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.     

Intraguild predation: fiction or reality?

Intraguild predation has become a major research topic in biological control. Quantification of multipredator interactions and an understanding of the consequences on target prey populations are needed, which only highlights the importance of population dynamics models in this field. However, intraguild predation models are usually based on Lotka–Volterra equations, which have been shown not to be adequate for modeling population dynamics of aphidophagous insects and their prey. Here we use a simple model developed for simulation of population dynamics of aphidophagous insects, which is based on the type of egg distribution made by predatory females, to estimate the real strength of intraguild predation in the aphidophagous insects. The model consists of two components: random egg distribution among aphid colonies, and between-season population dynamics of the predatory species. The model is used to estimate the proportion of predatory individuals that face a conflict with a heterospecific competitor at least once during their life. Based on this, predictions are made on the population dynamics of both predatory species. The predictions are confronted with our data on intraguild predation in ladybirds.     

Dynamics of maturing populations and their asymptotic behaviour

Summary It is well known that the partial differential equation of the traditional model describing the dynamics of an age-dependent population is of the first order hyperbolic type. An equation of that type cannot simultaneously accommodate a renewal type birth boundary condition and a death boundary condition by old age (accumulation of aging injury) and thus lacks biological realism (mortality by old age). In this paper a governing equation of a parabolic type is derived to represent the expected size of a stochastically maturing population. Using techniques well known for the solution of parabolic partial differential and Volterra integral equations, the asymptotic behaviour of such a maturing population is discussed. Due to a non-local boundary condition, the boundary value problem encountered appears to be new.     

State-dependent neutral delay equations from population dynamics

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE—shift system) is a limiting case of a system of two standard delay equations.     

Modelling and analyzing density-dependent population processes: Comparison between wild and laboratory strains of the bean weevil,Callosobruchus chinensis (L.)

Three models were constructed for analyzing the population characteristics ofC. chinensis on stored beans; model A describing the whole reproductive process with a single equation, model B describing the three age-specific processes (oviposition, egg survival and larval survival) with separate equations, and model C which describes all these processes not for the whole habitat but for the individual beans comprizing it. The logit equation was employed here as a common basis to describe the density-response relationship involved. All three models showed very good fit to the experimental data obtained for both laboratory and wild strains of the weevil. The parameter values characterizing the population dynamics were, however, widely different between the two strains; the laboratory one which had been reared for some 500 generations showed significantly higher reproductive capacity, less sensitive and gentler response to crowding in both adult and egg stages, and more uniform egg distribution among individual beans, as compared with the wild strain newly introduced. Sensitivity analyses using these models suggested that these changes in population characteristics have been attained by the process of domestication or adaptation to stable laboratory conditions through a long period of time. This process seemed in effect to have optimized the population's performances in the laboratory environment. Evolutionary significance of such optimization was discussed with reference to the selection pressure which may have acted upon individuals.     

Bifurcation theory, adaptive dynamics and dynamic energy budget-structured populations of iteroparous species

In this paper, we describe a technique to evaluate the evolutionary dynamics of the timing of spawning for iteroparous species. The life cycle of the species consists of three life stages, embryonic, juvenile and adult whereby the transitions of life stages (gametogenesis, birth and maturation) occur at species-specific sizes. The dynamics of the population is studied in a semi-chemostat environment where the inflowing food concentration is periodic (annual). A dynamic energy budget-based continuous-time model is used to describe the uptake of the food, storage in reserves and allocation of the energy to growth, maintenance, development (embryos, juveniles) and reproduction (adults). A discrete-event process is used for modelling reproduction. At a fixed spawning date of the year, the reproduction buffer is emptied and a new cohort is formed by eggs with a fixed size and energy content. The population consists of cohorts: for each year one consisting of individuals with the same age which die after their last reproduction event. The resulting mathematical model is a finite-dimensional set of ordinary differential equations with fixed 1-year periodic boundary conditions yielding a stroboscopic map. We will study the evolutionary development of the population using the adaptive dynamics approach. The trait is the timing of spawning. Pairwise and mutual invasibility plots are calculated using bifurcation analysis of the stroboscopic map. The evolutionary singular strategy value belonging to the evolutionary endpoint for the trait allows for an interpretation of the reproduction strategy of the population. In a case study, parameter values from the literature for the bivalve Macoma balthica are used.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

Complementary predation on metamorphosing species promotes stability in predator–prey systems

Functionally redundant predation and functionally complementary predation are both widespread phenomena in nature. Functional complementary predation can be found, for example, when predators feed on different life stages of their prey, while functional redundant predation occurs when different predators feed on all life stages of a shared prey. Both phenomena are common in nature, and the extent of differential life-stage predation depends mostly on prey life history; complementary predation is expected to be more common on metamorphosing prey species, while redundant predation is thought to be higher on non-metamorphosing species. We used an ordinary differential equation model to explore the effect of varying degree of complementary and redundant predation on the dynamic properties of a system with two predators that feed on an age-structured prey. Our main finding was that predation on one stage (adult or juvenile) resulted in a more stable system (i.e., it is stable for a wider range of parameters) compared to when the two predators mix the two prey developmental stages in their diet. Our results demonstrate that predator–prey dynamics depends strongly on predators' functionality when predator species richness is fixed. Results also suggest that systems with metamorphosing prey are expected to be more diverse compared to systems with non-metamorphosing prey.     

