Growth rate, population entropy, and perturbation theory

This paper is concerned with the connection between two classes of population variables: measures of population growth rate--the Malthusian parameter, the net reproduction rate, the gross reproduction rate, and the mean life expectancy; and measures of demographic heterogeneity--population entropy. It is shown that the entropy functions predict the response of the growth rate parameters to perturbations in the age-specific fecundity and mortality schedule. These results are invoked to introduce the notion of environmental intensity. The intensity function, expressed in terms of the entropy parameters, is applied to give a comparative study of the effect of environmental factors on the dynamics of Swedish and French populations.     

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.     

The effects of food level and social density on reproduction in the Least Killifish,Heterandria formosa

The feedbacks from population density to demographic parameters, which drive population regulation, are the accumulated results of several ecological processes. The compensatory feedback from increased population density to fertility includes at least two distinct factors, the effects of decreases in per capita food level and increases in the social density (the number of interacting individuals). Because these effects have been studied separately, their relative importance is unknown. It is also unclear whether food limitation and social density combine additively to influence fertility. We investigated these questions with two factorial experiments on reproduction in the Least Killifish, Heterandria formosa. In one experiment, we crossed two levels of density with two levels of a total food ration that was distributed to all individuals. In the other experiment, we crossed two levels of density with two levels of per capita food. Whereas the first experiment suggested that the effects of variation in food level and density were synergistic, the second experiment indicated that they were not. The apparent synergism—the statistical interaction of food and density levels—was the result of confounding per capita food with social density in that design. In the second experiment, the effects of social density on reproductive rate were stronger than the effects of food level, whereas the effects of food level were stronger on offspring size at parturition than those of social density. The results suggest that the social stresses that emerge at higher densities play an important role in the compensatory response of fertility to density, a role, that is, at least as important as that of decreased per capita food levels.     

Experimental evidence for density-dependent reproductive output in a coral reef fish

We tested for density-dependent reproduction in a small coral reef fish using field manipulations of density and observational data. Males of the study species, the bridled goby (Coryphopterus glaucofraenum Gill), defend benthic nest sites, within which they spawn with females, and females can spawn repeatedly over an extended breeding season. In small areas, usually only a single male nested at any one time regardless of how many males were present, so the probability of nesting was inversely proportional to density. Nesting males were almost always the largest in the vicinity, suggesting that, for males whose home ranges overlap, social interactions dictate opportunities to nest. Both the per capita rate at which clutches were laid and the number of eggs produced per clutch declined with increasing density, so the per capita rate of egg production was also density dependent. All three measures of fecundity were better predicted by numerical density (numbers per unit area) than biomass (mass of fish per unit area), and were well described as an inverse function of the number of gobies in the vicinity. A simple hypothesis consistent with these results is that a constant number of females spawn, regardless of density. Alternately, the effect of crowding may depend primarily on the number of interacting individuals and affect all females relatively equally. This density dependence could thus contribute to population regulation at the spatial scale over which populations become reproductively closed.     

Non-linear density dependence in time series is not evidence of non-logistic growth

Time series of population density are often used to seek deviations from logistic regulation by testing for a non-linear decline in per capita growth rate with density. Here I show that this method fails when the interval between observations is not matched by the timing of density impacts on growth. Time series overestimate instantaneous density impacts at low density and underestimate them at high density. More generally, logistic growth produces a deterministically decelerating decline in per capita growth with density if the interval between measures of population size exceeds any lag in density response. Deceleration arises independently out of stochastic density fluctuations, and under-compensating regulation. These multiple influences lead to the conclusion that sequential density estimates provide insufficient information on their own to reveal the identity of non-logistic growth processes. They can yield estimates of density compensation, however, which may suggest time lags in density dependence. Analysis of an empirical time series illustrates the issues.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Effects of early horn growth on reproduction and hunting mortality in female chamois

