Identifying the density-dependent structure underlying ecological time series

A central problem in ecology is explaining the causes of population fluctuations, and an important step in the solution is determining the structure of the negative feedback (density dependent) process regulating population dynamics. The conventional way to determine the dimension or order of density dependence in a time series is to calculate the partial autorcorrelation function (PACF). We maintain, however, that PACF is not designed with biological populations in mind and has the wrong null model for detecting the structure of density dependence. We suggest an alternative diagnostic, the partial rate correlation function (PRCF), which is specifically designed for biological populations and has an appropriate null model for detecting their density dependent structures. Tests with simulated data show PRCF to be superior to PACF in detecting the underlying density dependent structure of two simple mathematical models.     

Foundations of stochastic development

A consistent mathematical theory of stochastic poikilotherm development has been derived based upon a minimum set of biological assumptions obtained from the literature. In the subsequent analysis, the resulting developmental rate can be justifiably represented as a random variable. Three cases are considered: (1) developmental rates dependent only on temperature, (2) rates dependent on both temperature and age, and (3) rates dependent on a general function of temperature and time. The analysis provides a mathematical foundation for the current practice of superimposing a probability distribution function on a biological time scale to describe the development of individuals from a population.     

The impact of random frequency-dependent mutations on the average population fitness

ABSTRACT: BACKGROUND: In addition to selection, the process of evolution is accompanied by stochastic effects, such as changing environmental conditions, genetic drift and mutations. Commonly it is believed that without genetic drift, advantageous mutations quickly fixate in a halpoid population due to strong selection and lead to a continuous increase of the average fitness. This conclusion is based on the assumption of constant fitness. However, for frequency dependent fitness, where the fitness of an individual depends on the interactions with other individuals in the population, this does not hold. RESULTS: We propose a mathematical model that allows to understand the consequences of random frequency dependent mutations on the dynamics of an infinite large population. The frequencies of different types change according to the replicator equations and the fitness of a mutant is random and frequency dependent. To capture the interactions of different types, we employ a payoff matrix of variable size and thus are able to accommodate an arbitrary number of mutations. We assume that at most one mutant type arises at a time. The payoff entries to describe the mutant type are random variables obeying a probability distribution which is related to the fitness of the parent type. CONCLUSIONS: We show that a random mutant can decrease the average fitness under frequency dependent selection, based on analytical results for two types, and on simulations for n types. Interestingly, in the case of at most two types the probabilities to increase or decrease the average fitness are independent of the concrete probability density function. Instead, they only depend on the probability that the payoff entries of the mutant are larger than the payoff entries of the parent type.     

Density-dependent dispersal in host-parasitoid assemblages

Most spatial population models assume constant rates of dispersal. However, in a given community, dispersal may not only depend on the density of conspecifics, i.e. density‐dependent dispersal, but also on the density of other species, a phenomenon we term ‘community‐dependent dispersal’. We co‐vary the densities of both the beetle host Callosobruchus chinensis and its parasitoid wasp, Anisopteromalus calandrae, in a laboratory study and record the proportions of each species that disperse within a two‐hour period. The parasitoid in these systems exhibits community‐dependent dispersal – dispersing more frequently when parasitoid density is high and larval host density is low. This supported our prediction that individuals should disperse according to competition for available resources. However, in this study the host's dispersal was independent of density. We suggest that this may be due to less intense selection acting on host dispersal strategies than on the parasitoid. We consider some possible consequences of community‐dependent dispersal for a number of spatial population processes. A well‐known host‐parasitoid metapopulation model is expanded so that it includes a greater range of dispersal functions. When the model is parameterised with the parasitoid community‐dependent dispersal function observed in the empirical study, similar population dynamics are obtained as when fixed‐rate dispersal functions are applied. The importance of dispersal functions for invasions of both competitive and host‐parasitoid systems is also considered. The model results demonstrate that understanding how individuals disperse in response to different species’ population densities is important in determining the rate of spread of an invasion. We suggest that more empirical studies are needed to establish what determines dispersal rate and distance in a range of species, combined with theoretical studies investigating the role of the dispersal function in determining spatial population processes.     

