具有经济阈值的植物疾病模型

研究了具有经济阈值和人文控制策略的植物疾病模型．根据某一参数的三种情况分析了唯一的正的周期解的存在性,并利用定性理论给出了在该参数某种范围下周期解全局稳定的充分条件,同时得到在其它两种情况下周期解的不稳定性．文章所得结论推广了综合疾病管理中植物疾病模型的经典结论．     

Optimal control of a vectored plant disease model for a crop with continuous replanting

ABSTRACT

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.     

Dynamics of plant mosaic disease propagation and the usefulness of roguing as an alternative biological control

In this research article, an epidemiological model is formulated for mosaic disease considering plant and vector populations. Plant host population has been divided into three compartments namely healthy, latently infected and infected ones, and vector population is divided into two compartments: non-infective and infective vectors. The system possesses three equilibria: plant-only, disease-free and endemic equilibrium. Plant-only equilibrium is always unstable; disease-free equilibrium is stable when the basic reproduction number, R₀, is less than unity and unstable for when it crosses unity, and ensure existence of an endemic equilibrium which may be stable or can undergo a Hopf bifurcation. Finally, impulse periodic roguing with varied rate and time interval is adopted for cost effective and eco-friendly disease control and future direction of agriculture management. The dynamics of the impulsive system has also been analysed. Detailed numerical simulations are employed to support the analytical results. We found that roguing is most cost effective and useful management for mosaic disease eradication of plants if applied at proper rate and interval.     

Pest management through continuous and impulsive control strategies

In this paper, we propose two mathematical models concerning continuous and, respectively, impulsive pest control strategies. In the case in which a continuous control is used, it is shown that the model admits a globally asymptotically stable positive equilibrium under appropriate conditions which involve parameter estimations. As a result, the global asymptotic stability of the unique positive equilibrium is used to establish a procedure to maintain the pests at an acceptably low level in the long term. In the case in which an impulsive control is used, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than some critical value. 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 susceptible pest-eradication solution loses its stability. Further, the existence of a nontrivial periodic solution is also studied by means of numerical simulation. Finally, the efficiency of continuous and impulsive control policies is compared.     

Non-smooth plant disease models with economic thresholds

In order to control plant diseases and eventually maintain the number of infected plants below an economic threshold, a specific management strategy called the threshold policy is proposed, resulting in Filippov systems. These are a class of piecewise smooth systems of differential equations with a discontinuous right-hand side. The aim of this work is to investigate the global dynamic behavior including sliding dynamics of one Filippov plant disease model with cultural control strategy. We examine a Lotka–Volterra Filippov plant disease model with proportional planting rate, which is globally studied in terms of five types of equilibria. For one type of equilibrium, the global structure is discussed by the iterative equations for initial numbers of plants. For the other four types of equilibria, the bounded global attractor of each type is obtained by constructing appropriate Lyapunov functions. The ideas of constructing Lyapunov functions for Filippov systems, the methods of analyzing such systems and the main results presented here provide scientific support for completing control regimens on plant diseases in integrated disease management.     

Dynamic analysis of lactic acid fermentation in membrane bioreactor

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. This novel theoretical framework could result in an objective criterion on how to control the substrate concentration in order to keep a sustainable and steady output of lactic acid. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincaré-Bendixson Theorem, we obtain the conditions for the globally asymptotical stability of the equilibrium. Secondly, impulsive input substrate is also considered. Using Floquet's theorem and small-amplitude perturbation, we obtain the biomass-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

SIR-SVS epidemic models with continuous and impulsive vaccination strategies

Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.     

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.     

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 dynamics of infectious diseases in orchards with roguing and replanting as control strategy

 Roguing and replanting is a widely adopted control strategy of infectious diseases in orchards. Little is known about the effect of this type of management on the dynamics of the infectious disease. In this paper we analyze a structured population model for the dynamics of an S-I-R type epidemic under roguing and replanting management. The model is structured with respect to the total number of infections and the number of post-infectious infections on a tree. Trees are assumed to be rogued, and replaced by uninfected trees, when the total number of infections on the tree reaches a threshold value. Stability analysis and numerical exploration of the model show that for specific parameter combinations the internal equilibrium can become unstable and large amplitude periodic fluctuations arise. Several hypothesis on the mechanism causing the destabilisation of the steady-state are considered. The mechanism leading to the large amplitude fluctuations is identified and biologically interpreted. Received 2 September 1994     

ANTHER SMUT DISEASE IN DIANTHUS SILVESTER (CARYOPHYLLACEAE): NATURAL SELECTION ON FLORAL TRAITS

Mating opportunities, pollination intensity, and pollen dispersal ability may vary with variation in floral traits such as color, size, and shape. Where these traits are selected by pollinators for enhanced elaboration, they should evolve toward the equilibrium between selection for further elaboration and selection against this through reduced fecundity or vitality. Here we show that pollinator-borne fungal diseases of plants may be a factor influencing the position of this equilibrium. Populations of the rock pink, Dianthus silvester often contain individuals infected with the anther smut fungus Microbotryum violaceum (= Ustilago violacea). In a naturally infected population in the Alps of eastern Switzerland we investigated how intrapopulation variation in flower size and nectar rewards influenced spore deposition and how floral traits varied with disease status. We found that spore deposition increased with increasing petal size, suggesting that large-flowered plants were at a greater risk of disease. Spore deposition was also higher for plants growing in patches with many or a high proportion of diseased neighbors. Multiple regression analyses showed that petal size or nectar reward influenced spore deposition when the effects of neighborhood disease abundance were controlled statistically. In sequential analyses, after removing the effects of disease density or frequency and plant gender, petal length explained significant variation in spore deposition. Diseased plants had reduced female reproductive organs, but calyx size was intermediate between that of healthy perfect and female flowers of this gynodioecious-gynomonoecious species, and diseased plants bore flowers with the largest petals. This may reflect a symptom of this disease or the cause, if larger-flowered plants are more likely to become infected. We conclude that investment to pollinator attraction may bring an enhanced risk of contracting this sterilizing pollinator-borne disease, so natural selection by the fungus M. violaceum acts to lower attractiveness to pollinators.     

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．     

具有阶段结构和脉冲控制的时滞生物防治害虫模型

基于昆虫病毒防治害虫的策略,建立具有脉冲效应的时滞微分方程模型,利用脉冲微分方程的Floquet乘子理论及比较定理,证明该模型害虫灭绝T周期解的全局吸引性.     

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.     

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．     

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．     

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.     

具周期脉冲的周期差分系统的全局性态

本文建立了具有常数脉冲和周期脉冲的周期差分系统,得到了常数脉冲系统全局稳定周期解存在的充分条件,并证明了周期脉冲的周期系统的周期解是全局吸引的。     

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.     

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.