On Ebenman's model for the dynamics of a population with competing juveniles and adults

A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical valuer=1 of the inherent net reproductive rater. Stability is studied in this general setting near the bifurcation point and conditions are derived that determine which of these two bifurcating branches is the stable branch. These general results are supplemented by numerical studies of the asymptotic dynamics over wider parameter ranges where various other bifurcations and stable attractors are found. The implications of these results are discussed with respect to the effects on stability that age class competition within a population can have and whether such competition is stabilizing or destabilizing. Supported by National Science Foundation Grant No. DMS-8714810.     

Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model

In this paper, we develop a new approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems and discuss its application to the MSEIR epidemic model. For the homogeneous system, there is no attracting nontrivial equilibrium, instead we have to examine existence and stability of persistent solutions. Assuming that the host population dynamics can be described by the stable population model, we rewrite the basic system into the system of ratio age distribution, which is the age profile divided by the stable age profile. If the host population has the stable age profile, the ratio age distribution system is reduced to the normalized system. Then we prove the stability principle that the local stability or instability of steady states of the normalized system implies that of the corresponding persistent solutions of the original homogeneous system. In the latter half of this paper, we prove the threshold and stability results for the normalized system of the age-structured MSEIR epidemic model.      

A model approach to the population dynamics of the rotifer Brachionus rubens in two-stage chemostat culture

Summary A model, based on energy-flow considerations, is presented which describes the population dynamics of Brachionus rubens in the second stage of a two-stage algalrotifer chemostat. The rotifers are foodlimited with substrate-inhibition occurring at high algal densities. The model shows two stable states: steady state with constant density of rotifers and washout of the animals. Which one of the stable states is reached depends on the initial conditions.Empirical data are in general agreement with the model. Deviations may be explained by the fact that the data underlying the model calculations are based on a different food alga (Chlorella vulgaris) than the one used in the experiments (Monoraphidium minutum).The observed population growth rate reaches a maximum value of 0.84 (1/day) at algal densities of 3–4. 10⁶ cells/ml. It decreases at higher algal densities. The egg ratio is related linearily to algal density without being reduced at high algal densities.This study is dedicated to the memory of Prof. Dr. Udo Halbach     

Generalized Linear Dynamics of a Plant-Parasitic Nematode Population and the Economic Evaluation of Crop Rotations

In 1-year experiments, the final population density of nematodes is usually modeled as a function of initial density. Often, estimation of the parameters is precarious because nematode measurements, although laborious and expensive, are imprecise and the range in initial densities may be small. The estimation procedure can be improved by using orthogonal regression with a parameter for initial density on each experimental unit. In multi-year experiments parameters of a dynamic model can be estimated with optimization techniques like simulated annealing or Bayesian methods such as Markov chain Monte Carlo (MCMC). With these algorithms information from different experiments can be combined. In multi-year dynamic models, the stability of the steady states is an important issue. With chaotic dynamics, prediction of densities and associated economic loss will be possible only on a short timescale. In this study, a generic model was developed that describes population dynamics in crop rotations. Mathematical analysis showed stable steady states do exist for this dynamic model. Using the Metropolis algorithm, the model was fitted to data from a multi-year experiment on Pratylenchus penetrans dynamics with treatments that varied between years. For three crops, parameters for a yield loss assessment model were available and gross margin of the six possible rotations comprising these three crops and a fallow year were compared at the steady state of nematode density. Sensitivity of mean gross margin to changes in the parameter estimates was investigated. We discuss the general applicability of the dynamic rotation model and the opportunities arising from combination of the model with Bayesian calibration techniques for more efficient utilization and collection of data relevant for economic evaluation of crop rotations.     

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.     

Memory in idiotypic networks due to competition between proliferation and differentiation

A model employing separate dose-dependent response functions for proliferation and differentiation of idiotypically interacting B cell clones is presented. For each clone the population dynamics of proliferating B cells, non-proliferating B cells and free antibodies are considered. An effective response function, which contains the total impact of proliferation and differentiation at the fixed points, is defined in order to enable an exact analysis. The analysis of the memory states is restricted in this paper to a two-species system. The conditions for the existence of locally stable steady states with expanded B cell and antibody populations are established for various combinations of different field-response functions (e.g. linear, saturation, log-bell functions). The stable fixed points are interpreted as memory states in terms of immunity and tolerance. It is proven that a combination of linear response functions for both proliferation and differentiation does not give rise to stable fixed points. However, due to competition between proliferation and differentiation saturation response functions are sufficient to obtain two memory states, provided proliferation preceeds differentiation and also saturates earlier. The use of log-bell-shaped response functions for both proliferation and differentiation gives rise to a “mexican-hat” effective response function and allows for multiple (four to six) memory states. Both a primary response and a much more pronounced secondary response are observed. The stability of the memory states is studied as a function of the parameters of the model. The attractors lose their stability when the mean residence time of antibodies in the system is much longer than the B cells' lifetime. Neither the stability results nor the dynamics are qualitatively chanbed by the existence of non-proliferating B cells: memory states can exist and be stable without non-proliferating B cells. Nevertheless, the activation of non-proliferating B cells and the competition between proliferation and differentiation enlarge the parameter regime for which stable attractors are found. In addition, it is shown that a separate activation step from virgin to active B cells renders the virgin state stable for any choice of biologically reasonable parameters.     