A stochastic discrete generation birth,continuous death population growth model and its approximate solution

This paper describes a single species growth model with a stochastic population size dependent number of births occurring at discrete generation times and a continuous population size dependent death rate. An integral equation for a suitable transformation of the limiting population size density function is not in general soluble, but a Gram-Charlier representation procedure, previously used in storage theory, is successfully extended to cover this problem. Examples of logistic and Gompertz type growth are used to illustrate the procedure, and to compare with growth models in random environments. Comments on the biological consequences of these models are also given.Currently at Department of Mathematics, University of MarylandWork partially supported by the Danish Natural Science Research Council and Monash University     

Modeling ecological traps for the control of feral pigs

Ecological traps are habitat sinks that are preferred by dispersing animals but have higher mortality or reduced fecundity compared to source habitats. Theory suggests that if mortality rates are sufficiently high, then ecological traps can result in extinction. An ecological trap may be created when pest animals are controlled in one area, but not in another area of equal habitat quality, and when there is density‐dependent immigration from the high‐density uncontrolled area to the low‐density controlled area. We used a logistic population model to explore how varying the proportion of habitat controlled, control mortality rate, and strength of density‐dependent immigration for feral pigs could affect the long‐term population abundance and time to extinction. Increasing control mortality, the proportion of habitat controlled and the strength of density‐dependent immigration decreased abundance both within and outside the area controlled. At higher levels of these parameters, extinction was achieved for feral pigs. We extended the analysis with a more complex stochastic, interactive model of feral pig dynamics in the Australian rangelands to examine how the same variables as the logistic model affected long‐term abundance in the controlled and uncontrolled area and time to extinction. Compared to the logistic model of feral pig dynamics, the stochastic interactive model predicted lower abundances and extinction at lower control mortalities and proportions of habitat controlled. To improve the realism of the stochastic interactive model, we substituted fixed mortality rates with a density‐dependent control mortality function, empirically derived from helicopter shooting exercises in Australia. Compared to the stochastic interactive model with fixed mortality rates, the model with the density‐dependent control mortality function did not predict as substantial decline in abundance in controlled or uncontrolled areas or extinction for any combination of variables. These models demonstrate that pest eradication is theoretically possible without the pest being controlled throughout its range because of density‐dependent immigration into the area controlled. The stronger the density‐dependent immigration, the better the overall control in controlled and uncontrolled habitat combined. However, the stronger the density‐dependent immigration, the poorer the control in the area controlled. For feral pigs, incorporating environmental stochasticity improves the prospects for eradication, but adding a realistic density‐dependent control function eliminates these prospects.     

Density dependent selection in parthenogenetic and self-mating populations

The consequences of density dependent selection on genetically heterogeneous, diploid populations reproducing by self-mating or various parthenogenetic mechanisms is investigated. A logistic fitness function that depends upon both the genotype of an individual and the density of the population is used. Such a fitness function simultaneously determines the population size and the genotype frequencies. The equilibrium solutions to a one locus and two locus model are given as well as some generalizations to n loci and nonlogistic fitness functions. Conditions are found that maintain several different genotypes simultaneously in the equilibrium population. The interaction of such selection with the genetic mechanisms which determine mode of reproduction in parthenogenetic populations is also discussed.     

Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model

The present work employs a nonlocal delay reaction-diffusion model to study the impacts of the density dependent birth function, maturation time delay and population dispersal on single species dynamics (i.e., extinction, survival, extinction-survival). It is shown that the maturation time and the birth function are two major factors determining the fate of single species. Whereas the dispersal acts as a subsidiary factor that only affects the spatial patterns of population densities. When the birth function has a compensating density dependence, maturation time delay cannot destabilize the population survival at the positive equilibrium. Nevertheless, when the birth function has an over-compensating density dependence, the population densities of single species fluctuate in the spatial domain due to the increased maturation time delay. With the Allee effect and over-compensating density dependence, the increases in the maturation time may cause extinction of the single species in the entire spatial domain. The numerical simulations suggest that the solutions of the general model may temporarily remain nearby a stationary wave pulse or a stationary wavefront of the reduced model. The former indicates the survival of single species in a narrow region of the spatial domain. Whereas the latter represents the survival in the entire left-half or right-half of the spatial domain.     