Stochastic modeling of aphid population growth with nonlinear, power-law dynamics

This paper develops a deterministic and a stochastic population size model based on power-law kinetics for the black-margined pecan aphid. The deterministic model in current use incorporates cumulative-size dependency, but its solution is symmetric. The analogous stochastic model incorporates the prolific reproductive capacity of the aphid. These models are generalized in this paper to include a delayed feedback mechanism for aphid death. Whereas the per capita aphid death rate in the current model is proportional to cumulative size, delayed feedback is implemented by assuming that the per capita rate is proportional to some power of cumulative size, leading to so-called power-law dynamics. The solution to the resulting differential equations model is a left-skewed abundance curve. Such skewness is characteristic of observed aphid data, and the generalized model fits data well. The assumed stochastic model is solved using Kolmogrov equations, and differential equations are given for low order cumulants. Moment closure approximations, which are simple to apply, are shown to give accurate predictions of the two endpoints of practical interest, namely (1) a point estimate of peak aphid count and (2) an interval estimate of final cumulative aphid count. The new models should be widely applicable to other aphid species, as they are based on three fundamental properties of aphid population biology.     

脉冲-扩散周期竞争系统解的渐近行为

研究了在周期变化环境中具有扩散及种群密度可能发生突变的两竞争种群动力系统的数学模型．模型由反应扩散方程组以及初边值及脉冲条件组成．文章建立了研究模型的上下解方法,获得了一些比较原理．利用脉冲常微分方程的比较定理以及利用相应的脉冲常微分方程的解控制和估计所讨论模型的解,研究了系统模型的解的渐近性质．     

Modeling Spruce Budworm Population Revisited: Impact of Physiological Structure on Outbreak Control

Understanding the dynamics of spruce budworm population is very important for the protection of spruce and balsam fir trees of North American forests, and a full understanding of the dynamics requires careful consideration of the individual physiological structures that is essential for outbreak control. A model as a delay differential equation is derived from structured population system, and is validated by comparing simulation results with real data from the Green River area of New Brunswick (Canada) and with the periodic outbreaks widely observed. Analysis of the equilibrium stability and examination of the amplitudes and frequencies of periodic oscillations are conducted, and the effect of budworm control strategies such as mature population control, immature population control and predation by birds are assessed. Analysis and simulation results suggest that killing only budworm larvae might not be enough for the long-term control of the budworm population. Since the time required for development during the inactive stage (from egg to second instar caterpillar) causes periodic outbreak, a strategy of reducing budworms in the inactive stage, such as removing egg biomass, should also be implemented for successful control.     

Maternal effects mediated by maternal age: from life histories to population dynamics

1. Maternal effects describe how mothers influence offspring life histories. In many taxa, maternal effects arise by differential resource allocation to young, often identified by variation in propagule size, and which affects individual traits and population dynamics. 2. Using a laboratory model system, the soil mite Sancassania berlesei, we show that, controlling for egg size, older mothers lay eggs that hatch later, develop more slowly, and mature at larger body sizes. 3. Such differences in life histories lead to marked population dynamical effects lasting for multiple generations, as evidenced by an experiment initiated with similarly sized eggs that came from young or old mothers. Differences in maturation from the initial cohort led to differences in population structure and life history that propagated the initial differences over time. 4. Maternal-age effects, which are not related to gross provisioning of the egg and are therefore phenotypically cryptic, can have profound implications for population dynamics, especially if environmental variation can affect the age structure of the adult population.     

Simplifying a physiologically structured population model to a stage-structured biomass model

We formulate and analyze an archetypal consumer-resource model in terms of ordinary differential equations that consistently translates individual life history processes, in particular food-dependent growth in body size and stage-specific differences between juveniles and adults in resource use and mortality, to the population level. This stage-structured model is derived as an approximation to a physiologically structured population model, which accounts for a complete size-distribution of the consumer population and which is based on assumptions about the energy budget and size-dependent life history of individual consumers. The approximation ensures that under equilibrium conditions predictions of both models are completely identical. In addition we find that under non-equilibrium conditions the stage-structured model gives rise to dynamics that closely approximate the dynamics exhibited by the size-structured model, as long as adult consumers are superior foragers than juveniles with a higher mass-specific ingestion rate. When the mass-specific intake rate of juvenile consumers is higher, the size-structured model exhibits single-generation cycles, in which a single cohort of consumers dominates population dynamics throughout its life time and the population composition varies over time between a dominance by juveniles and adults, respectively. The stage-structured model does not capture these dynamics because it incorporates a distributed time delay between the birth and maturation of an individual organism in contrast to the size-structured model, in which maturation is a discrete event in individual life history. We investigate model dynamics with both semi-chemostat and logistic resource growth.     