Age- and density-dependent population dynamics in static and variable environments

We consider a general model of a single-species population with age- and density-dependent per capita birth and death rates. In a static environment we show that if the per capita death rate is independent of age, then the local stability of any stationary state is guaranteed by the requirement that, in the region of the steady state, the density dependence of the birth rate should be negative and that of the death rate positive. In a variable environment we show that, provided the system is locally stable, small environmental fluctuations will give rise to small age structure and population fluctuations which are related to the driving environmental fluctuations by a simple “transfer function.” We illustrate our general theory by examining a model with a per capita death rate which is age and density independent and a per capita birth rate which is zero up to some threshold age a₀, adopts a finite density-dependent value up to a maximum age a_o + α, and is zero thereafter. We conclude from this model that resonance due specifically to single-species age-structure effects will only be of practical importance in populations whose members have a life cycle consisting of a long immature phase followed by a short burst of intense reproductive effort (α a_o).     

A size dependent predator-prey interaction: who pursues whom?

We investigate the properties of an (age, size) -structured model for a population of Daphnia that feeds on a dynamical algal food source. The stability of the internal equilibrium is studied in detail and combined with numerical studies on the dynamics of the model to obtain insight in the relation between individual behaviour and population dynamical phenomena. Particularly the change in the (age, size)-relation with a change in the food availability seems to be an important behavioural mechanism that strongly influences the dynamics. This influence is partly stabilizing and partly destabilizing and leads to the coexistence of a stable equilibrium and a stable limit cycle or even coexistence of two stable limit cycles for the same parameter values. The oscillations in this case are characterized by drastic changes in the size-structure of the population during a cycle. In addition the model exhibits the usual predator-prey oscillations that characterize Lotka-Volterra models.     

祁连山大野口流域青海云杉种群数量动态

种群数量动态揭示了种群的结构特征及其潜在的驱动机制,有助于预测种群未来的动态,进而为森林生态系统的保护与恢复提供理论依据。本研究基于10.2 hm²青海云杉动态监测样地数据,以种群径级结构代替年龄结构,编制静态生命表,绘制径级结构图、存活曲线、死亡率曲线、消失率曲线和4个生存分析函数曲线,分析青海云杉种群数量特征,并利用种群数量动态变化指数和时间序列模型对种群数量动态进行预测。结果表明：（1）青海云杉种群的年龄结构近似于倒"J"型,幼苗和小树储量丰富;（2）种群存活曲线趋近于Deevey-Ⅱ型,为稳定型种群,死亡率曲线和消失率曲线变化趋势基本一致,均在第2、8龄级出现高峰期;（3）生存率曲线呈下降趋势,累计死亡率曲线呈上升趋势,死亡密度曲线缓慢下降,而危险率曲线逐渐上升,该种群具有：前期减少、中期稳定、后期衰退的生长特点;（4）种群数量变化动态指数V_pi>0,表明该种群属于增长型种群,V_pi''>0且趋近于0,则表明该种群趋近于稳定型;（5）时间序列预测分析表明,在未来2、4、6、8个龄级时间后,种群呈稳定增长趋势。研究显示,祁连山大野口流域青海云杉种群为稳定增长型种群,只要未来不遭受强烈干扰,种群数量会保持逐渐增长。针对该种群幼龄个体在前期的更新过程死亡率较高情况,建议在今后的经营管理中应重点加强对第1、2龄级植株生存环境的保护和改善,提高幼苗和小树的存活率。     

A stage structured predator-prey model and its dependence on maturation delay and death rate

Many of the existing models on stage structured populations are single species models or models which assume a constant resource supply. In reality, growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and large values of maturation time delays, the population dynamics takes the simple form of a globally attractive steady state. Our linear stability work shows that if the resource is dynamic, as in nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.Work is partially supported by NSF grant DMS-0077790.Mathamatics Subject Classification (2000):92D25, 35R10Revised version: 26 February 2004     

