on the imprimitivity index.     

Population growth with stochastic fluctuations in the life table

Monte Carlo simulations with the Leslie matrix and similar population models show that as the variance in survivorship or fecundity increases, the expected population growth rate decreases. This is attributed to Jensen's inequality with the observation that the rate of increase is a concave function of age-specific survivorship and fertility rates. Applications of this observation are advised for demographic studies, population simulation, optimal harvest strategies, and natural selection for variance in fertility and survivorship rates.     

Risk spreading as an adaptive strategy in iteroparous life histories

It has long been conjectured, though without satisfactory proof, that life tables with a long reproductive span are advantageous in an environment where fecundity or immature survival rates fluctuate randomly. In the present analysis we recast the nonlinear Leslie matrix problem as an autoregressive time series model for the birth rate, with random addition and removal of newborn. This transformation renders the model linear with respect to the environmental variation, allowing ready solution for the ultimate population size and for the conditions resulting in stationarity of the population distribution. We show that for life tables where the fecundities of all adult age classes are the same (no restrictions are put on the survivorship schedule, or on the age at first reproduction), and where density dependence operates via total adult density, the realized growth rate is less than the growth rate calculated from the mean Leslie matrix associated with the population's growth history. The degree of the discrepancy increases with the environmental variability, and decreases with iteroparity, thus completing a proof which confirms the correctness of the initial conjecture for a class of biologically reasonable lifetable models.     

Compensation in individual growth rates and its influence on lake trout population dynamics in the Michigan waters of Lake Superior

The role of compensatory mechanisms in the population dynamics of lake trout in the Michigan waters of Lake Superior was explored during three time periods: the pre-sea lamprey period, prior to 1950 when lake trout were at a relatively high abundance and the fishery was the primary source of lake trout mortality; the sea lamprey dominant period, from 1951 to 1961 when lake trout were at a very low abundance due to sea lamprey predation and overexploitation; and currently, from 1985 to 1993 when wild lake trout abundance was at a moderate level. The role of compensatory changes in growth and fecundity rates of lake trout in the Michigan waters of Lake Superior was evaluated using a life table approach. Individual growth and fecundity rates were calculated and compared between time periods. These rates were used to determine age-specific fecundity which, along with age-specific survival, were incorporated into a Leslie projection matrix to calculate the finite rate of population increase (λ). Individual growth rates and age-specific fecundity rates changed in response to the different levels of lake trout abundance during each of the study periods. Lake trout during the sea lamprey dominant period, which experienced the lowest abundance and highest mortality levels, exhibited the fastest individual growth rates and the highest age-specific fecundity. These high rates contributed to the relatively large compensatory scope exhibited by lake trout during the sea lamprey dominant period as compared to lake trout during the pre-sea lamprey or the current periods which are associated with higher levels of abundance.     

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. The level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.     

Modeling managed monkey populations: sustainable harvest of longtailed macaques on a natural habitat island

Computer simulation of population dynamics can be useful in managing harvested populations of monkeys on islands. Between 1988 and 1991, 420 adult female and 58 adult male simian retrovirus-free Macaca fascicularis were released onto Tinjil Island, Indonesia, to provide the nucleus for a free-ranging breeding colony. Natural habitat breeding facilities are excellent alternatives to wild trapping and compound breeding, maximizing the health and well-being of animals destined for essential biomedical research. To avoid a population crash, the number of offspring that can be harvested annually must be based on life table characteristics such as age-specific natality and mortality. We used a modified Leslie matrix to model changes in female population size over 26 years. First, we assumed that all 420 females were released simultaneously and varied the annual birth rate (50%, 60%, 70%), survival rate, and number of offspring harvested per year. Assuming high survival and birth rates vs. low rates, about four times as many female offspring could be harvested annually from a stable population (87 vs. 20 offspring). Terminal population size after 26 years did not differ much across rates modeled (568–696 females). Second, we modeled the number of females actually released (including the recent addition of 42 new female breeders) and harvested (averaging 49 annually 1991–1994), and projected the population through 2014. This indicated that threshold harvest rates and terminal population sizes increased considerably over the first model, assuming intermediate (78 harvested, 952 females) and high (152 harvested, 1,331 females) rates of survivorship and natality, but were unchanged assuming low rates (20 harvested, 559 females). A review of the literature and field observations on Tinjil suggest that actual birth and survival rates resemble the intermediate values modeled. If so, the present density on the island, projected to be ∼215 males and females per square kilometer, is approaching carrying capacity. The high values are realistic upper limits. If actual survivorship and birth rates are at the high end of those modeled, the island's population may be on the verge of rapid expansion, requiring increased harvest and provisioning. © 1996 Wiley-Liss, Inc.     

