Periodic solution of a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control

In this paper, a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型的周期解

研究了一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型．证明了无脉冲系统的负向全局渐近稳定性及带有脉冲状态反馈控制系统具有阶一周期解,并且给出阶一周期解存在和稳定的充分条件．数值模拟验证了理论结果．     

Modelling and analysis of impulsive releases of sterile mosquitoes

To study the impact of releasing sterile mosquitoes on mosquito-borne disease transmissions, we propose two mathematical models with impulsive releases of sterile mosquitoes. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We also establish thresholds for the control of the wild mosquito population by selecting the release rate and the release period. In the second model, the impulsive releases are determined by the closely monitored wild mosquito density, or the state feedback. We prove the existence of an order one periodic solution and find a relatively small attraction region, which ensures the wild mosquito population is under control. We provide numerical analysis which shows that a smaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback releases.     

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincare map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers' favour.     

状态依赖脉冲微分方程的周期解的存在性及其在害虫治理中的应用

在文献[1]中研究的状态依赖脉冲微分方程的基础上,推广了其中的判定一般性平面自治状态依赖脉冲微分方程的准则,并利用它得到了文献[1]中所没有涉及到的情况下的状态依赖脉冲微分方程的阶一周期解☆栌在性．之后本文以此为基础并结合数值模拟的手段讨论了系统在农业害虫治理中的一些应用意义．     

基于脉冲扰动作用下的一个捕食者-食饵模型的动力学性质

基于害虫的生物控制和化学控制策略,考虑到化学杀虫剂对天敌的影响,利用脉冲微分方程建立了在不同的固定时刻分别喷洒杀虫剂和释放天敌的具有依氏(Ivlev)功能性反应的捕食者-食饵脉冲动力系统．证明了当脉冲周期小于某个临界值时,系统存在一个渐近稳定的害虫根除周期解,否则系统是持续生存的．通过分析表明如果采取有效的化学控制策略,那么这种综害虫合控制策略更有效．     

固定周期脉冲微分方程到状态依赖脉冲的转化及应用

本文研究了一类二维状态依赖脉冲微分方程的阶1周期解存在性和轨道稳定性条件．然后,将一维固定周期脉冲的微分方程转化为二维状态依赖脉冲微分方程,研究其阶一周期解的存在性和稳定性．作为应用,我们研究了固定周期常数收获的Logistic方程的动力学性质,以及两个固定周期注射药物单室扩散模型的动力学性质．     

The dynamics of plant disease models with continuous and impulsive cultural control strategies

Plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are proposed and investigated. This novel theoretical framework could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Firstly, continuous replanting of healthy plants and removing of infected plants is taken. The existence and stability of disease-free equilibrium and positive equilibrium are studied and continuous cultural control strategy is given. Secondly, plant disease model with impulsive replanting of healthy plants and removing of infected plants is also considered. Using Floquet's theorem and small amplitude perturbation, the sufficient conditions under which the infected plant free periodic solution is locally stable are obtained. Moreover, permanence of the system is investigated. Under certain parameter spaces, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. The modeling methods and analytical analysis presented can serve as an integrating measure to identify and design appropriate plant disease control strategies.     

一类食饵具时滞与扩散的非线性脉冲捕食系统正周期解存在性

主要讨论了一类食饵具有时滞与扩散的非线性脉冲捕食系统正周期解的存在性问题,应用迭合度理论,得到系统存在正周期解的充分条件,推广了没有脉冲时的情形.数值模拟进一步验证了结论的正确性.     

具有阶段结构和群体防卫的脉冲捕食系统

讨论了食饵具有群体防卫和捕食者具有阶段结构的脉冲控制捕食系统,根据Floquet乘子理论和脉冲比较定理,获得了食饵（害虫）灭绝周期解局部稳定与系统持续生存的充分条件．利用Matlab软件对害虫灭绝周期解和害虫周期爆发现象进行了数值模拟,并揭示了诸如高倍周期振荡,混沌,吸引子突变等复杂的动力学现象．得出的结论为害虫治理提供了可靠的策略依据．     

Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control

In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one.     

The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects

In this paper, we consider the prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects. By applying the Floquet theory of linear periodic impulsive equation, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, that is, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, and thus, we further prove that the system is uniformly permanent if the impulsive period is larger than some critical value, and meanwhile we also give the conditions for the extinction of one of the two preys and permanence of the remaining species. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population's intrinsic growth rate. The resulting bifurcation diagrams show that with the varying of parameters, the system experiences process of cycles, periodic windows, periodic-doubling cascade, symmetry breaking bifurcation as well as chaos.     

