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.     

Optimal control problem on insect pest populations

In this article we present a model of insect infestation of grape vines and consider the optimal control of the pest through egg population removal. Existence and uniqueness of solutions are proved.     

A study on time-limited control of single-pest with stage-structure

Two kinds of time-limited pest control models of single-pest with stage-structure, which can be described by the boundary value problem of ordinary differential equation and impulsive differential equation, are presented according to the ways of artificial control (continuous control and impulsive control). The conditions under which the corresponding model has a solution are given. If the model has a solution, the corresponding aim of pest control can be achieved. The theoretical results show that both the mature and the immature pest should be controlled synchronously, otherwise the aims of pest control can not be achieved in a finite time. Finally, some discussions and numerical simulations show that the impulsive control is more practical than the continuous control.     

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.     

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 STUDY OF A PEST CONTROL MODEL INCORPORATING STERILE INSECT TECHNIQUE

The menace of insect pests is a topic of major concern throughout the world. Chemical pesticides are conventionally used to control these insect pests. However, the adverse effects of these synthetic pesticides, such as high toxicity from residues in food, contamination of water and the environment resulting in human health hazard and resistance of the pest to the pesticides have necessitated development of some nonconventional approaches of biological pest control. In this research, we have focused on a mathematical model of biological pest control using the sterile insect release technique. Unlike most of the existing modeling studies in this field that mainly deal with the pest population only, we have incorporated the crop population as a distinct dynamical equation together with the fertile and sterile insect pests. Local stability analysis is performed around the crop and fertile insect free axial equilibrium, the fertile‐insect‐free boundary equilibrium, the crop‐free boundary equilibrium and the equilibrium point of coexistence. From the study we have derived a number of thresholds for the SIRR (the main parameter for our study) that cause existence and or extinction of the crop population as well as the fertile insect pests. A global study of the model system using comparison arguments revealed existence of a global attractor for the system. Numerical simulations are done to support and augment analytical results.     

Time-limited pest control of a Lotka–Volterra model with impulsive harvest

A kind of time-limited pest control of a Lotka–Volterra model with impulsive harvest, described by the initial and boundary value problem of impulsive differential equation, is presented. The aim of pest control can be achieved if the model has a solution, otherwise the aim cannot be achieved. By the comparison principle, the conditions under which the model has a solution are found by a series of the upper solutions and the conditions under which the model has no solution are also given by a series of the lower solutions. Furthermore, if the other parameters are given, the times of harvesting pest in the given time is estimated. The theoretical results and the numerical simulations show that the density of the natural enemy will decrease when the pest decreases although the control measures to the pest do not directly affect the natural enemy. Finally, some discussions are given.     

Dynamics analysis of a pest management prey–predator model by means of interval state monitoring and control

In this work, a new pest management strategy by means of interval state monitoring is introduced into a prey–predator model, i.e. when the pest density exceeds the slightly harmful level but is below the damage level, the biological control is adopted in case of the predator density below a maintainable level, once the pest density exceeds the damage level, the chemical control is adopted. In order to determine the frequency of the chemical control and yield of releases of the predator, analysis on the existence of order-1 or order-2 periodic orbit is carried out by the construction of Poincaré map. The results could make the pest control strategy to be a periodic one without real-time monitoring the species. In addition, the stability and attractiveness of the periodic orbit are obtained by geometry approach, which ensures a certain robustness of control, i.e., even though the species densities are detected inaccurately or with a deviation, the system will be eventually stable at the periodic orbit under the control action. Furthermore, to obtain the optimum chemical control strength and yield releases of the predator, an optimization problem is constructed. The analytical results presented in the work are validated by numerical simulations for a specific model.     

Extinction and permanence in delayed stage-structure predator–prey system with impulsive effects

Based on the classical stage-structured model and Lotka–Volterra predator–prey model, an impulsive delayed differential equation to model the process of periodically releasing natural enemies at fixed times for pest control is proposed and investigated. We show that the conditions for global attractivity of the ‘pest-extinction’ (‘prey-eradication’) periodic solution and permanence of the population of the model depend on time delay. We also show that constant maturation time delay and impulsive releasing for the predator can bring great effects on the dynamics of system by numerical analysis. As a result, the pest maturation time delay is considered to establish a procedure to maintain the pests at an acceptably low level in the long term. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy-pest) model with age structure, exhibit a new modelling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

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.     

