昆虫种群动态非线性建模理论与应用

本文以非线性动力学为基础,对自然界中昆虫种群动态的复杂性、不确定性进行了建模方法的探讨,在讨论了昆虫种群动态的混沌与非线性时间序列预测方法的前提下,以山东省玉米螟等种群动态资料进行了实例分析。     

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％。     

一类随机SIRS传染病模型的持久性和灭绝性

讨论了一类非单调传染率的随机SIRS传染病模型.通过构造Liapunov函数并利用Ito公式得到了该模型全局唯一正解的存在性、p阶矩稳定、几乎必然指数稳定、随机系统的解围绕确定性模型正解的扰动、随机平均持续存在和随机灭绝等种群动力学性质的充分条件.最后,对文中的主要结论进行数值仿真.     

一个生物种群非线性泛函微分方程模型整体解的聚集性

自然界的种群,如鱼类、鸟类和有蹄兽类有聚集成群的现象,研究种群的聚集性是生物数学中的一个有趣的课题．本文讨论了一个有关生物种群的非线性泛函微分方程模型整体解的聚集性和稳定性问题．     

种群生存力分析:准确性和保护应用

目前已提出了五类估计濒危物种绝灭风险的种群生存力分析模型 ,即 :分析模型、单种群确定性模型、单种群随机模型、异质种群模型和显空间模型。模型的选择取决于物种的生活史特征和可用的数据。与用于保护实践的其他方法相比 ,种群生存力分析 (PVA)是相对准确的量化工具。然而 ,一些濒危物种种群统计学数据质量差和种群动态的有关假说模糊不清可能影响到模型预测的准确性 ,因此 ,要谨慎地使用PVA。在西方国家 ,PVA在濒危物种保护计划和管理中应用越来越广泛。它主要用于 :( 1)预测濒危物种未来的种群大小 ;( 2 )估计一定时间内物种的绝灭风险 ;( 3 )评估一套保护措施 ,确定哪个能使种群的存活时间最长 ;( 4)探索不同假说对小种群动态的影响 ;( 5 )指导濒危动物野外数据的搜集工作。我国的濒危物种很多 ,然而开展PVA研究的濒危物种却很少。应大力发展适合于模拟我国特有濒危物种及其保护问题的PVA模型     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

人口增长的数学模型探讨

本文根据人类种群与生物种群的共性与特点,给出一种人口增长的数学模型,并从生态学意义上作一些探讨。一、人口增长的确定性模型在人类种群的动态变化过程中,由于人类世代之间有重叠,因此宏观上可以认为种群数量是连续变化的,可以用一个实值连续函数x=x(t)表示t时刻人类种群的大小。人口运动的基本动态模型可表示为:     

种群增长的分段指数模型及其参数估计

本文给出了种群增长的分段指数模型其中N（t）是在时刻t种群的密度,No=N（t0）,r0和rl是群群的内禀增长率,t0是转变点,H（t-t0）=1,t≥t0,H（t-t0）=0,t＜t0．利用非线性模型的正割法（DUD,Doesn’tusederivatives）,可同时确定模型的所有参数（包括交点t0在内）．并用于描述长爪沙鼠种群动态．     

一个受到开发的扩散种群的持续与灭绝条件

本文讨论了一类服从Logistic增长规律,且受到开发或捕食的扩散种群动态模型：的渐近性质,得到了具有重要生物学意义的结果．即当开发或捕食率时,种群稳定持续,否则趋于灭绝．     

生长季节和非生长季节交替出现的种群动态模型及环境变化的影响

有些生物的生长季节和非生长季节交替出现,本文建立了描述这种生物种群动态的方程,并研究了在环境稳定、随机波动、定向变化的情况下种群的变化方式,还讨论了种群的危害及濒危情况．生长季节延长时种群增大,有害生物的危害加重,濒危生物濒危程度减轻;生长季节缩短时种群减小,有害生物的危害减轻,濒危生物更加濒危或灭绝．     

Stochastic instability and Liapunov stability

Summary The relationship between the deterministic stability of nonlinear ecological models and the properties of the stochastic model obtained by adding weak random perturbations is studied. It is shown that the expected escape time for the stochastic model from a bounded region with nonsingular boundary is determined by a Liapunov function for the nonlinear deterministic model. This connection between stochastic and deterministic models brings together various notions of persistence and vulnerability of ecosystems as defined for deterministically perturbed or randomly perturbed models.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Remarks on branching-extinction evolutionary cycles

We show in this paper that the evolution of cannibalistic consumer populations can be a never ending story involving alternating levels of polymorphism. More precisely, we show that a monomorphic population can evolve toward high levels of cannibalism until it reaches a so-called branching point, where the population splits into two sub-populations characterized by different, but initially very close, cannibalistic traits. Then, the two traits coevolve until the more cannibalistic sub-population undergoes evolutionary extinction. Finally, the remaining population evolves back to the branching point, thus closing an evolutionary cycle. The model on which the study is based is purely deterministic and derived through the adaptive dynamics approach. Evolutionary dynamics are investigated through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the evolutionary model. The general conclusion emerging from this study is that branching-extinction evolutionary cycles can be present in wide ranges of environmental and demographic parameters, so that their detection is of crucial importance when studying evolutionary dynamics.     

Demography of Verreaux’s sifaka in a stochastic rainfall environment

In this study, we use deterministic and stochastic models to analyze the demography of Verreaux’s sifaka (Propithecus verreauxi verreauxi) in a fluctuating rainfall environment. The model is based on 16 years of data from Beza Mahafaly Special Reserve, southwest Madagascar. The parameters in the stage-classified life cycle were estimated using mark-recapture methods. Statistical models were evaluated using information-theoretic techniques and multi-model inference. The highest ranking model is time-invariant, but the averaged model includes rainfall-dependence of survival and breeding. We used a time-series model of rainfall to construct a stochastic demographic model. The time-invariant model and the stochastic model give a population growth rate of about 0.98. Bootstrap confidence intervals on the growth rates, both deterministic and stochastic, include 1. Growth rates are most elastic to changes in adult survival. Many demographic statistics show a nonlinear response to annual rainfall but are depressed when annual rainfall is low, or the variance in annual rainfall is high. Perturbation analyses from both the time-invariant and stochastic models indicate that recruitment and survival of older females are key determinants of population growth rate.     

The probability of extinction in a bovine respiratory syncytial virus epidemic model

Backward bifurcation is a relatively recent yet well-studied phenomenon associated with deterministic epidemic models. It allows for the presence of multiple subcritical endemic equilibria, and is generally found only in models possessing a reasonable degree of complexity. One particular aspect of backward bifurcation that appears to have been virtually overlooked in the literature is the potential influence its presence might have on the behaviour of any analogous stochastic model. Indeed, the primary aim of this paper is to investigate this possibility. Our approach is to compare the theoretical probabilities of extinction, calculated via a particular stochastic formulation of a deterministic model exhibiting backward bifurcation, with those obtained from a series of stochastic simulations. We have found some interesting links in the behaviour between the deterministic and stochastic models, and are able to offer plausible explanations for our observations.     

Models for spatially distributed populations: The effect of within-patch variability

This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are nonlinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study.     

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.     

Sexual reproduction and parent-offspring conflict in the RNA "Tumour" virus/host relationship: implications for vertebrate oncogene evolution.

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.     

A stochastic model for facilitated transport of oxygen through tissue slices

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.     

Predicting Unobserved Exposures from Seasonal Epidemic Data

We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious individuals, our method enables a data-based model prediction of the number of unobserved exposed individuals over very long times.