具非线性传染率与生物化学控制的害虫管理S-I模型

讨论了具有非线性传染率与脉冲控制的害虫管理S-I传染病模型,此模型考虑的是脉冲投放病虫和喷洒农药.不但得到了系统的所有解的一致完全有界,而且得到了害虫灭绝的边界周期解的全局渐进稳定和系统的一致持久的条件.为实际的害虫管理提供了可靠的理论依据.     

Nonlinear incidence rate of a pest management SI model with impulsive control strategy

In this work, we consider a pest management SI model with concerns about releasing of infective pests and spraying pesticides at different fixed moments. We prove that all solutions the investigated system are uniformly ultimately bounded, and there exists globally asymptotic stable pest‐extinction boundary periodic solution when certain condition is satisfied. Furthermore, the permanent condition of the system is also obtained. It is concluded that the approach, which combines releasing infective pests with spraying pesticides in different fixed moments, provides reliable tactic basis for the practical pest management. Copyright © 2009 John Wiley & Sons, Ltd.     

基于喷洒杀虫剂及释放病虫的脉冲控制害虫模型

基于喷洒杀虫剂及释放病虫的综合控制害虫策略,建立了具有脉冲控制的微分方程模型.利用脉冲微分方程的F loquet理论、比较定理,证明了害虫灭绝周期解的全局渐近稳定性与系统的持久性.     

A delayed epidemic model with stage-structure and pulses for pest management strategy

From a biological pest management standpoint, epidemic diseases models have become important tools in control of pest populations. This paper deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive, provided that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than another critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its attractivity. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.     

A impulsive infective transmission SI model for pest control

In this paper, we propose a model with impulsive control of epidemics for pest management. By using Floquet's theorem, small‐amplitude perturbation skills and comparison theorem, we show that there exists a globally asymptotically stable susceptible pest‐eradication periodic solution when the release amount of infective pests is larger than some critical value. However, when the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial periodic susceptible pest‐eradication solution loses its stability. Further, the existence of a positive periodic endemic solution and other rich dynamics are also studied by numerical simulation. Therefore, we can use the amount of release of infective pests to control susceptible pests at desirable low levels. Copyright © 2007 John Wiley & Sons, Ltd.     

Modelling and analysis of an impulsive SI model with Monod-Haldane functional response

An impulsive SI model with Monod-Haldane functional response for pest control is proposed and investigated. First, we have proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the above system can be permanent. Then, influences of impulsive perturbation including impulse period, the time of spraying pesticide and the quantity of releasing infective pests on the above system have been studied. Moreover, numerical simulations show that the system has rich dynamical behaviors. Finally, it is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying is more effective than the classical one if the chemical control is adopted rationally.     

An impulsive predator–prey model with disease in the prey for integrated pest management

In this paper, an impulsive predator–prey model with disease in the prey is investigated for the purpose of integrated pest management. In the first part of the main results, we get the sufficient condition for the global stability of the susceptible pest-eradication periodic solution. This means if the release amount of infective prey and predator satisfy the condition, then the pest will be doomed. In the second part of the main results, we also get the sufficient condition for the permanence of the system. This means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. In the last section, we interpret our mathematical results. We also point out some possible future work.     

The dynamics of pest control pollution model with age structure and time delay

In this paper, by using pollution model and impulsive delay differential equation, we investigate the dynamics of a pest control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. We assume only the pests are affected by pesticide. We show that there exists a global attractive pest-extinction periodic solution when the periodic natural enemies release amount μ₁ and pesticide input amount μ₂ are larger than some critical value. Further, the condition for the permanence of the system is also given. By numerical analyses, we also show that constant maturation time delay, pulse pesticide input and pulse releasing of the natural enemies can bring obvious effects on the dynamics of system. We believe that the results will provide reliable tactic basis for the practical pest management.     

The onset of positive periodic solutions for a biochemical pest management model

In this paper, the bifurcation of nontrivial periodic solutions for an impulsively perturbed system of ordinary differential equations which models an integrated pest management strategy is studied by means of a fixed point approach. A biological control, consisting in the periodic release of infective pests, and a chemical control, consisting in pesticide spraying, are employed to maintain susceptible pests below an acceptable level. It is assumed that the biological and chemical control act with the same periodicity, but not in the same time. It is then shown that if the constant amount of infective pests released each time reaches a certain threshold value, then the trivial susceptible pest-eradication periodic solution loses its stability, which is transferred to a newly emerging nontrivial periodic solution.     

Dynamics on a pest management SI model with control strategies of different frequencies

Based on spraying pesticide and introducing infected pest and natural enemy for pest control, an SI ecological epidemic model with different frequencies of pesticide applications and infected pests and natural enemy releases is proposed and studied. With spraying either more or less frequently than the releases, the threshold condition of existence and global attractiveness of susceptible pest extinction periodic solution is obtained. We investigate the effects of the pest control tactics on the threshold conditions. We also show that the system has rich dynamics including period-doubling bifurcations and chaos as the release period increases, which implies that the presence of impulsive intervention makes the dynamic behavior more complex. Finally, to see how the pesticide applications can be reduced, we develop a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reaches the given Economic Threshold. It indicates that the hybrid method is the most effective method to control pest and the frequency of pesticide applications largely depends on the initial densities and the control tactics.     

Mathematical modelling to control a pest population by infected pests

In this paper, we formulate and investigate the pest control models in accordance with the mathematical theory of epidemiology. We assume that the release of infected pests is continuous and impulsive, respectively. Therefore, our models are the ordinary differential equations and the impulsive differential equations. We study the global stability of the equilibria of the ordinary differential equations. The permanence of the impulsive differential equations is proved. By means of numerical simulation, we obtain the critical values of the control variable under different methods of release of infected pests. Thus, we provide a mathematical evidence in the management of an epidemic controlling a pest.     