脉冲捕获捕食者与食饵具阶段结构的捕食-食饵模型

研究脉冲捕获捕食者与食饵具阶段结构的捕食-食饵模型.利用频闪映射理论,得到食饵灭绝的周期解是全局吸引的;运用时滞脉冲微分方程理论,证明了此系统是持久的.本文的结论为生态保护提供了可靠的策略依据.     

The geometrical analysis of a predator-prey model with two state impulses

Using successor functions and Poincaré-Bendixson theorem of impulsive differential equations, the existence of periodical solutions to a predator-prey model with two state impulses is investigated. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are given. Some simulations are exerted to prove the results.     

具收获与脉冲的时滞捕食模型研究

研究了与生物资源管理相关的食饵具脉冲扰动与成年捕食者具连续收获的阶段结构时滞捕食-食饵模型.利用离散动力系统的频闪映射和脉冲时滞微分方程理论,得到了捕食者灭绝周期解的全局吸引和系统持久的充分条件,也证明了系统的所有解的一致完全有界.结论为现实的可再生生物资源管理提供了可靠的策略依据.     

Integrated pest management models and their dynamical behaviour

Two impulsive models of integrated pest management (IPM) strategies are proposed, one with fixed intervention times and the other with these unfixed. The first model allows natural enemies to survive but under some conditions may lead to extinction of the pest. We use a simple prey-dependent consumption model with fixed impulsive effects and show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than certain critical values. The effects of pest resistance to pesticides are also studied. The second model is constructed in the light of IPM practice such that when the pest population reaches the economic injury level (EIL), a combination of biological, cultural, and chemical tactics that reduce pests to tolerable levels is invoked. Using analytical methods, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given Economic Threshold (ET). The complete expression for this periodic solution is given and the ET is evaluated for given parameters.We also show that in some cases control costs can be reduced by replacing IPM interventions at unfixed times with periodic interventions. Further, we show that small perturbations of the system do not affect the existence and stability of the periodic solution. Thus, we provide the first demonstration using mathematical models that an IPM strategy is more effective than classical control methods.     

研究脉冲竞争系统周期解新的单调迭代方法(英文)

文章研究了一类正常细胞和癌细胞相互作用的竞争系统周期解的存在性.数学模型包括竞争型的Lotka-Volterra方程组与描述周期性化疗的脉冲条件.文章建立了一类新的单调迭代方法,该方法是构造性的,周期解可以由一个线性迭代过程得到,每一步迭代只需求解一个脉冲微分方程初值问题.文章获得了系统至少存在一个严格正的周期解的充分条件.     

一类具有两次脉冲和非线性传染率的害虫管理SI模型的灭绝性与持久性

讨论了一类在两个不同固定时刻分别释放染病害虫和喷洒农药且具有HollingⅡ类传染率的SI模型.通过脉冲微分方程的Floquet理论和小幅扰动技巧,证明了当释放的染病害虫数量超过某个临界值时,系统存在一个渐进稳定的易感害虫根除周期解,否则系统是持续生存的.通过数值模拟,验证了所得结论的正确性及系统动力学行为的复杂性,分析说明了所提出的脉冲控制策略的有效性.     

The dynamics of a Lotka–Volterra predator–prey model with state dependent impulsive harvest for predator

According to the economic and biological aspects of renewable resources management, we propose a Lotka–Volterra predator–prey model with state dependent impulsive harvest. By using the Poincaré map, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three under some conditions. Numerical results are carried out to illustrate the feasibility of our main results. The bifurcation diagrams of periodic solutions are obtained by using the numerical simulations, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations, which implies that the presence of pulses makes the dynamic behavior more complex.     

具有脉冲生育和脉冲收获的时滞SEI害虫治理模型的动力学分析

根据控制害虫的生物策略,考虑了一个不同时刻害虫具有脉冲生育和对害虫进行脉冲收获的时滞的SEI害虫治理模型.我们证明了系统所有的解都是一致有界的,同时获得了无病周期解是全局吸引的充分条件.进一步,得到了具有时滞的系统持续生存的充分条件.基于研究所得到的结果,作者提出了害虫治理的一些合理建议.