On two-sex models leading to stable populations

A mathematical, 2-sex, stable-population model that treats sex and age simultaneously was developed. The birth function is expressed in the form of an integral formula which when solved yields the intrinsic growth rate. Some of the concepts involved include the intrinsic sex ratio, the intrinsic age-specific birthrates, and the gross and net reproduction rates for both sexes. The new model demonstrates that both sexes can coexist in a stable population, whereas in the 1-sex model the intrinsic rates are internally inconsistent in regard to sex. 7 forms of the model are discussed and applied to data for the United States from 1963.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Patterns of age-specific means and genetic variances of mortality rates predicted by the mutation-accumulation theory of ageing

A general quantitative genetic model of mutations with age-specific deleterious effects is developed. It is shown that, for the simplest case of a species with age-independent reproductive rates and extrinsic adult mortality rates, and no pleiotropic effects of age-specific mutations, exponential increases with age of both the mean and additive genetic variance of age-specific mortality rates are expected. Models where age-specific mutations have pleiotropic effects on mortality that extend either throughout adult life, or are confined to juvenile stages, produce equilibria with exponential increases in the mean and additive variance of mortality rates during much of adult life. However, the rates of increase diminish late in life, and can even become zero. Predictions concerning the additive genetic correlations in mortality rates between different ages are also developed. The predictions of the models are compared with data on humans and Drosophila.     

Age-dependent models of population growth

P. H. Leslie (1945, Biometrika, 33, 183–212) (and others) introduced vector-matrix growth difference equations to model populations in which birth and death rates are age-dependent. We develop differential versions of these equations in both the unrestricted growth and logistic cases. We find that the vector logistic equations (difference and differential) are explicitly solvable in terms of solutions of the unrestricted equations, even when vital rates vary with time. These explicit solution formulas make it easy to determine the behavior of solutions as time goes on.     

Periodic and quasi-periodic behavior in resource-dependent age structured population models

To describe the dynamics of a resource-dependent age structured population, a general non-linear Leslie type model is derived. The dependence on the resources is introduced through the death rates of the reproductive age classes. The conditions assumed in the derivation of the model are regularity and plausible limiting behaviors of the functions in the model. It is shown that the model dynamics restricted to its ω-limit sets is a diffeomorphism of a compact set, and the period-1 fixed points of the model are structurally stable. The loss of stability of the non-zero steady state occurs by a discrete Hopf bifurcation. Under general conditions, and after the loss of stability of the structurally stable steady states, the time evolution of population numbers is periodic or quasi-periodic. Numerical analysis with prototype functions has been performed, and the conditions leading to chaotic behavior in time are discussed.     

Modeling Age-Specific Mortality for Countries with Generalized HIV Epidemics

Background

In a given population the age pattern of mortality is an important determinant of total number of deaths, age structure, and through effects on age structure, the number of births and thereby growth. Good mortality models exist for most populations except those experiencing generalized HIV epidemics and some developing country populations. The large number of deaths concentrated at very young and adult ages in HIV-affected populations produce a unique ‘humped’ age pattern of mortality that is not reproduced by any existing mortality models. Both burden of disease reporting and population projection methods require age-specific mortality rates to estimate numbers of deaths and produce plausible age structures. For countries with generalized HIV epidemics these estimates should take into account the future trajectory of HIV prevalence and its effects on age-specific mortality. In this paper we present a parsimonious model of age-specific mortality for countries with generalized HIV/AIDS epidemics.

Methods and Findings

The model represents a vector of age-specific mortality rates as the weighted sum of three independent age-varying components. We derive the age-varying components from a Singular Value Decomposition of the matrix of age-specific mortality rate schedules. The weights are modeled as a function of HIV prevalence and one of three possible sets of inputs: life expectancy at birth, a measure of child mortality, or child mortality with a measure of adult mortality. We calibrate the model with 320 five-year life tables for each sex from the World Population Prospects 2010 revision that come from the 40 countries of the world that have and are experiencing a generalized HIV epidemic. Cross validation shows that the model is able to outperform several existing model life table systems.

Conclusions

We present a flexible, parsimonious model of age-specific mortality for countries with generalized HIV epidemics. Combined with the outputs of existing epidemiological and demographic models, this model makes it possible to project future age-specific mortality profiles and number of deaths for countries with generalized HIV epidemics.     

Backwrd population projection by a generalized inverse

The Leslie population projection matrix may be used to project forward in time the age distribution or age-sex distribution of a population. As it is a singular matrix, it does not have an inverse, and so it is not clear that there is a corresponding procedure for backward projection. In terms of the eigenvalues and eigenvectors of the Leslie matrix, certain generalized inverses are constructed that can sometimes be used advantageously for backward projection.     