An impulsively controlled predator–pest model with disease in the pest

In this paper, we consider an integrated pest management model with disease in the pest and a stage structure for its natural predator, which is subject to impulsive and periodic controls. A nonlinear incidence rate expressed in an abstract form, is used to describe the propagation of the disease, which is spread through the periodic release of infective pests, the functional response of the mature predator also being given in an abstract, unspecified form. Sufficient conditions for the local and global stability of the susceptible pest-eradication periodic solution are found by means of Floquet theory and comparison methods, the permanence of the system also being discussed. These stability conditions are shown to be biologically significant, being reformulated as balance conditions for the susceptible pest class.     

Dynamics of an impulsive control system which prey species share a common predator

In an ecosystem multiple prey species often share a common predator and the interactions between the preys are neutral. In view of these facts and based on a multiple species prey–predator system with Holling IV and II functional responses, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a locally asymptotically stable pest-eradication periodic solution under the assumption that the impulsive period is less than some critical value (or the release amount of the predator is greater than another critical value). Permanence conditions are established when the impulsive period is greater than another critical value (or the release amount of the predator is less than some critical value). Numerical results show that the system we consider has more complex dynamics including period solution, quasi-periodic oscillation, chaos, intermittency and crises.     

Modelling approach for biological control of insect pest by releasing infected pest

Models of biological control have a long history of theoretical development that have focused on the interactions between a predator and a prey. Here we have extended the classical epidemic model to include a continuous and impulsive pest control strategies by releasing the infected pests bred in laboratory. For the continuous model, the results imply that the susceptible pest goes to extinct if the threshold condition R₀ < 1. While R₀ > 1, the positive equilibrium of continuous model is globally asymptotically stable. Similarly, the threshold condition which guarantees the global stability of the susceptible pest-eradication periodic solution is obtained for the model with impulsive control strategy. Consequently, based on the results obtained in this paper, the control strategies which maintain the pests below an acceptably low level are discussed by controlling the release rate and impulsive period. Finally, the biological implications of the results and the efficiency of two control strategies are also discussed.     

Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with saturation incidence concerning impulsive control strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. When the impulsive period is larger than some critical value, the stability of the pest-eradication periodic solution is lost; the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Numerical results show that the system we consider can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by period-doubling cascade, symmetry-breaking pitchfork bifurcation, quasi-periodic oscillate, chaos, and non-unique dynamics.     

An integrated pest management model with delayed responses to pesticide applications and its threshold dynamics

Pulse-like pest management actions such as spraying pesticides and killing a pest instantly and the release of natural enemies at critical times can be modelled with impulsive differential equations. In practice, many pesticides have long-term residual effects and, also, both pest and natural enemy populations may have delayed responses to pesticide applications. In order to evaluate the effects of the duration of the residual effectiveness of pesticides and of delayed responses to pesticides on a pest management strategy, we developed novel mathematical models. These combine piecewise-continuous periodic functions for chemical control with pulse actions for releasing natural enemies in terms of fixed pulse-type actions and unfixed pulse-type actions. For the fixed pulse-type model, the stability threshold conditions for the pest eradication periodic solution and permanence of the model are derived, and the effects of key parameters including killing efficiency rate, decay rate, delayed response rate, number of pesticide applications and number of natural enemy releases on the threshold values are discussed in detail. The results indicate that there exists an optimal releasing period or an optimal number of pesticide applications which maximizes the threshold value. For unfixed pulse-type models, the effects of the killing efficiency rate, decay rate and delayed response rate on the pest outbreak period, and the frequency of control actions are also investigated numerically.     

Global attractivity and permanence of a stage-structured pest management SI model with time delay and diseased pest impulsive transmission

In this paper, we consider a stage-structured pest management SI model with time delay and diseased pests impulsive transmission. We obtain the sufficient conditions of the global attractivity of pest-extinction boundary periodic solution and the permanence of the system. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide a reliable tactic basis for the practice of pest management.     

A predator–prey model with disease in the prey and two impulses for integrated pest management

In this paper, a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. In the first part of the main results, the sufficient condition for the global stability of the susceptible pest-eradication periodic solution is obtained, which means if the release amount of infective prey and predator satisfy the condition, then the pest will be controlled. The sufficient condition for the permanence of the system is also obtained subsequently, which means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. At last, we interpret our mathematical results.     

椰心叶甲虫综合防治脉冲动力学模型的周期解

椰心叶甲虫是棕榈科植物最主要的害虫之一.论文针对两类寄生蜂攻击椰心叶甲虫不同年龄阶段的特点,建立了阶段结构的脉冲定期喷洒药物和释放天敌的综合防治模型.通过重合度理论和分析工具,证明了该模型周期解的存在性,给出了周期解存在的充分条件,并通过数值模拟验证了理论结果的有效性.     

Geometrical analysis of a pest management model in food-limited environments with nonlinear impulsive state feedback control

In this paper, a nonlinear impulsive state feedback control system is proposed to model an integrated pest management in food-limited environments. In the system, impulsive feedback control measures are implemented to control pests on the basis of the quantitative state of pests. Mathematically, an intuitive geometric analysis is used to indicate the existence of periodic solutions. The stability of periodic solutions is investigated by using Analogue of Poincar\''{e} Criterion. At last, numerical simulations are given to verify the theoretical analysis.