Mathematical modeling and control of population systems: Applications in biological pest control

The aim of this paper is to apply methods from optimal control theory, and from the theory of dynamic systems to the mathematical modeling of biological pest control. The linear feedback control problem for nonlinear systems has been formulated in order to obtain the optimal pest control strategy only through the introduction of natural enemies. Asymptotic stability of the closed-loop nonlinear Kolmogorov system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. Numerical simulations for three possible scenarios of biological pest control based on the Lotka–Volterra models are provided to show the effectiveness of this method.     

OPTIMIZING SPATIAL AND DYNAMIC POPULATION-BASED CONTROL STRATEGIES FOR INVADING FOREST PESTS

This paper explores spatial optimization approaches to managing invading exotic forest pests. The relevant land planning area is divided into cells, and pest population growth and dispersal are modeled with linear, continuous-variable formulations. Management actions are limited to those that apply directly to pest extermination, as opposed to host-oriented management actions. A stylized case example is used to demonstrate the potential application of the model formulation and to show that simple spatial strategies such as “barrier zone” approaches to slowing an organism's invasion may not always be optimal.     

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.     

Complex dynamics and switching transients in periodically forced Filippov prey–predator system

By employing threshold policy control (TPC) in combination with the definition of integrated pest management (IPM), a Filippov prey–predator model with periodic forcing has been proposed and studied, and the periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of the prey. This study aims to address how the periodic forcing and TPC affect the pest control. To do this, the sliding mode dynamics and sliding mode domain have been addressed firstly by using Utkin’s equivalent control method, and then the existence and stability of sliding periodic solution are investigated. Furthermore, the complex dynamics including multiple attractors coexistence, period adding sequences and chaotic solutions with respect to bifurcation parameters of forcing amplitude and economic threshold (ET) have been investigated numerically in more detail. Finally the switching transients associated with pest outbreaks and their biological implications have been discussed. Our results indicate that the sliding periodic solution could be globally stable, and consequently the prey or pest population can be controlled such that its density falls below the economic injury level (EIL). Moreover, the switching transients have both advantages and disadvantages concerning pest control, and the magnitude and frequency of switching transients depend on the initial values of both populations, forcing amplitude and ET.     

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.     

Direct spatial optimization in natural resource management: Four linear programming examples

This paper discusses methods for directly incorporating spatial relationships in natural resource optimization models. The approaches are contrasted with the conventional emphasis on adjacency restrictions. Four formulations are presented that address a variety of problems using linear (continuous-variable) programming, which allows reliable solvability of even very large models. The problems addressed include: habitat placement for the black-footed ferret (an aggressive disperser), habitat placement for the black-tailed prairie dog (a gregarious, recalcitrant disperser), placement of control efforts directed at restricting the spread of an exotic pest, and the semi-permanent placement of habitat for the northern spotted owl (a static problem).     

具有脉冲效应和综合害虫控制的捕食系统

本文通过生物控制和化学控制提出了具有周期脉冲效应与害虫控制的捕食系统. 系统保护天敌避免灭绝,在一些条件下可以使害虫灭绝.就是说当脉冲周期小于某一临界值时,存在全局稳定害虫灭绝周期解.脉冲周期增大大于临界值时,平凡害虫灭绝周期解失去稳定性并产生正周期解,利用分支理论来研究正周期解的存在性.进而,利用李雅普诺夫函数和比较定理确定了持续生存的条件.     

无公害害虫治理策略的数学研究

首先应用状态脉冲反馈控制的理论,建立了无公害害虫治理中的数学模型,并且对所建的模型进行定性分析,利用微分方程几何理论中后续函数法得到系统的阶一周期解存在的充分条件,证明该周期解是轨道渐近稳定的,同时利用数值模拟的手段讨论了系统在害虫治理中的应用意义.     

害虫综合防治建模驱动“半连续动力系统理论”兴起

以害虫综合防治数学建型为启迪,对生物数学研究的一些相关问题开展了一系列的研究.从实际问题出发,我们分别建立了"常微分方程模型"以及对应的"周期脉冲控制模型";随着害虫综合防治常态化管理和环境污染常态化防治的兴起,我们建立了"状态脉冲反馈控制系统"模型,简称"半连续动力系统"模型,提出了"半连续动力系统"相关的概念,创建了其基本理论,并且作了系统性的研究,例如:半连续动力系统的周期解以及周期解的稳定性、同宿轨和同宿分支、异宿轨和异宿分支以及"双边控制系统"等概念及其判定定理的研究;进一步将"半连续动力系统"相应的理论和方法应用于生物数学其他方面的一些相关问题的研究.本文以数学模型为载体,归纳总结了近十多年来对生物数学的研究历程,指出了当前研究中尚待解决的问题.     

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.