The dynamics of density dependent population models

Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a strange attractor is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).Supported by NSF Grant No. BMS 74-21240.     

我国人口增长的总和生育率模型及人口预警

运用数学公式推导人类种群在"一夫一妻"基本制度下的总和生育率的"世纪更替"水平,或实现人口零增长的参数计算公式为 TFR=1+S/100(S/100为性别比,当男女性别比为1 ∶ 1时TFR=2.0.当性别比因女少而严重失调时, TFR>2.0).在对我国人口增长及总和生育率的变化进行分析研究后发现:实行了30a的计划生育政策取得巨大成就,奇迹般地将我国总和生育率从6.0左右迅速降下来.但是,近年来的总和生育率呈急剧下降趋势,目前全国TFR指标在1.8以下,有可能已经接近1.1.根据北京、上海的统计数据及人口普查分析,这两个地区的TFR已经小于1.0.根据建立的人口增长总和生育率(TFR)模型,按照目前提倡的TFR=1的指标,继续严格执行下去,300年内我国的人口将从现在的13亿急剧下降到398万人.     

Maximum sustainable yield of a density dependent sex-age-structured population model

The maximum sustainable yield of an age-structured, density dependent, sex-differentiated population is investigated. The model is based on the nonlinear version of the McKendrick [11] model introduced by Gurtin and MacCamy [8], modified to include sex-differentiated dynamics. It is determined that the maximum sustainable yield is attainable by an age specific harvesting policy in which the number of harvesting ages for males and the harvesting ages for females total at most five.     

Global asymptotic stability of density dependent integral population projection models

Many stage-structured density dependent populations with a continuum of stages can be naturally modeled using nonlinear integral projection models. In this paper, we study a trichotomy of global stability result for a class of density dependent systems which include a Platte thistle model. Specifically, we identify those systems parameters for which zero is globally asymptotically stable, parameters for which there is a positive asymptotically stable equilibrium, and parameters for which there is no asymptotically stable equilibrium.     

Modeling the population dynamics of annual plants with seed bank and density dependent effects

A model is proposed for the population dynamics of an annual plant (Sesbania vesicaria) with a seed bank (i.e. in which a proportion of seeds remain dormant for at least one year). A simple linear matrix model is deduced from the life cycle graph. The dominant eigenvalue of the projection matrix is estimated from demographic parameters derived from field studies. The estimated values for population growth rate () indicates that the study population should be experiencing a rapid exponential increase, but this was not the case in our population.The addition of density dependent effects on seedling survivorship and adult fecundity, effects for which field studies provide evidence, considerably improves our model. Depending on the demographic parameters, the model leads to stable equilibrium, oscillations, or chaos. Study of the behaviour of this model in the parameter space shows that the existence of a seed bank allows higher among-year variation of adult fecundity, without leaving the region of demographic stability. Field data obtained over 3 years confirm this prediction.     

Coexistence resulting from an alternation of density dependent and density independent growth

When two species compete with each other, one is likely to displace or exclude the other. Several circumstances under which they may coexist indefinitely have been presented in the literature; the present contribution presents examples of one more. Under circumstances where both populations are repeatedly decreased (for example because of annual environmental changes) then subsequent to each decrease both species grow unrestrictedly and then interact with each other in a competitive fashion. If the species that grows more rapidly under unrestricted conditions is at a disadvantage during competitive phases of growth, this effect prolongs coexistence but may not prevent eventual extinction of one or the other species. However, it is shown that there are certain broad ranges of conditions for population growth that lead to permanent cyclical stability. The stability described here is such that the ecosystem will return to the same dynamic balance even when severely perturbed. It is also shown that this kind of stability can be either favored or prevented in certain cases by random fluctuations in the environment affecting season length, kill factor, etc.     

Density dependent diffusional model of an infective population

In this work, we present a density-dependent diffusional model which, coupled to three different types of growth, permitted us to study the infective potential of a bacteria species. The results show that those species with strong internal competency have the higher colonizing capacity in terms of invasion speed. Here, we also advanced a model for the static spatial inhomogeneous distribution that some species establish after migration. It is proposed that the origin of these patterns is the result of a balance between the dispersal tendency and the attractive behavior. The results obtained were compared with the observed behavior of Rhizobium spp. during infection of leguminous roots. A possible explanation of the observed morphologies of nodule development in different legumes is suggested.     