Fitness versus longevity in age-structured population dynamics

 We examine the dynamics of an age-structured population model in which the life expectancy of an offspring may be mutated with respect to that of the parent. While the total population of the system always reaches a steady state, the fitness and age characteristics exhibit counter-intuitive behavior as a function of the mutational bias. By analytical and numerical study of the underlying rate equations, we show that if deleterious mutations are favored, the average fitness of the population reaches a steady state, while the average population age is a decreasing function of the average fitness. When advantageous mutations are favored, the average population fitness grows linearly with time t, while the average age is independent of the average fitness. For no mutational bias, the average fitness grows as $t^{2/3}$. Received: 21 December 1999 / Revised version: 31 October 2001 / Published online: 14 March 2002     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Modeling the adaptive immune response in HBV infection

The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R ₀ < 1, (2) and the existence of four endemic steady states when R ₀ > 1. The local stability of these steady states depends on functions of R ₀. Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.     

Time-delayed model of RNA interference

RNA interference (RNAi) is a fundamental cellular process that inhibits gene expression through cleavage and destruction of target mRNA. It is responsible for a number of important intracellular functions, from being the first line of immune defence against pathogens to regulating development and morphogenesis. In this paper we consider a mathematical model of RNAi with particular emphasis on time delays associated with two aspects of primed amplification: binding of siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in the expanded production of dsRNA responsible for RNA silencing. Analytical and numerical stability analyses are performed to identify regions of stability of different steady states and to determine conditions on parameters that lead to instability. Our results suggest that while the original model without time delays exhibits a bi-stability due to the presence of a hysteresis loop, under the influence of time delays, one of the two steady states with the high (default) or small (silenced) concentration of mRNA can actually lose its stability via a Hopf bifurcation. This leads to the co-existence of a stable steady state and a stable periodic orbit, which has a profound effect on the dynamics of the system.     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications. This work was performed under the auspices of the U.S. Atomic Energy Commission.     

The dynamics of a model of a plankton-nutrient interaction

A stability analysis is given for a model of plankton dynamics introduced by Wroblewskiet al. (Global Biogeochem. Cycles 2, 199–218, 1988). The detailed dependence of the steady-states and their stability on the various model parameters is explicitly presented and analysed. It is shown that under certain conditions the coexistence of phytoplankton and zooplankton occurs in an orbitally stable oscillatory mode. A distinguished parameter is varied and the steady-states computed. The significance of the lack of stable steady-states leading to periodic population levels is investigated and related to certain oceanographic data. Partially supported by NSF Grant DMS-8902712. Supported by New Zealand University Grants Committee Postdoctoral Fellowship.     

新疆野核桃自然保护区不同坡向丛生野核桃年龄结构及数量动态

通过对新疆野核桃自然保护区丛生野核桃(Juglans regia)进行普查,以径级结构代替年龄结构,采用匀滑技术编制种群静态生命表,并运用时间序列模型预测种群数量动态,从而获得丛生野核桃树的龄级结构与动态。结果表明,不同坡向丛生野核桃以中龄级为主,年龄结构呈两头小中间大的纺锤型;4个坡向种群年龄结构的动态指数Vpi虽大于0,但最大仅为19.10%,且Vpi’极低,最大为1.12%,阳坡种群增长趋势最高,抗干扰能力最强,种群稳定性最好,半阳坡种群增长趋势最弱,抗干扰能力最弱,种群稳定性最差;各坡向种群均存在多个死亡高峰,阳坡最高峰在15龄级,半阳坡在15和18龄级,阴坡和半阴坡均在9龄级,存活曲线更接近Deevey-Ⅱ型;在未来2、5、10龄级后,中龄级个体数量减少,老龄个体呈增加趋势,丛生野核桃种群稳定性难以长期维持。因此对丛生野核桃幼苗的保护非常关键。     

Cell division and the stability of cellular populations

 This paper couples a general d-dimensional (d arbitrary) model for the intracellular biochemistry of a generic cell with a probabilistic division hypothesis and examines the consequence of division for stability of cell function and structure. We show rather surprisingly that cell division is capable of giving rise to a stable population of cells with respect to function and structure even if, in the absence of cell division, the underlying biochemical dynamics are unstable. In the context of a simple example, our stability condition suggests that rapid cell proliferation plays a stabilizing role for cellular populations. Received: 15 January 1996 / Revised version: 31 July 1998