1.?Environmental conditions during early development can affect the growth patterns of vertebrates, influencing future survival and reproduction. In long-lived mammals, females that experience poor environmental conditions early in life may delay primiparity. In female bovids, annual horn growth increments may provide a record of age-specific reproduction and body growth. Horn length, however, may also be a criterion used by hunters in selecting animals to harvest, possibly leading to artificial selection. 2.?We studied three populations of chamois (Rupicapra rupicapra) in the western Alps to explore the relationships between female horn length and early growth, age of primiparity and age-specific reproduction. We also compared the risk of harvest to reproductive status and horn length. 3.?Early horn growth was positively correlated with body mass in pre-reproductive females and with reproduction in very young and senescent adults. Females with strong early horn growth attained primiparity at an earlier age than those with weak early growth. Horn length did not affect hunter selection, but we found a strong hunter preference for nonlactating females. 4.?Our research highlights the persistent effects of early development on reproductive performance in mammals. Moderate sport harvests are unlikely to affect the evolution of phenotypic traits and reproductive strategies in female chamois. A policy of penalizing hunters that harvest lactating females, however, may increase the harvest of 2-year-old females, which have high reproductive potential.     

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.     

基于生态足迹方法的南京可持续发展研究

20世纪90年代末以来,随着南京城市化进程的加快和城市建成区面积的扩展,城市生态安全问题日益突出。在运用生态足迹方法分析南京市1999—2009年人均生态足迹及生态承载力变化的基础上,采用Mann-kendall法对南京市人均生态足迹突变点进行分析,应用灰色关联模型从经济、社会和生态子系统方面分析不同阶段人均生态足迹变化的原因。研究表明:1999—2009年间南京市人均生态足迹呈增长趋势,人均生态承载力呈减小趋势,单位万元GDP生态足迹和生态多样性指数呈逐年下降趋势,生态经济系统发展能力较好,但生态压力较大。南京市人均生态足迹变化分为1999—2002年的低位增长阶段和2003—2009年高位波动阶段。通过影响南京市生态足迹变化因素分析发现,2003—2009年与1999—2002年相比,经济子系统依然处于主要地位,但影响因素由农业向以三产为主转变,同时生态子系统和社会子系统各因素的影响力提升。据此,提出优化产业结构,控制人口数量,加强基础设施建设,改善居民居住环境,加大城市绿化建设,提高城市污物处理能力,减少污染物排放的南京可持续发展对策。     

Demography of the Pacific walrus (Odobenus rosmarus divergens): 1974–2006

Global climate change may fundamentally alter population dynamics of many species for which baseline population parameter estimates are imprecise or lacking. Historically, the Pacific walrus is thought to have been limited by harvest, but it may become limited by global warming‐induced reductions in sea ice. Loss of sea ice, on which walruses rest between foraging bouts, may reduce access to food, thus lowering vital rates. Rigorous walrus survival rate estimates do not exist, and other population parameter estimates are out of date or have well‐documented bias and imprecision. To provide useful population parameter estimates we developed a Bayesian, hidden process demographic model of walrus population dynamics from 1974 through 2006 that combined annual age‐specific harvest estimates with five population size estimates, six standing age structure estimates, and two reproductive rate estimates. Median density independent natural survival was high for juveniles (0.97) and adults (0.99), and annual density dependent vital rates rose from 0.06 to 0.11 for reproduction, 0.31 to 0.59 for survival of neonatal calves, and 0.39 to 0.85 for survival of older calves, concomitant with a population decline. This integrated population model provides a baseline for estimating changing population dynamics resulting from changing harvests or sea ice.     

Regulation of local populations of a coral reef fish via joint effects of density- and number-dependent mortality

Density-dependent mortality can regulate local populations - effectively minimizing the likelihood of local extinctions and unchecked population growth. It is considered particularly important for many marine reef organisms with demographically open populations that lack potential regulatory mechanisms tied to local reproduction. While density-dependent mortality has been documented frequently for reef fishes, few studies have explored how the strength of density-dependence varies with density, or how density-dependence may be modified by numerical effects (i.e., number-dependent mortality). Both issues can have profound effects on spatial patterns of abundance and the regulation of local populations. I address these issues through empirical studies in Moorea, French Polynesia, of the six bar wrasse (Thalassoma hardwicke), a reef fish that settles to isolated patch reefs. Per capita mortality rates of newly settled wrasse increased as a function of density and were well approximated by the Beverton-Holt function for both naturally formed and experimentally generated juvenile cohorts. Average instantaneous mortality rates were a decelerating function of initial densities, indicating the per capita strength of density-dependence decreased with density. Results of a factorial manipulation of density and group size indicate that per capita mortality rates were simultaneously density- and number-dependent; fish at higher densities and/or in groups had higher probabilities of disappearing from patch reefs compared with fish that were solitary and/or at lower densities. Mortality rates were ~30% higher for fish at densities of 0.5 fish/m² than at 0.25 fish/m². Similarly, mortality rates increased by ~45% when group size was increased from 1 to 2 individuals per patch, even when density was kept constant. These observations suggest that the number of interacting individuals, independent of patch size (i.e., density-independent effects) can contribute to regulation of local populations. Overall, this work highlights a greater need to consider numerical effects in addition to density effects when exploring sources of population regulation.     

