Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Year-class fluctuations in vendace,Coregonus albula (Linnaeus): Who's got the upper hand in intraspecific competition?

The vendace Coregonus albula (L.) populations in the lakes Mjøsa and Osensjøen exhibited fluctuating year-class strength. In Mjøsa, a strong year-class emerged every third year, except for the four year period between the strong year-classes 1969 and 1973. The difference between the strong and weak year-classes decreased from the 1960s, through the 1970s to the 1980s. The Mjøsa vendace matured sexually at age 2 +, and more than ten sexually mature age-groups were present in the population. Growth ceased at maturation, and asymptotic length was 23.6 cm. In Osensjøen, one strong year-class (1969) dominated the population during the period 1976–1987. The Osensjøen vendace matured sexually at age 3, and more than 15 sexually mature age-groups were present in the population. Growth ceased at maturation, and asymptotic length was 28.4 cm. In both lakes, vendace fed on crustacean zooplankton in the epilimnion throughout summer and autumn. Our data indicate that regular year-class oscillations occur as a result of the juvenile survival being negatively correlated to the number of adults.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

Asymptotic distribution of an index for disease clustering

Tango (1984, Biometrics 40, 15-26) proposed an index for disease clustering in time, applicable to grouped data with the assumption that the population at risk remains fairly uniform over the study period. However, the asymptotic distribution of the index derived under the hypothesis of no clustering was rather complex for simple use. Recently, Whittemore and Keller (1986, Biometrics 42, 218) and Whittemore et al. (1987, Biometrika 74, 631-635) proved that the distribution of the index is asymptotically normal. The present paper indicates that their approximation may be poor for moderately large sample sizes and suggests a central chi-square distribution as a better approximation to the asymptotic distribution of this index.     

Patterns of growth and body condition in sea otters from the Aleutian archipelago before and after the recent population decline

1. Growth models for body mass and length were fitted to data collected from 1842 sea otters Enhydra lutris shot or live-captured throughout south-west Alaska between 1967 and 2004. Growth curves were constructed for each of two main year groups: 1967-71 when the population was at or near carrying capacity and 1992-97 when the population was in steep decline. Analyses of data collected from animals caught during 2004, when the population density was very low, were precluded by a small sample size and consequently only examined incidentally to the main growth curves. 2. Growth curves demonstrated a significant increase in body mass and body length at age in the 1990s. Asymptotic values of body mass were 12-18% higher in the 1990s than in the 1960s/70s, and asymptotic values for body length were 10-11% higher between the same periods. Data collected in 2004 suggest a continued increase in body size, with nearly all data points for mass and length falling significantly above the 1990s growth curves. 3. In addition to larger asymptotic values for mass and length, the rate of growth towards asymptotic values was more rapid in the 1990s than in the 1960s/70s: sea otters reached 95% of asymptotic body mass and body length 1-2 years earlier in the 1990s. 4. Body condition (as measured by the log mass/log length ratio) was significantly greater in males than in females. There was also an increasing trend from the 1960s/70s through 2004 despite much year-to-year variation. 5. Population age structures differed significantly between the 1960s/70s and the 1990s with the latter distribution skewed toward younger age classes (indicating an altered lx function) suggesting almost complete relaxation of age-dependent mortality patterns (i.e. those typical of food-limited populations). 6. This study spanned a period of time over which the population status of sea otters in the Aleutian archipelago declined precipitously from levels at or near equilibrium densities at some islands in the 1960s/70s to < 5% of estimated carrying capacity by the late 1990s. The results of this study indicate an improved overall health of sea otters over the period of decline and suggest that limited nutritional resources were not the cause of the observed reduced population abundance. Our findings are consistent with the hypothesis that the decline was caused by increased killer whale predation.     

Genetic steady-state under BLUP selection for an infinite and homogeneous population with discrete generations

A matrix derivation is proposed to analytically calculate the asymptotic genetic variance-covariance matrix under BLUP selection according to the initial genetic parameters in a large population with discrete generations. The asymptotic genetic evolution of a homogeneous population with discrete generations is calculated for a selection operating on an index including all information (pedigree and records) from a non-inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is independent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction error variance depends on selection. Furthermore, records of several generations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order not to overestimate the asymptotic genetic gains and not to underestimate the asymptotic genetic variances.     

