Periodic solutions of population models in a periodically fluctuating environment.

A periodically fluctuating environment is assumed in a population-modeling process that generates nonautonomous difference equations. The existence and uniqueness of periodic solutions are studied. A sufficient condition for existence and a necessary condition for uniqueness are obtained. Stability of the periodic solutions is investigated. Several numerical examples are given to illustrate the basic results, and a brief discussion is presented.     

具有无限时滞和扩散项的非自治竞争系统的正周期解(英文)

研究了一类具有无限时滞和扩散项的非自治竞争系统,利用重合度的廷拓定理,得到了该系统正周期解存在的充分性条件.     

一类食饵具有S-型变收获率的非自治离散捕食系统渐近行为

建立并研究了一类具有S-型变收获率的非自治离散捕食系统,应用差分不等式、Brouwer不动点定理和分析技巧分别建立了系统的持久生存性,正周期解的存在性及全局渐近稳定性的充分条件,我们还给出了具体验证例子及相应的数值模拟结果.     

一类带参数的泛函微分方程的正周期解的存在性(英文)

利用锥上不动点理论,本文研究了一类非线性泛函微分方程正周期解的存在多样性和ω-周期解的不存在性.获得了一些新的结果.应用这些新的结果,讨论了一类带参数的血细胞生成模型,给出该模型的正周期解的存在性,此结果利用以前的方法是无法得到的.     

Asymptotic behaviour of the non-autonomous competing two-species Lotka-Volterra models with impulsive effect

In this paper, the nonautonomous competing two-species Lotka-Volterra models with impulsive effect are considered, where all the parameters are time-dependent and asymptotically approach the corresponding periodic functions. Under some conditions, it is shown that the semi-trivial positive solutions of the models asymptotically approach the semi-trivial positive periodic solutions of the corresponding periodic system. It is also shown that the positive solution of the models asymptotically approach the positive periodic solution of the corresponding periodic system.     

Asymptotic behaviour of the non-autonomous competing two-species Lotka–Volterra models with impulsive effect

In this paper, the nonautonomous competing two-species Lotka–Volterra models with impulsive effect are considered, where all the parameters are time-dependent and asymptotically approach the corresponding periodic functions. Under some conditions, it is shown that the semi-trivial positive solutions of the models asymptotically approach the semi-trivial positive periodic solutions of the corresponding periodic system. It is also shown that the positive solution of the models asymptotically approach the positive periodic solution of the corresponding periodic system.     

具性别和阶段结构的非自治二种群捕食者-食饵系统研究

讨论了一类食饵具有性别结构,捕食者具有阶段结构的非自治捕食者．食饵系统,运用Liapunov函数方法,得到了该系统一致持续生存的充分条件．对于该模型的周期系统,在适当条件下,存在唯一、全局渐近稳定的周期解．对更具普遍意义的概周期现象,也得出了概周期正解唯一存在且全局渐近稳定的充分条件．     

一类捕食-被捕食系统周期解的存在性

对于二维非自治捕食-被捕食系统,本文讨论了它的解的有界性,进而得到周期解和正周期解的存在性。     

关于一类非自治阶段结构捕食系统的持久性与周期解

本文研究了一类非自治阶段结构捕食系统的渐近性质,得到在适当的条件下系统的持久性,对应周期系统正周期解的存在性、唯一性以及全局渐近稳定性。     

稳定有界的Logistic方程的最优捕获策略

考虑单种群非自治的Logistic方程的开采问题．在R^ 中都存在均值的意义下,作为周期和概周期函数的推广,首先给出稳定有界函数的概念．然后定义一个新的最终最优收获策略用于处理我们的问题．选择单位时间的最大持久收益的极限均值作为管理目标。同时得到了最佳的种群水平．作为应用,我们以概周期系数的Logistic方程为例,表明我们的结果不仅推广了经典的Clark关于自治的Logistic方程的收获问题,而且推广了范猛和王克的关于周期的Logistic方程的收获问题的结果．     

非自治LOTKA-VOLTERRA生态系统的有界性及周期解的存在性

本文利用图论和Liapunov方法研究了非自治LOTKA-VOLTERRA生态系统的有界性和周期解的存在性,得到了一些新的结果．     

一类具有三个阶段结构的扩散捕食系统研究

讨论了一类捕食者具有三个阶段结构和Beddington型功能性反应,食饵可以在两个斑块间扩散的非自治捕食者-食饵系统.运用Liapunov函数方法,得到了该系统一致持续生存的充分条件.对于该模型的周期系统,讨论了存在唯一、全局渐近稳定的周期解的条件.     

Mathematical analysis of a model for a plant-herbivore system

The apple twig borer (Amphicerus bicaudatus) is an insect pest of the grape vine, causing considerable damage to the grape vine in early spring. A simple difference equation model is formulated and analysed for this plant-herbivore system based on two control strategies, cane removal and pesticide application. The system has two equilibria, one where the pest is present and one where the pest is absent. Regions are found in parameter space for global stability of the equilibria and in the absence of global stability it is shown that there exist periodic or quasiperiodic solutions.     

On periodic solutions of a delay integral equation modelling epidemics

Summary A delay-integral equation, proposed by Cooke and Kaplan in [1] as a model of epidemics, is studied. The focus of this work is on the qualitative behavior of solutions as a certain parameter is allowed to vary. It is shown that if a certain threshold is not exceeded then solutions tend to zero exponentially while if this threshold is exceeded, periodic solutions exist. Many features of the numerical studies in [1] are explained.     

An analysis of a dendritic neuron model with an active membrane site

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.This work was supported in part by NSF Grants MCS 83-00562 and MDS 85-01535     

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.     

A simple parasite model with complicated dynamics

 During their first year of life sheep acquire parasites through grazing, and simultaneously build up an immunity to infection. At the beginning of each year non-immune lambs are introduced onto contaminated pasture. We represent this process by differential equations describing the within-year dynamics, and defining a difference equation that describes the between-year dynamics. An example with two system parameters is analysed in detail. It is shown that regions exist in parameter space where periodic (between-year) or aperiodic solutions occur. Parasite control schemes could change the system dynamics from a stable equilibrium to complicated long-term fluctuations. Received: 11 August 1997 / Revised version: 4 December 1997     

具时滞的n斑块捕食-食饵扩散系统的正周期解

讨论了具时滞和功能反应的n斑块捕食—食饵扩散系统,利用新的方法,得到了该系统正周期解存在性的判别准则．     

An equation of growth of a single species with realistic dependence on crowding and seasonal factors

Various biological phenomena lead to single species models where the relative rate of increase is a non-monotone function of the density, i.e. depensation models. A brief survey of the literature and some new models are given. A nonlinear nonautonomous O.D.E. is proposed as a general depensation model in a periodically fluctuating environment. Results on existence, multiplicity and global stability of periodic solutions are given.     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001