Deconstructing Ixodes ricinus: a partial matrix model allowing mapping of tick development,mortality and activity rates

A stage‐structured Leslie matrix model of a partial, discrete population of Ixodes ricinus (Linnaeus) (Ixodida: Ixodidae) ticks was developed to elucidate the impact of climate trends on the distribution and phenology of this species in the western Palaearctic. The model calculates development and mortality rates for each instar and evaluates recruitment rates based on the development of the tick population. The model captures the changes in development and mortality rates, providing a coherent index of performance correlated with the tick's geographic range. Maximum development rates are recorded for latitudes south of 36 °N and are spatially correlated with sites of maximum temperature, highest saturation deficit and highest mortality. The maximum available developmental time (the total annual time during which temperature allows development) for I. ricinus in the western Palaearctic is < 45% of the total year. North of 60 °N, available developmental time decreases sharply to only 15% of the year. The latitudinal boundary at which survival rates sharply drop is 43–46 °N, clearly delimiting the classically recognized extent of the main tick populations. The pattern of activity for larval–nymphal synchrony shows a clear west–east pattern. The model demonstrates the impact of climate according to tick stage and geographic location, and provides a practical framework for testing how the tick's lifecycle is affected by climate change.     

Age-specific birth rates of California sea lions (Zalophus californianus) in the Gulf of California, Mexico

Estimates of demographic parameters are essential for assessing the status of populations and assigning conservation priority. In light of the difficulties associated with obtaining such estimates, vital rates are rarely available even for well-studied species. We present the first estimates of age-specific birth rates for female California sea lions ( Zalophus californianus ) >10 yr of age. These rates were estimated from the reproductive histories of five cohorts of animals branded as pups between 1980 and 1984 at Los Islotes colony in the Gulf of California, Mexico. Age-specific birth rates varied among age classes and ranged between 0.06 and 0.80. The highest birth rates were observed for females between 10 and 15 yr of age, with decreased birth rates among older females. The effect of age, year, and resighting effort were explored using logistic regression analysis. Based on Akaike Information Criteria, birth rates were best explained by female age, while year and resighting effort did not have a significant effect. The odds ratio of producing a pup decreased with age but did not change significantly for middle-aged females. Our estimates of age-specific birth rates are consistent with general patterns observed for other large vertebrates.     

The relative importance to population increase of fluctuations in mortality,fecundity and the time variables of the reproductive schedule

Summary Previous authors have used simple models to investigate the relative importance to population increase of variations in the total and age-specific reproductive rates. But while acknowledging that the latter were the product of the age specific birth and death rates, they have used their models only to investigate changes in total or age-specific birth rates and have not been concerned with variations in death rates. This paper extends the use of Lewontin's (1965) model, to a wide range of values of r, the exponential rate of population increase. It shows how the relative importance of changes in certain life-history features can change with r and be reversed when r is near to zero. It is also shown that variations in mortality rate are not necessarily best expressed in analogous terms to variations in birth rate. If more suitable terms are used it is seen that changes in mortality rate can be of varying importance depending on the existing mortality rate. They can be overwhelmingly important when the mortality rate is high.     

The structure and dynamics of woodreed Calamagrostis canescens population: a modelling approach

A scale of ontogenetic states has been developed for woodreed Calamagrostis canescens, a perennial species dominating the grass layer of fell forest areas. The population structure is considered as a set of age-stage groups of individuals differing both in the ontogenetic stage and the chronological age measured in years. to describe the dynamics through years a special kind of matrix formalism has been proposed which is reducible neither to the classic Leslie matrix for an age-structured population, nor to the well-known Lefkovitch matrix for a stage-structured one, and which does not suffer from excessiveness of the "two-dimensional" representation for the structure implying the projection matrix of a block pattern. It has been shown however that the protection matrix corresponding to C. canescens life-history graph embodies the canonical features of matrix formalism for structured population dynamics, such as the exponential population growth or decline, the convergence to a stable equilibrium structure, the calculable indicator of growth/decline/equilibrium (i.e., a measure of the population reproductive potential) as well as possibility to determine the relative reproductive value of each group. On the other hand, "left-sidedness of the age spectrum", a property that is often observed in real populations and is inherent in Leslie models of growing populations, may fail in the age-stage-structured model. The aggregation of age-stage groups into the age classes is possible only under special strict relationship among the age-stage-specific vital rates of the population. The both circumstances serve a methodical indication that an additional dimension such as the stages, for example, ought to be introduced into the age structure of the model population.     

Predator-prey models in periodically fluctuating environments

A class of ordinary or integrodifferential equations describing predator-prey dynamics is considered under the assumption that the coefficients are periodic functions of time. This class is characterized by the logistic behaviour of the prey in the absence of predators and it includes the Leslie model. We show that there exists a periodic solution provided that the average of the predator's intrinsic rate of increase is greater than a critical value. We use well-known results in bifurcation theory for nonlinear eigenvalue problems, as well as an extension to the case of non-globally defined operators of some recent results on the global nature of branches of solutions.