Asymptotic worst-case mixing in simple demographic models of HIV/AIDS.

The majority of published modeling work regarding the impact of mixing patterns among subgroups on the spread of HIV infection assumes either that the overall population size remains constant, the aggregate immigration to the population occurs at a constant annual rate, or that no immigration occurs and the population in question declines due to HIV or other causes. In this paper, immigration rates are modeled as simple functions of population size and may be interpreted as aggregate birth rates. This assumption implies asymptotic exponential growth in the disease-free population as long as per capita birth rates exceed per capita mortality rates. The introduction of HIV infection to such a population may change this situation, and the asymptotic population growth rate can be reduced substantially as a result. The specific manner in which this occurs depends in part upon difficult to observe mixing patterns among those with different sexual activity rates. Rather than attempting to explicitly model a variety of mixing patterns, a bound on the impact of worst-case mixing is produced, where "worst case" refers to the mixing pattern that maximizes the asymptotic prevalence of infection, which is equivalent to minimizing the asymptotic population growth rate. These new techniques are illustrated with a numerical example.     

The generality of management recommendations across populations of an invasive perennial herb

Demographic models are widely used to produce management recommendations for different species. For invasive plants, current management recommendations to control local population growth are often based on data from a limited number of populations per species, and the assumption of stable population structure (asymptotic dynamics). However, spatial variation in population dynamics and deviation from a stable structure may affect these recommendations, calling into question their generality across populations of an invasive species. Here, I focused on intraspecific variation in population dynamics and investigated management recommendations generated by demographic models across 37 populations of a short-lived, invasive perennial herb (Lupinus polyphyllus). Models that relied on the proportional perturbations of vital rates (asymptotic elasticities) indicated an essential role for plant survival in long-term population dynamics. The rank order of elasticities for different vital rates (survival, growth, retrogression, fecundity) varied little among the 37 study populations regardless of population status (increasing or declining asymptotically). Summed elasticities for fecundity increased, while summed elasticities for survival decreased with increasing long-term population growth rate. Transient dynamics differed from asymptotic dynamics, but were qualitatively similar among populations, that is, depending on the initial size structure, populations tended to either increase or decline in density more rapidly than predicted by asymptotic growth rate. These findings indicate that although populations are likely to exhibit transient dynamics, management recommendations based on asymptotic elasticities for vital rates might be to some extent generalised across established populations of a given short-lived invasive plant species.     

Size Matters: Individual Variation in Ectotherm Growth and Asymptotic Size

Body size, and, by extension, growth has impacts on physiology, survival, attainment of sexual maturity, fecundity, generation time, and population dynamics, especially in ectotherm animals that often exhibit extensive growth following attainment of sexual maturity. Frequently, growth is analyzed at the population level, providing useful population mean growth parameters but ignoring individual variation that is also of ecological and evolutionary significance. Our long-term study of Lake Erie Watersnakes, Nerodia sipedon insularum, provides data sufficient for a detailed analysis of population and individual growth. We describe population mean growth separately for males and females based on size of known age individuals (847 captures of 769 males, 748 captures of 684 females) and annual growth increments of individuals of unknown age (1,152 males, 730 females). We characterize individual variation in asymptotic size based on repeated measurements of 69 males and 71 females that were each captured in five to nine different years. The most striking result of our analyses is that asymptotic size varies dramatically among individuals, ranging from 631–820 mm snout-vent length in males and from 835–1125 mm in females. Because female fecundity increases with increasing body size, we explore the impact of individual variation in asymptotic size on lifetime reproductive success using a range of realistic estimates of annual survival. When all females commence reproduction at the same age, lifetime reproductive success is greatest for females with greater asymptotic size regardless of annual survival. But when reproduction is delayed in females with greater asymptotic size, lifetime reproductive success is greatest for females with lower asymptotic size when annual survival is low. Possible causes of individual variation in asymptotic size, including individual- and cohort-specific variation in size at birth and early growth, warrant further investigation.     