Harvesting in a resource dependent age structured Leslie type population model

We analyse the effect of harvesting in a resource dependent age structured population model, deriving the conditions for the existence of a stable steady state as a function of fertility coefficients, harvesting mortality and carrying capacity of the resources. Under the effect of proportional harvest, we give a sufficient condition for a population to extinguish, and we show that the magnitude of proportional harvest depends on the resources available to the population. We show that the harvesting yield can be periodic, quasi-periodic or chaotic, depending on the dynamics of the harvested population. For populations with large fertility numbers, small harvesting mortality leads to abrupt extinction, but larger harvesting mortality leads to controlled population numbers by avoiding over consumption of resources. Harvesting can be a strategy in order to stabilise periodic or quasi-periodic oscillations in the number of individuals of a population.     

Intensity of grass invasion negatively correlated with population density and age structure of an endangered dune plant across its range

Biological Invasions - Invasive species are a global threat to ecosystem biodiversity and function; non-native grass invasion has been particularly problematic in sparsely vegetated ecosystems such...     

ABO blood groups and fertility in an Indian population

A total of 589 compatible mating couples could be investigated against 432 incompatible mating couples in order to determine the selective mechanism operating on ABO blood groups. There appears to be no striking difference in the proportion of childless couples between the two groups. The mean number of living children presents a significant difference. There is 21% deficiency of 'A' children in the two groups. Similarly, there is 16% deficiency of 'B' children in the two groups. It appears that there is 31.9% fetal wastage in incompatible matings as compared with 17.15% in compatible matings.     

Local stability of a population with density-dependent fertility

The local stability of an equilibrium population configuration is investigated. The population is governed by a density-dependent, age-structured, nonlinear renewal equation. Local stability is shown to depend essentially on the rate of change of the net reproduction rate as a function of the population size. Examples illustrating the results and the characteristics promoting stability are discussed.     

The effect of population density on the growth of an animal population

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Contribution from the Department of Fisheries, Kyoto University.

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Contribution from the Entomological Laboratory, Kyoto University, No. 201.     

Low mineral density of a weight-bearing bone among adult women in a high fertility population

Evolutionary theories of aging posit that greater reproductive effort causes somatic decline given a fundamental trade-off between investing energy in reproduction and repair. Few studies in high fertility human populations support this hypothesis, and problems of phenotypic correlation can obscure the expected trade-off between reproduction and somatic condition. This cross-sectional study investigates whether greater reproductive effort is associated with reduced calcaneal bone mineral density (BMD) among female Tsimane forager-farmers of lowland Bolivia. We also investigate whether female Tsimane BMD values are lower than sex- and age-matched US reference values, despite the fact that Tsimane engage in higher physical activity levels that can increase mechanical loading. To measure calcaneal BMD, quantitative ultrasonography was performed on 130 women (mean ± SD age = 36.6 ± 15.7, range = 15–75) that were recruited regardless of past or current reproductive status. Anthropometric and demographic data were collected during routine medical exams. As predicted, higher parity, short inter-birth interval, and earlier age at first birth are associated with reduced BMD among Tsimane women after adjusting for potential confounders. Population-level differences are apparent prior to the onset of reproduction, and age-related decline in BMD is greater among Tsimane compared with American women. Greater cumulative reproductive burden may lower calcaneal BMD individually and jointly with other lifestyle and heritable factors. Fitness impacts of kin transfers in adulthood may determine the value of investments in bone remodeling, and thus affect selection on age-profiles of bone mineral loss. Am J Phys Anthropol 156:637–648, 2015. © 2014 Wiley Periodicals, Inc.     

