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.     

A Stage-Structured Stochastic Model of Sterile Male Under Immigration

Abstract

This study deals with control of pest population in agricultural ecosystem using sterile insect technique (SIT). A three-dimensional stage-structured model of the pest under the release of sterile male has been considered. This article also considers the effect of this technique under immigration of wild insects in the control area. Moreover, the deterministic model is extended to a stochastic one allowing random fluctuations around the positive interior equilibrium. The stochastic stability properties of the model are investigated, both analytically and numerically. The thresholds of sterile males that obtained from our study might be helpful to understand and implement the technique properly.     

Simple models for biological control of crop pests and their application

In this work, we construct simple models in terms of differential equations for the dynamics of pest populations and their management using biological pest control. For the first model used, the effect of the biological control is modelled by a function of repeated infinite impulses. And, our second model uses a periodic function proportional to the population to model the effect of biological control. In both cases, we present analytical solutions and derive a discrete version of them. Moreover, convergence conditions are given for periodic solutions. Finally, an application of such models is described for diamondback moth in a plot of broccoli to be controlled by the application of biological pesticides and beneficial parasitoids.     

非线性脉冲状态依赖捕食-被捕食模型的定性分析

由于资源的有限性以及害虫群体对杀虫剂的抗性发展等因素,使得杀虫剂对害虫的杀死率具有饱和效应．因此,当害虫的数量达到经济阈值时, 杀虫剂对害虫的杀死率与经济阈值有关．为了刻画上述饱和效应,建立了一类非线性脉冲状态依赖捕食被捕食模型．利用Lambert W函数和脉冲半动力系统的相关技巧,分析了模型阶1正周期解的存在性和稳定性, 得到了相应的充分条件．进而讨论了非线性脉冲与线性脉冲对阶1周期解存在性的影响.     

OPTIMAL STRATEGY FOR ERADICATION OF MULTIPLE POPULATIONS USING THE STERILE INSECT TECHNIQUE

Sterile insect technique (SIT) provides an attractive alternative to chemical pesticides for insect pest eradication. Because mass rearing facilities are expensive both to construct and to maintain, project managers have a limited number of sterile insects to use in any single eradication campaign. The problem of availability of sufficient numbers of sterile insects can be magnified when two or more independent pest infestations occur at the same time. In this paper we present the results of a modeling study in which dynamic programming is used to compute the optimal allocation strategy for release of sterile insects to control two independent infestations of Mediterranean fruit fly, Ceratitis capitata (Wied.) (medfly). We assume that a fixed number of sterile flies are available per week. Both deterministic and stochastic analyses are presented.     

Impulsive control strategy of a pest management SI model with nonlinear incidence rate

In this work, we consider a pest management SI model with impulsive release of infective pests and spraying pesticides. We prove that all solutions of the investigated system are uniformly ultimately bounded and the pest-extinction periodic solution is globally asymptotically stable when some condition is satisfied. We also obtain the permanent condition of the system. It is concluded that the approach of combining impulsive release of infective pests with impulsive spraying pesticides provides reliable tactic basis for the practical pest management.     

Sterile insect release method as a control measure of insect pests: A mathematical model

Recently non-conventional approaches of pest control are getting much more importance in different parts of the world. The main reason behind this is the long list of side effects of conventional approaches (use of pesticides etc.). The present paper focuses on one such extremely useful method of insect pest control, namely the Sterile Insect Release Method (SIRM), by using a mathematical model. A blend of dynamical behaviours of the model is studied critically, which, in turn, indicates the relevance of the method. The effect of uncertain environmental fluctuations on both fertile and sterile insects is also investigated. Our analytical findings are verified through computer simulation. Some important restrictions on the parameters of the system are mentioned, which may be implemented for a better performance of SIRM.     

Effects of population dispersal and impulsive control tactics on pest management

In this paper, we firstly consider a Lotka–Volterra predator–prey model with impulsive constant releasing for natural enemies and a proportion of killing or catching pests at fixed moments, and we have proved that there exists a pest-eradication periodic solution which is globally asymptotically stable. Further, we extend the model for the population to move in a two-patch environment. The effects of population dispersal and impulsive control tactics are investigated, i.e. we chiefly address the question of whether population dispersal is beneficial or detrimental for pest persistence. To do this, some special cases are theoretically investigated and numerical investigations are done for general case. The results indicate that for some ranges of dispersal rates, population dispersal is beneficial to pest control, but for other ranges, it is harmful. These clarify that we can get some new effective pest control strategies by controlling the dispersal rates of pests and natural enemies.     

