Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

Equivalence relationships between stage-structured population models

Matrix population models are widely applied in conservation ecology to help predict future population trends and guide conservation effort. Researchers must decide upon an appropriate level of model complexity, yet there is little theoretical work to guide such decisions. In this paper we present an analysis of a stage-structured model, and prove that the model's structure can be simplified and parameterised in such a way that the long-term growth rate, the stable-stage distribution and the generation time are all invariant to the simplification. We further show that for certain structures of model the simplified models require less effort in data collection. We also discuss features of the models which are not invariant to the simplification and the implications of our results for the selection of an appropriate model. We illustrate the ideas using a population model for short-tailed shearwaters (Puffinus tenuirostris). In this example, model simplification can increase parameter elasticity, indicating that an intermediate level of complexity is likely to be preferred.     

Alternative dynamical states in stage-structured consumer populations

The population dynamics of a consumer population with an internal structure is investigated. The population is divided into juvenile and adult individuals that consume different resources and do not interfere with each other. Over a broad range of external conditions (varying mortality and different resource levels), alternative stable states exist. These population states correspond to domination of juveniles and domination of adults, respectively. When mortality is varied, hysteresis between the alternative states only occurs if juveniles have more resources than adults. In the opposite case the juvenile-dominated state is stable for all values of mortality, but the adult-dominated state is not. When the population is modelled with more than one juvenile stage, the adult-dominated state becomes a periodic orbit due to a delay in the regulatory mechanism of the population dynamics. It is shown numerically that the stage-structured model converges to a model with continuous size structure for very large numbers of successive juvenile stages.     

Harvesting on a stage-structured single population model with mature individuals in a polluted environment and pulse input of environmental toxin

In the natural world, there are many species whose individual members have a life history that they take them with two distinct stages: immaturity and maturity. In particular, we have in mind mammalian populations and some amphibious animals. We improve the assumption of a single population as a whole. It is assumed that the immature individuals and mature individuals are divided by a fixed period. This paper concentrates on the study of a stage-structured single population model with mature individuals in a polluted environment and pulse input of environmental toxin at fixed moments. Furthermore, the mature individuals are harvested continuously. We show that the population goes extinct if the harvesting rate is beyond a critical threshold. Conditions for the extended permanence of the population are also examined. From the biological point of view, it is easy to protect species by controlling the harvesting amount, impulsive period of the exogenous input of toxin and toxin impulsive input amount, etc. Our results provide reasonable tactics for biological resource management.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

A density-dependent model of Cirsium vulgare population dynamics using field-estimated parameter values

Two versions of a stage-structured model of Cirsium vulgare population dynamics were developed. Both incorporated density dependence at one stage in the life cycle of the plant. In version 1 density dependence was assumed to operate during germination whilst in version 2 it was included at the seedling stage. Density-dependent parameter values for the model were estimated from annual census data in a factorial grazing experiment. Version 1 of the model produced significant estimates of density dependence under field conditions. The estimated values, when included in a simulation of the dynamics, produced two-point limit cycles under conditions of hard grazing. The limit cycles were most pronounced at the early rosette stage. Comparison of the effects of density dependence at the two different stages in the life cycle revealed a strong difference in predicted dynamics. This emphasizes the importance of determining where density dependence operates under field conditions and the potential problems of arbitrarily assigning it to particular life-history stages. Version 1 of the model produced a good prediction of observed mean plant density across the different grazing treatments (r ²=0.81, P<0.001).     

A model of physiologically structured population dynamics with a nonlinear individual growth rate

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.This work was partially supported by DGICYT PB90-0730-C02-01 and PB91-0497.     

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.     

Steady-state analysis of structured population models

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.     

Population dynamics of thrips prey and their mite predators in a refuge

Prey refuges are expected to affect population dynamics, but direct experimental tests of this hypothesis are scarce. Larvae of western flower thrips Frankliniella occidentalis use the web produced by spider mites as a refuge from predation by the predatory mite Neoseiulus cucumeris. Thrips incur a cost of using the refuge through reduced food quality within the web due to spider mite herbivory, resulting in a reduction of thrips developmental rate. These individual costs and benefits of refuge use were incorporated in a stage-structured predator-prey model developed for this system. The model predicted higher thrips numbers in presence than in absence of the refuge during the initial phase. A greenhouse experiment was carried out to test this prediction: the dynamics of thrips and their predators was followed on plants damaged by spider mites, either with or without web. Thrips densities in presence of predators were higher on plants with web than on unwebbed plants after 3 weeks. Experimental data fitted model predictions, indicating that individual-level measurements of refuge costs and benefits can be extrapolated to the level of interacting populations. Model-derived calculations of thrips population growth rate enable the estimation of the minimum predator density at which thrips benefit from using the web as a refuge. The model also predicted a minor effect of the refuge on the prey density at equilibrium, indicating that the effect of refuges on population dynamics hinges on the temporal scale considered.     