Limited inbreeding depression in a bottlenecked population is age but not environment dependent

Inbreeding depression is known to vary greatly between populations and among species. Some of this variation is due to differences in genetic load between populations, while some is due to differences in the environment (e.g. local weather conditions) or demography of the population (e.g. age structure and breeding experience) in which inbreeding is expressed. Although the effects of these factors in isolation are well understood, there is still relatively little known about the interface between inbreeding on one hand, and environment and demography on the other in wild populations. We examined how environmental and demographic factors mediated the effects of inbreeding in a threatened species of bird. The Stewart Island robin, Petroica australis rakiura, has been subjected to a prolonged bottleneck for over 150 years. A complete pedigree of a reintroduced island population, extending back seven seasons to its founding, was available for analysis along with survival data (at the level of the brood) obtained from intensive monitoring over two breeding seasons. We found no strong support that the degree to which a brood was inbred affected its survival at either the hatching, fledging or recruitment stages. The inbreeding coefficient of the mother did have an effect on brood survival when analysed over all three life history stages, but only as a result of an interaction with female age, with broods of one‐year‐old inbred females suffering greater mortality than those of older inbred females. Although habitat type, temperature, rainfall and year were the best predictors of brood survival for most life history stages, their effects were weak and there were no interactions with inbreeding. Furthermore, there was no strong evidence of inbreeding depression associated with two periods of severe weather. This population is atypical in that inbreeding depression appears to be weak even under severe environmental conditions, and may be indicative that this bottlenecked population has either reduced genetic load or has fixed deleterious alleles.     

A density dependent model describing Salmo trutta population dynamics in an arborescent river network. Effects of dams and channelling

Our aim is to model the Salmo trutta population dynamics (three age-classes) in an arborescent river network (four levels, 15 patches), by considering both migrations (fast time scale) and demography (slow time scale). We study how the environmental management can influence the global population dynamics. We present a general model coupling both a linear discrete model for constant migrations and a non-linear density-dependent Leslie model for the demography, with (15 × 3) difference equations (15 patches, three age-classes). The variable aggregation method applied to discrete time models allows us to aggregate the previous model into a new one with only three equations. We assume fecundity and survival gradients with respect to the river network levels. The Salmo trutta whole population tends towards an equilibrium state depending on the environmental structure, and we show that dams have a stronger influence than channelling on this equilibrium.     

Fertilization in an egg-brooding colonial ascidian does not vary with population density

The possibility that free-spawning marine organisms may be subject to fertilization failure at low population density (due to the effects of sperm dilution) has sparked much interest, but these effects have been demonstrated only in a few species that broadcast their eggs. Some egg-brooding species may overcome dilution effects by filtering low concentrations of sperm from seawater and fertilizing eggs throughout an extended period of time. We examined the effects of population density and size on fertilization in Botryllus schlosseri, a hermaphroditic colonial ascidian that free-spawns sperm, but broods eggs. We experimentally manipulated the size and density of mating groups and surveyed fertilization levels in natural populations that varied in density. Fertilization was not affected by variation in population size or density in either the experimental or natural populations. Near the end of the reproductive season, some eggs may have been fertilized too late to complete development, suggesting a temporal form of sperm limitation that has not been considered in other systems. We also detected greater variability in fertilization levels at lower population density. Nevertheless, these results suggest that caution must be used in extrapolating reported density effects on fertilization to all taxa of free-spawners; density effects may be reduced in brooders that have efficient sperm collection mechanisms.     

Optimal life histories with age dependent tradeoff curves

Following the approach of Schaffer (1974, Ecology 55, 291-303.) and Charlesworth & Leon (1976, Am. Nat. 110, 449-459.) the tradeoff between fecundity and survival/growth is investigated in an age-structured population with density independent life history parameters. The results of the above authors are generalized by allowing the tradeoff curve to vary with age; the life cycle is assumed to have two stages: an initial stage during which the organism generally improves in her capacity to reproduce, grow and survive, and a final stage during which her general performance remains constant or declines. The principal result is that, during the final stage, RV per unit size should decrease and over the course of the entire life, should either decrease, or increase at first and then decrease. With the additional assumption that the tradeoff curves at different ages are similar in shape, it is shown that unit fecundity should increase throughout the reproductive life of the organism.