High larval density does not induce a prophylactic immune response in a butterfly

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Life table studies on the paddy field population of the green rice leafhopper,Nephotettix cincticeps Uhler (Hemiptera: Cicadellidae), with special reference to the mechanism of population regulation

The population growth of the green rice leafhopper, Nephotettix cincticeps, in the paddy field was analyzed based on the life table data accumulated for six years. The paddy field population, which stems from the invading adults of the first generation (G-I), repeats two complete generations, and the hatchlings of the fourth generation (G-IV) enter diapause and overwinter as the fourth instar nymphs in fallow paddy fields. It was clarified that the density dependent reduction in the mean longevity and oviposition rate of adult females in G-II and III played a primary role in stabilizing the annual population densities. The annual average of the mean longevity of G-II females (3.9 days) was much shorter than that of G-III ones (7.7 days) and thus the density dependent reduction in the mean longevity induced a more prompt regulatory effect on the oviposition of G-II females compared with G-III ones. As the result, two equilibrium densities of eggs were obtained, e.g., ca 100 and 700 eggs per hill in G-III and IV, respectively. Density dependent decrease in the proportion of mature females in the adult population was especially conspicuous in G-II, and this was closely associated with the density dependent reduction in the mean longevity and fecundity. Thus, the density dependent dispersal (emigration) of the adult females by flight in G-II and III was the most convincing factor in the process of population regulation. The density dependent dispersal of the adult females is effective in avoiding the deleterious effects of nymphal crowding in a breeding habitat unit (a paddy field), and may result in a more even distribution of the population over a continuous habitat units in a locality than otherwise.     

The influence of size-dependent life-history traits on the structure and dynamics of populations and communities

Individual organisms often show pronounced changes in body size throughout life with concomitant changes in ecological performance. We synthesize recent insight into the relationship between size dependence in individual life history and population dynamics. Most studies have focused on size‐dependent life‐history traits and population size‐structure in the highest trophic level, which generally leads to population cycles with a period equal to the juvenile delay. These cycles are driven by differences in competitiveness of differently sized individuals. In multi‐trophic systems, size dependence in life‐history traits at lower trophic levels may have consequences for both the dynamics and structure of communities, as size‐selective predation may lead to the occurrence of emergent Allee effects and the stabilization of predator–prey cycles. These consequences are linked to that individual development is density dependent. We conjecture that especially this population feedback on individual development may lead to new theoretical insight compared to theory based on unstructured or age‐dependent models. Density‐dependent individual development may also cause individuals to realize radically different life histories, dependent on the state and dynamics of the population during their life and may therefore have consequences for individual behaviour or the evolution of life‐history traits as well.     

杭州石荠苎(Mosla hangchowensis)种群密度制约实验的统计分析

本文研究了我国特有、分布区极狭窄的一年生草本植物——杭州石荠苎(Mosla hangchowensis)种群的密度制约规律。结果表明：在生长季内,种群的死亡率与密度密切相关。种群的最适密度为200～1000株/m²左右。不同密度种群的平均株高、开花数等性状随时间的动态关系均符合“logistic”模型。高密度种群中60%左右的个体能完成生活史;低密度种群中80%以上的个体能完成生活史。种群密度较高制约杭州石荠苎的植株形态和繁殖投资。     

Frequency dependence and competition

Intraspecific competition implies interaction among the individuals of a population, so natural selection on genotypic variation in characters related to the competition will necessarily be frequency dependent. Intraspecific antagonistic competition exhibits properties similar to other behavioural interactions between individuals. In exploitative intraspecific competition the interactions among individuals are less direct. Exploitation modifies the abundance of the various limiting resources according to the use of these resources by the individual members of the population. The amount of resource available to an individual is therefore a function of the phenotypes present in the population, through their density and frequency.     

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.     