Local dynamics and dispersal in a structured population of the whirligig beetle Deneutus assimilis

The study illustrates the ecological determinants and evolutionary consequences of dispersal in the pond-living water beetle Dineutus assimilis (Coleoptera: Gyrinidae). Over 2 years, local populatiopn dynamics were studied in 51 ponds within a 60-km² study area. In most of the 31 occupied ponds, and even in large populations, abundances changed dramatically from one year to the next. Nine extinction and nine colonisation events were observed. These temporal patterns show no sign of spatial autocorrelation. Such a habitat distribution should favour high dispersal rates. Indeed, D. assimilis was found to be a very effective coloniser of newly available sites (mean propagule size: 23). A mark-recapture study showed that most dispersal occurred after diapause and over distances ranging from 100 m to at least 20 km. Yet despite frequent movement, the local variability in environmental conditions maintiins a large variance in average reproductive success per pond. Furthermore, immigration rates vary widely within a season. The apparent lack of correlation between these two sources of variation should greatly strengthen the role of drift in this system. A companion paper (Nürnberger and Harrison 1995) documents a non-random distribution of mitochondrial haplotypes due to recent population bottlenecks.     

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.     

Asymptotic behaviour of a model of hierarchically structured population dynamics

 A hierarchically structured population model with a dependence of the vital rates on a function of the population density (environment) is considered. The existence, uniqueness and the asymptotic behaviour of the solutions is obtained transforming the original non-local PDE of the model into a local one. Under natural conditions, the global asymptotical stability of a nontrivial equilibrium is proved. Finally, if the environment is a function of the biomass distribution, the existence of a positive total biomass equilibrium without a nontrivial population equilibrium is shown. Received 16 February 1996; received in revised form 16 September 1996     

An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

Observation and control in a model of a cell population affected by radiation

The effect of radiation on a cell population is described by a two-dimensional nonlinear system of differential equations. If the radiation rate is not too high, the system is known to have an asymptotically stable equilibrium. First, for the monitoring of this effect, the concept of observability is applied. For the case when the total number of cells is observed, without distinction between healthy and affected cells, a so-called observer system is constructed, which, at least near the equilibrium state, makes it possible to recover the dynamics of both the healthy and the affected cells, from the observation of the total number of cells without distinction.     

Estimation of the proliferation and maturation functions in a physiologically structured model of thymocyte development

The thymus provides a stable microenvironment for post-natal thymocyte development that is finely regulated by a complicated network of cytokines, chemokines, cell-cell contacts, etc., the dysregulations of which contribute to many immunologic diseases including malignant lymphomas. A physiologically structured model in the form of first order partial differential equation (PDE) was developed to simulate the whole process. The combined effects of the thymic microenvironment were conceptualized into two (proliferation and differentiation) fields to serve as kernels of the PDE. In this paper, a novel method is developed to estimate the maturity-time structures of the two fields based solely on cell population data that are experimentally viable. Numerical examples demonstrate the effectiveness of the present method in revealing the two-dimensional (maturity and time) landscapes of the thymic microenvironment.      

大鹏湾反曲原甲藻种群动态机理模型辨识

建立了我国南海大鹏湾反曲原甲藻种群动态机理模型,在温度、盐度、溶解氧（ＤＯ）、可溶性无机磷（ＤＩＰ）、可溶性亚硝态氮、可溶性硝态氮和酸碱度（ＰＨ值）等７个因子的分析中,辨识出温度为反曲原甲藻的限制因子;种群数量变动中引入自回归平稳随机模拟,并建立３个站位６个层面的６个自回归与非线性回归联立模型,以动态递阶的方式对反曲原甲藻种群动态进行回代,研究模型对实测值的拟合结果,拟合率达８１．７％。     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.