Density dependent population growth of the two-spotted spider mite, Tetranychus urticae, on the host plant Leonurus cardiaca

We observed Tetranychus urticae (Koch), a polyphagous spider mite herbivore, on Leonurus cardiaca (L.) at several sites in eastern North America at variable density, ranging from extremely dense to sparse. To understand the nature of T. urticae 's population dynamics we experimentally manipulated population densities on L. cardiaca and assessed per capita growth after 1 to 2 generations in laboratory and field experiments. In particular, we took a 'bottom-up' approach, manipulating both plant size and quality to examine effects on mite dynamics. Per capita growth was strongly dependent on the initial density of the mite population. Spider mite populations grew (1) in a negatively density dependent manner on small plants and (2) unhindered by density dependence on large plants. Mean per capita growth was 59% higher on small plants compared to large plants, irrespective of mite density. We also found evidence for density dependent induced susceptibility to spider mites in small plants and density dependent induced resistance in large plants. Hence, spider mite populations grew at a relatively fast rate on small plants, and this was associated with negative density dependence due to factors that depress population growth, such as food deterioration or limitation. On large plants, spider mite populations grew at a relatively slow rate, apparently resulting in herbivore densities that may not have been high enough to cause intraspecific competition or other forms of negative density dependence.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

Food web structure and climate effects on the dynamics of small mammals and owls in semi-arid Chile

Population dynamics of small mammals and predators in semi-arid Chile is positively correlated with rainfall associated with incursions of El Niño (El Niño Southern Oscillation: ENSO). However, the causal relationships between small mammal fluctuations, predator oscillations, and climatic disturbances are poorly understood. Here, we report time series models for three species of small mammal prey and two species of owl predators. The large differences in population fluctuations between the three small mammal species are related to differences in their respective feedback structures. The analyses reveal that per capita growth rate of the leaf-eared mouse is a decreasing function of log density and of log barn owl abundance together with a positive rainfall effect. In turn, per capita population growth rate ( R -function) of the barn owl is a negative function of log barn owl abundance and a positive function of leaf-eared mouse abundance, suggesting a predator–prey interaction. The dramatic population fluctuations exhibited by leaf-eared mouse ( Phyllotis darwini ) are caused by climate effects coupled with a complex food web architecture.     

Female Fecundity Is Dependent on Substrate, Rather Than Male Abundance, in the Wind-Pollinated, Dioecious Understory Palm Chamaedorea radicalis

We examined the impact of substrate, population density, and sexual composition on female reproductive success in harvested populations of the tropical understory palm Chamaedorea radicalis . Leaf harvest for the international floral greenery trade does not kill palms, but has been shown to reduce flowering frequency and has been projected to reduce population growth. Because C. radicalis is dioecious and wind-pollinated, a reduction in flowering density from leaf harvest or other anthropogenic factors may lower pollen movement between flowering conspecifics and lower fruit production. Such reduced fecundity in harvested populations can contribute to an Allee effect, where the per capita rate of population growth declines at low density. We tested for these effects by sampling C. radicalis along transects in ten populations, spanning a range of densities, during May (flowering peak) and August (fruiting peak) 2002. We applied path analysis using structural equation modeling (SEM) to test a set of hypothesized relationships between substrate, female size, several population parameters, and female fecundity. SEM models revealed that female fecundity was not dependent on any measure of population density or sexual composition, providing no evidence for an Allee effect. The strongest model ( R ²= 92.9%) related C. radicalis reproductive output only to substrate, female size, flower number, and proportional fruit set. This model revealed that palms on rock outcrops were larger, produced more flowers, and had higher proportional fruit set than palms on the forest floor. This finding suggests that protection of females on outcrops should enhance population growth and viability.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Connecting geographical distributions with population processes