一类具扰动种群模型的渐近性(英文)

研究了一类具有扰动的非自治Ｋｏｌｍｏｇｏｒｏｖ模型的渐近性．得到了扰动系戎的解渐近稳定的若干充分条件．     

Pushed beyond the brink: Allee effects,environmental stochasticity,and extinction

To understand the interplay between environmental stochasticity and Allee effects, we analyse persistence, asymptotic extinction, and conditional persistence for stochastic difference equations. Our analysis reveals that persistence requires that the geometric mean of fitness at low densities is greater than one. When this geometric mean is less than one, asymptotic extinction occurs with high probability for low initial population densities. Additionally, if the population only experiences positive density-dependent feedbacks, conditional persistence occurs provided the geometric mean of fitness at high population densities is greater than one. However, if the population experiences both positive and negative density-dependent feedbacks, conditional persistence only occurs if environmental fluctuations are sufficiently small. We illustrate counter-intuitively that environmental fluctuations can increase the probability of persistence when populations are initially at low densities, and can cause asymptotic extinction of populations experiencing intermediate predation rates despite conditional persistence occurring at higher predation rates.     

Robust design and the use of auxiliary variables in estimating population size for an open population

We consider the problem of estimating the population size for an open population where the data are collected over secondary periods within primary periods according to a robust design suggested by Pollock (1982, Journal of Wildlife Management 46, 757-760). A conditional likelihood is used to estimate the parameters associated with a generalized linear model in which the capture probability is assumed to have a logistic form depending on individual covariates. A Horvitz-Thompson-type estimator is used to estimate the population size for each primary period and the survival probabilities between primary periods. The asymptotic properties of the proposed estimators are investigated through simulation and are found to perform well. A data set for such a robust design of a small-mammal capture-recapture study conducted at Dummy Bottom within Browns Park National Wildlife Refuge is analyzed.     

Asymptotic Properties of Spearman’s Rank Correlation for Variables with Finite Support

The asymptotic variance and distribution of Spearman’s rank correlation have previously been known only under independence. For variables with finite support, the population version of Spearman’s rank correlation has been derived. Using this result, we show convergence to a normal distribution irrespectively of dependence, and derive the asymptotic variance. A small simulation study indicates that the asymptotic properties are of practical importance.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread

Summary In this paper we use Aronson's and Weinberger's [1–4] concept of asymptotic speed to estimate the asymptotic behaviour of the solution of a nonlinear integral equation (with the nonlinearity not being monotone), which describes the development of a spatially distributed population.     

Unexplained variability among spatial replicates in transient elasticity: implications for evolutionary ecology and management of invasive species

Understanding actual and potential selection on traits of invasive species requires an assessment of the sources of variation in demographic rates. While some of this variation is assignable to environmental, biotic or historical factors, unexplained demographic variation also may play an important role. Even when sites and populations are chosen as replicates, the residual variation in demographic rates can lead to unexplained divergence of asymptotic and transient population dynamics. This kind of divergence could be important for understanding long- and short- term differences among populations of invasive species, but little is known about it. We investigated the demography of a small invasive tree Psidium cattleianum Sabine in the rainforest of Hawaiʻi at four sites chosen for their ecological similarity. Specifically, we parameterized and analyzed integral projection models (IPM) to investigate projected variability among replicate populations in: (1) total population size and annual per capita population growth rate during the transient and asymptotic periods; (2) population structure initially and asymptotically; (3) three key parameters that characterize transient dynamics (the weighted distance of the structure at each time step from the asymptotic structure, the strength of the sub-dominant relative to the dominant dynamics, and inherent cyclicity in the subdominant); and (4) proportional sensitivity (elasticity) of population growth rates (both asymptotic and transient) to perturbations of various components of the life cycle. We found substantial variability among replicate populations in all these aspects of the dynamics. We discuss potential consequences of variability across ecologically similar sites for management and evolutionary ecology in the exotic range of invasive species.     

Dynamic models of infectious diseases as regulators of population sizes

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.Research supported by Centers for Disease Control contract 200-87-0515. Support services provided at the University of Iowa Center for Advanced Studies     

Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.