The dynamics of density dependent population models

Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a strange attractor is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).Supported by NSF Grant No. BMS 74-21240.     

The Krylov-Bogoliubov-Mitropolskii method applied to models of population dynamics

In this study the possibility of applying the asymptotic method of Krylov-Bogoliubov-Mitropolskii to problems of population dynamics is shown. Especially a general Volterra-Gause-Witt type model for prey-predator interaction is investigated. A discussion on the results obtained is given for the general model and for a particular case as well.     

Linking population dynamics models with empirically derived models through phytoplankton primary production

There are alternative methods for estimation of phytoplankton primary production (PP) that are fundamentally different in the calculation approach. The process‐oriented PP model is a mechanistic, empirically derived method based on the photosynthesis–light relationships. The population dynamics‐based PP calculation, which is a synthetic method, provides a production estimate based on population dynamics of phytoplankton. These alternative methods were here compared with regard to production estimates and linked to enhance the performance of the existing models of population dynamics applied to a wide variety of lakes worldwide in terms of morphometry, nutrient status, and light environments. Estimates of PP were shown to be sensitive to changes in phytoplankton sinking and zooplankton grazing rates in both methods. Production estimates in the process‐oriented PP model were also sensitive to light‐associated parameters such as day length. Although the production estimated from the population dynamics‐based PP calculation tended to be lower than that from the process‐oriented PP model irrespective of lake morphometry, production estimates calculated from both methods with standard parameterization were comparable when production was estimated on an annual timescale. However, it was also shown that the alternative methods could produce different production estimates when estimated on shorter timescales such as cyanobacterial blooms in summer. Cyanobacteria with low mortality due to grazing and sinking losses have been considered as trophic bottlenecks, but there is increasing evidence that their mortality is, to a considerable extent, due to parasitic pathogens. In the case of cyanobacterial blooms, an addition of parasite‐related loss term (19%–33% of standing stock) resulted in a resolution of the difference in production estimates between the methods. These analyses theoretically support the critical role of parasitism and resolve the bottleneck problem in aquatic ecosystem metabolism.     

Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

高山姬鼠种群数量动态及预测预报模型

为了摸清高山姬鼠种群数量变节变动规律,探讨其种群数量预测方法,采用夹夜法调查逐月捕获率,用捕获率为预测指标,建立种群数量预测预报模型。对1996-2008年贵州省大方县高山姬鼠种群数量动态及种群数量进行分析预测,结果表明:高山姬鼠主要分布于稻田、旱地耕作区,是大方县农田害鼠优势种,占总鼠数的62.32%。10a平均捕获率为(2.58±1.27)%,全年种群数量变动曲线呈单峰型,各年度种群数量的变化曲线基本相似,一年内种群数量在6月份出现1个数量高峰,平均捕获率达(4.63±3.03)%。不同年度、不同月份、不同季节之间种群数量存在显著差异。根据历年高山姬鼠种群数量变动幅度及发生危害情况,结合当地鼠害防治指标,制定了高山姬鼠种群数量分级标准。分析1996-2008年高山姬鼠数量高峰期前各月捕获率、种群繁殖参数(性比、怀孕率、胎仔数、睾丸下降率、繁殖指数)与数量高峰期6月种群密度的关系后发现,4月份种群数量基数与6月份种群密度之间相关极显著,运用回归分析方法,建立了应用4月份种群数量基数(X)预测数量高峰期6月份种群密度(Y)的短期预测预报模型:Y=1.7558X+0.1442,可提前2个月预测当年数量峰种群密度和发生程度,经回测验证,数值和数量级预测值与实测值基本相符,数值预测和数量级预测平均吻合率为92.84%、100.00%,结果比较准确,故该预测预报模型具有一定的实用性和可行性。     

Steady states in population models with monotone,stochastic dynamics

Existence, uniqueness and asymptotic stability of stochastic equilibrium are established in multi-dimensional population models with monotone dynamics.     