The geographical distribution of a species is determined by a large number of complex processes operating over spatial scales spanning 10 orders of magnitude. Patterns in population processes have been described at numerous scales. We show that two patterns, measured at different scales, jointly allow us to infer heretofore unknown patterns in the distribution of demographic patterns across the geographical range of a species. The resulting model describes three fundamentally different modes of geographical variation in vital rates of populations. One mode is characterized by a positive nonlinear relationship between the maximum rate of population growth and the intensity of intraspecific competition across a geographical range. That is, populations that grow rapidly are also those where individuals experience the greatest per capita negative effect of the presence of other individuals. The second mode of behaviour is described by a negative nonlinear relationship between maximum growth rate and density dependence. Under this scenario, populations with low capacity to grow rapidly have highest intensities of intraspecific competitive effects. A third mode of behaviour is characterized by a weak positive relationship between growth rate and intraspecific competition, with very little geographical variation in maximum growth rate. A survey of studies relating temporal means and variances in population abundance for a variety of species indicate that the second mode of geographical variation in population dynamics across species ranges is the most common, though a few species appear to be characterized by the third mode.     

Nutritional availability and larval density dependence in Aedes aegypti

1. Density dependence is the effect of density on population growth. Density dependence is an aggregate term for a suite of complex interactions between animals and their environment. 2. Mechanistic studies of density dependence in mosquito ecology are sparse, and the role of environmental factors is poorly understood. 3. Two empirical study designs were compared to consider the interaction between nutritional availability and density in Aedes aegypti. First, larvae were fed per capita. Second, larvae were fed a fixed amount of food unadjusted for the number of individuals; therefore, at higher densities, individuals received less per capita. 4. Survivorship, wing length, and development rate were lower at high densities when larvae were fed a fixed, unadjusted amount of food. The opposite was observed when food was adjusted per capita, suggesting that high densities may be beneficial for larval development when per capita nutrition is held constant 5. These results demonstrate that negative associations between Ae. aegypti larval density and larval development are a manifestation of decreased per capita nutrient uptake at high densities. 6. Population regulation is a proportional response to environmental variability in Ae. aegypti. Increased survivorship at high densities when larvae were fed per capita demonstrates that nutritional availability is not the only mechanism of density dependence in mosquitoes. Further studies should characterise density dependence in mosquitoes by using mechanistic study designs across diverse environmental conditions.     

Interactions between Operophtera brumata and Tortrix viridana on oak: new evidence from time-series analysis

1. Yearly population estimates of the green oak leaf roller moth, Tortrix viridana (Lepidoptera: Tortricidae), and the winter moth, Operophtera brumata (Lepidoptera: Geometridae), were taken from the pedunculate oak, Quercus robur, from 1951 to 1966 in Wytham Woods, Oxfordshire, U.K. Larval mortality from parasitoids was recorded for both species over the same period. 2. Operophtera brumata population density fluctuated around a constant mean, whereas T. viridana population density exhibited a linear decline over the sampling period. Population counts were subjected to time-series analysis after the linear decline was removed from the T. viridana data by detrending. Multiple regression models were built so that variation in the per capita rate of increase of each population could be partitioned among (a) current and previous population density, (b) current and previous population density of the second moth species, and (c) rates of larval parasitism. 3. Multiple regression analysis suggested that variation in the per capita rate of increase of O. brumata could be explained by negative feedback from O. brumata density at T–2 (32%), a negative relationship with T. viridana density at T–1 (18%), and a positive relationship with parasitism suffered by T. viridana at T–1 (24%). 4. The T. viridana population time-series was dominated by a rapid feedback process such that per capita rate of increase at time T was negatively related to population density at time T–1, explaining 53% of variance in population growth rate. Per capita rate of increase was unrelated to larval parasitism or densities of O. brumata. 5. In light of previous life-table and experimental studies on O. brumata and T. viridana, the current analyses suggest that pupal predation (data not presented) and interspecific competition are significant determinants of O. brumata population growth rates in Wytham Woods. In contrast, T. viridana population growth rates appear to be dominated by rapid negative feedback consistent with intraspecific competition.