密度制约竞争二种群Volterra方程解的有界性及参数估计

本文给出密度制约且相互竞争二种群Volterra方程解的一种有界性及存在唯一性的证明。基于此,参考单种群Logistic方程反问题的方法”,给出了该Volterra方程参数的一种估计。     

Structured and Unstructured Continuous Models for <Emphasis Type="Italic">Wolbachia</Emphasis> Infections

We introduce and investigate a series of models for an infection of a diplodiploid host species by the bacterial endosymbiont Wolbachia. The continuous models are characterized by partial vertical transmission, cytoplasmic incompatibility and fitness costs associated with the infection. A particular aspect of interest is competitions between mutually incompatible strains. We further introduce an age-structured model that takes into account different fertility and mortality rates at different stages of the life cycle of the individuals. With only a few parameters, the ordinary differential equation models exhibit already interesting dynamics and can be used to predict criteria under which a strain of bacteria is able to invade a population. Interestingly, but not surprisingly, the age-structured model shows significant differences concerning the existence and stability of equilibrium solutions compared to the unstructured model.     

Effects of temperature on the immature development of the stone leek leafminer Liriomyza chinensis (Diptera: Agromyzidae)

The effect of nine constant temperatures (15, 17.5, 20, 22.5, 25, 27.5 30, 32.5, and 35 degrees C) on the development of the stone leek leafminer, Liriomyza chinensis (Kato), on Japanese bunching onion, Allium fistulosum L., was studied in the laboratory. Developmental times for immature stages were inversely proportional to temperature between 15 and 30 degrees C but increased at 32.5 degrees C. Total developmental times from egg to adult emergence decreased from 69.6 to 17.1 d for temperatures from 15 to 30 degrees C, with pupae requiring more time for development than the combined egg and larva stages. Both linear and nonlinear (Logan equation VI) models provided a reliable fit of development rates versus temperature for all immature stages. The lower developmental thresholds that were estimated from linear regression equations for the egg, first, second, and third instars, total larva, egg-larval, pupa, and total combined immature stages were 12.1, 10.6, 13.6, 8, 9.6, 11.3, 11.2, and 11.4 degrees C, respectively. The degree-day accumulation was calculated as 312.5 DD for development from egg to adult emergence. By fitting the nonlinear models to the data, the upper and optimal temperatures for egg, larva, pupa, and total immature stages were calculated as 37.8 and 31.7, 34.9 and 30.1, 35.8 and 30.6, and 35.0 and 30.9 degrees C, respectively. These data are useful for predicting population dynamics of L. chinensis under field conditions and determining the maximum proportion of susceptible individuals for facilitating improved timing of application of control measures.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

A stochastic model of the dynamics of host-pathogen systems with mutation

In this paper a stochastic model of the dynamics of host-pathogen systems with mutation is constructed. In previous works deterministic models of host-pathogen systems with no mutation were considered. The evolution of the pathogen population in any generation of the host is formulated as a multidimensional birth and death process, while the evolution of genotypic frequencies in successive generations of the host is described by a solution of a nonlinear vector difference equation. A general solution of the differential equations of the multidimensional birth and death process is presented and expressions for the stationary distribution, whenever it exists, and the mean time to extinction, when absorbing states are present, are derived. Some answers to questions raised in the discussion of a previous paper (Mode, 1962) are also contained in this paper. The research reported in this paper was supported by the United States Atomic Energy Comission, Division of Biology and Medicine Project AT(45-1)-1729.     

Egg banks in freshwater zooplankton: evolutionary and ecological archives in the sediment

Many representatives of freshwater zooplankton produce at some stage in their life cycle resting stages. A variable portion of the eggs of the previous growing period will hatch at the next occasion while the remaining ones are added to a persistent egg bank, where they can remain viable for decades or longer. The importance of the study of resting eggs and egg banks in general for such different disciplines as taxonomy, ecological biogeography, paleolimnology, nature conservation, evolutionary ecology and community and population ecology is generally appreciated. The major current and expected future developments in this rapidly expanding field of research are presented here. The structure and dynamics of the egg bank are determined by the life history characteristics of the species (or local population), the hatching phenology of their resting stages, and the characteristics of the habitat. The horizontal distribution of dormant stages is generally patchy, with a greater density in the deeper and/or windward parts of a pond or lake. In sediment cores, most viable (responsive) eggs occur in the upper centimeters, although vertical variation related to the history of fish predation or water quality occurs. The accumulation of resting stages of different species, generations and genotypes with variable regeneration niches results in a mixed egg bank with greater potential biodiversity than the active community sampled at any one moment. Through the benthic–pelagic coupling, this dormant reservoir may have considerable impact on the evolutionary potential of the organisms, the ecological dynamics of the community and the distribution of species. Egg banks can be considered the archive of the local habitat, since the pattern of changes in species assemblage and genotypes from the past up to the present reflect changes due to natural or anthropogenic impact that can be used to reconstruct evolutionary processes or even to restore the local habitat. Overlooking the egg bank as an important component of zooplankton communities may lead to erroneous interpretations in the analysis of community and population genetic structure. This review integrates technical and scientific information needed in the study of the structure and function of egg banks in zooplankton with special focus on the fascinating latest developments in the field.