昆虫种群动态模拟模型

昆虫是动物界中最大的类群,与人类有着密切的利害关系。对昆虫的数量预测与符合经济和生态规律的管理,一直都被国内外列入重点研究课题。种群动态模拟是害虫管理中重要的基础工作。近十年来,关于昆虫种群动态模型的理论和实验研究进展迅速。现分别从单种种群和多种种群两个方面对国内外近些年来昆虫种群动态模拟模型的研究进展进行了概括和总结。单种种群从两个方面阐述:一是最基本的种群动态模拟模型Log istic方程的研究成果,包括方程的修正、参数的拟合与最优捕获策略等;另一个方面是对种群动态模拟常用的矩阵模型的概述,主要介绍不等期年龄组、矩阵维数的变化、矩阵维数与历期的关系、个体之间的发育差异以及发育速率差异等等对昆虫种群动态模型的影响。多种群主要从建模和模型应用两个部分对国内外研究成果进行综述。最后,对种群动态模拟模型研究的发展方向做了深入地讨论,即在原有的数据采集工作的基础上,使用面向对象程序设计语言,把各种要素包括各种物种及各种环境条件抽象成类,用消息传递来表示昆虫种群内个体与个体、昆虫种群与环境之间的相互作用,再结合先进的数学算法,建立一个直观的、操作简单的昆虫种群动态模型库,使模型结构与现实世界有最大的相似性。这样就可以实现昆虫种群动态的可视化、立体化、实时化和精确化的监测及预测。     

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.      

On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton-Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.     

Dynamical and statistical models of vertebrate population dynamics

Population dynamics methodology now powerfully combines discrete time models (with constant parameters, density dependence, random environment, and/or demographic stochasticity) and capture-recapture models for estimating demographic parameters. Vertebrate population dynamics has strongly benefited from this progress: survival estimates have been revised upwards, trade-offs between life history traits have been demonstrated, analyses of population viability and management are more and more realistic. Promising developments concern random effects, multistate and integrated models. Some biological questions (density dependence, links between individual and population levels, and diversification of life histories) can now be efficiently attacked.     

Markov population processes as models of primate social and population dynamics

Thanks to recently developed theory of Markov population processes, models of how an individual primate migrates from one casual social group to another or from one breeding troop to another can now deal exactly with transition rates which depend nonlinearly on the sizes of both the group (or troop) left and the group (or troop) entered. Examples of such models presented here are consistent with existing observations of primate social and population dynamics and are more plausible as explanations of these data than previous linear models.     

Traveling waves in cross-diffusion models of population dynamics

"Traveling wave"--type solutions of some models with cross-diffusion were considered. It was shown that "cross-diffusion" terms, as opposed to "diffusion" terms, do not increase the dimensionality of the automodel system. The bifurcation approach was used to study the dependence of the wave solutions on the parameters of the models being considered, and the waves were classified. It was shown that, at the same parameter values, both "fast" and "slow" waves can exist and that these waves are described by different automodel systems.     

棉蚜种群动态预报模型

对1980-1989年河南棉区棉蚜种群动态分析表明,棉蚜为害可分为苗蚜和伏蚜两个时期,根据环境因子(气温、相对湿度、降雨量)和不同危害期,建立了2个模糊预报模型,并取得了较满意的预测效果。     

Generalized aphid population growth models with immigration and cumulative-size dependent dynamics

Mechanistic models in which the per-capita death rate of a population is proportional to cumulative past size have been shown to describe adequately the population size curves for a number of aphid species. Such previous cumulative-sized based models have not included immigration. The inclusion of immigration is suggested biologically as local aphid populations are initiated by migration of winged aphids and as reproduction is temperature-dependent. This paper investigates two models with constant immigration, one with continuous immigration and the other with restricted immigration. Cases of the latter are relatively simple to fit to data. The results from these two immigration models are compared for data sets on the mustard aphid in India.     

Complex dynamics of survival and extinction in simple population models with harvesting

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.