Direct negative density‐dependence in a pond‐breeding frog population

Understanding population dynamics is critical for the management of animal populations. Comparatively little is known about the relative importance of endogenous (i.e. density‐dependent) and exogenous (i.e. density‐independent) factors on the population dynamics of amphibians with complex life cycles. We examined the potential effects of density‐dependent and ‐independent (i.e. climatic) factors on population dynamics by analyzing a 15‐yr time series data of the agile frog Rana dalmatina population from Târnava Mare Valley, Romania. We used two statistical models: 1) the partial rate correlation function to identify the feedback structure and the potential time lags in the time series data and 2) a Gompertz state‐space model to simultaneously investigate direct and delayed density dependence as well as climatic effects on population growth rate. We found evidence for direct negative density dependence, whereas delayed density dependence and climate did not show a strong influence on population growth rate. Here we demonstrated that direct density dependence rather than delayed density dependence or climate determined the dynamics of our study population. Our results confirm the findings of many experimental studies and suggest that density dependence may buffer amphibian populations against environmental stress. Consequently, it may not be easy to scale up from individual‐level effects to population‐level effects.     

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.     

Population structure and the spread of disease

A common assumption of many mathematical models for the spread of disease is that there is random mixing among all individuals in the host population. This paper analyzes and develops a model for the spread of disease in a population consisting of several interacting subpopulations. The model considers 2 different types of interactions between individuals: 1) within a subpopulation because of geographic proximity, and 2) of the same or different subpopulations because of attendance at common social functions. A stability analysis performed on the equilibria of the model shows 2 stable states: 1) a population composed solely of susceptible individuals with no disease present, and 2) an interior point where there are susceptible, infective, and recovered individuals present at all times. The analysis shows that the threshold for disease maintenance is more easily exceed in centers that are members of a small local cluster than in randomly mixing centers, but that the spread of the disease throughout the population occurs more rapidly when the initial case attends a randomly mixing center. The conditions under which a disease will become established are dependent upon the transmission rate for the disease, the birth and death rate in each neighborhood, the recovery rate from the disease in each neighborhood, and the movement patterns of the individuals in the population. The study of the spread of disease in a population by means of mathematical models provides a valuable addition to the statistical data analyzed by epidemiologists. This model is relevant any time there is a division of the population into several interacting groups in which the probability of disease spread is a function both of neighborhood contact because of geographic proximity and of social interactions between groups.     

On the study of two interacting populations one of which acts independently of the other

Many ecological and biological systems can be studied in terms of a bivariate stochastic branching process, {X ₁ (t), X ₂ (t)}, each of whose components (or populations) varies in magnitude according to the laws of a generalized birth-death process. Of particular interest is such a model in which the birth and death rates of the first population,X ₁, are constant while those of the second population,X ₂, exhibit a functional dependence upon the magnitude of the first. It is shown, first, that the existence of the stochastic mean of a birth death process implies the existence of all higher moments. The values of all the factorial moments of such a process are then determined. The moments of the dependent population of the bivariate process are given in terms of its expectation and the joint probability density function of the process is determined. It is possible, therefore, to use Bayesian techniques to infer conclusions about the independent population, given information about the variation of the dependent one.     

Stability of an age specific population with density dependent fertility

A nonlinear version of the Lotka-Sharpe model of population growth is considered in which the age specific fertility is a function of the population size. The stability of an equilibrium population distribution is investigated with respect to both global and local perturbations. Sufficient conditions for such stability are presented, as are estimates for the rate of return of the population to the equilibrium configuration. Particular attention is paid to those situations in which the age dependent stability criteria coincide with those of age independent models.     

A kinetic analysis of spore inactivation in a composite heat and gamma radiation environment

This paper presents the analysis of a combined environment of dry heat and gamma radiation when applied for the purpose of spacecraft sterilization. The nature of the synergistic inactivation effect of dry heat and radiation on bacterial spores is explained using a semiempirical mathematical model, and the dependence of the inactivation rate upon a temperature dependent, nonlinear function of radiation dose rate is presented. An analysis of the temperature required for a defined population reduction with a defined upper limit on radiation dose and time is described. Also discussed is the dependency of the dose required for a defined population reduction on the radiation dose rate at any selected temperature.