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.     

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.     

COMBINING METHODS OF PEST CONTROL: COMPLEMENTARITY OF METHODS AND A GUIDING PRINCIPLE

Four sets of models are examined which represent various pairwise combinations of several methods of pest control. These methods involve the release of sterile male pests, the inundative release of parasitoids, insecticide application, pheromone trapping and food-baited trapping with either insecticides or sterilants. It was observed that two pest control methods will combine synergistically, and thus be complimentary, if their optimal action is at different pest densities and varies differently with pest density. The synergism thus generated by differing dependence on density, can however, be obscured if the two control methods interfere with each other in some other way, as occurs for example with the use of both insecticides and inundative release of parasitoids.     

Impulsive perturbations of a three-trophic prey-dependent food chain system

The dynamics of an impulsively controlled three-trophic food chain system with general nonlinear functional responses for the intermediate consumer and the top predator are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive controls act in a periodic fashion, the constant impulse (the biological control) and the proportional impulses (the chemical controls) acting with the same period, but not simultaneously. Sufficient conditions for the global stability of resource and intermediate consumer-free periodic solution and of the intermediate consumer-free periodic solution are established, the latter corresponding to the success of the integrated pest management strategy from which our food chain system arises. In this regard, it is seen that, theoretically speaking, the control strategy can be always made to succeed globally if proper pesticides are employed, while as far as the biological control is concerned, its global effectiveness can also be reached provided that the top predator is voracious enough or the (constant) number of top predators released each time is large enough or the release period is small enough. Some situations which lead to chaotic behavior of the system are also investigated by means of numerical simulations.     

Dynamical study of fractional order differential equations of predator‐pest models

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.     

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.     

MODELING INVASION OF PESTS RESISTANT TO Bt TOXINS PRODUCED BY GENETICALLY MODIFIED PLANTS: RECESSIVE VS. DOMINANT INVADERS

ABSTRACT. There is a growing public concern about the ecological and evolutionary consequence of the use of genetically modified organisms. We study the impact of Bt resistant pests on genetically modified Bt crops and compare exposure of Bt plants to recessive and dominant Bt resistant invaders. To simulate pest invasion we develop a conceptual reaction‐diffusion model of the Bt crop Bt susceptible insects Bt resistant insects for both the recessive and dominant pests. We show by means of computer simulations that there is a key parameter which we define as the growth number that characterizes the insects' fitness. We also show that the Bt resistant insects' invasion can lead to inhomogeneity in plant and insect spatial distributions. The plant and insect spatial patterns resulting from the Bt resistant insects' invasion are found to be dependent on the duration of the Bt resistant insect reproduction period. We compare averaged plant biomass resulting from the invasion of the dominant insects with the averaged plant biomass resulting from the invasion of the recessive insects. As a result, we show that in contrast to the recessive insects, the dominant ones initiate destruction of the plant population if the inflow of Bt susceptible insects is more than a critical value. In this case the plant biomass decays to zero. Otherwise, the plant biomass under the invasion of both the dominant and recessive insects depends on the duration of the insect reproduction period. We conclude that under invasion of dominant Bt resistant pests, the refuge strategy which has received wide acceptance in agricultural practice may not be scientifically sound practice.     

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.     

Hybrid approach for pest control with impulsive releasing of natural enemies and chemical pesticides: A plant–pest–natural enemy model

The agricultural pests can be controlled effectively by simultaneous use (i.e., hybrid approach) of biological and chemical control methods. Also, many insect natural enemies have two major life stages, immature and mature. According to this biological background, in this paper, we propose a three tropic level plant–pest–natural enemy food chain model with stage structure in natural enemy. Moreover, impulsive releasing of natural enemies and harvesting of pests are also considered. We obtain that the system has two types of periodic solutions: plant–pest-extinction and pest-extinction using stroboscopic maps. The local stability for both periodic solutions is studied using the Floquet theory of the impulsive equation and small amplitude perturbation techniques. The sufficient conditions for the global attractivity of a pest-extinction periodic solution are determined by the comparison technique of impulsive differential equations. We analyze that the global attractivity of a pest-extinction periodic solution and permanence of the system are evidenced by a threshold limit of an impulsive period depending on pulse releasing and harvesting amounts. Finally, numerical simulations are given in support of validation of the theoretical findings.     

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.     

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.