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.     

The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. 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 modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Analysis of a stage structured predator–prey Gompertz model with disturbing pulse and delay

In this paper, we introduce a general and robust prey-dependent consumption predator–prey Gompertz model with periodic harvesting for the prey and stage structure for the predator with constant maturation time delay and perform a systematic mathematical and ecological study. Sufficient conditions which guarantee the global attractivity of predator-extinction periodic solution and permanence of the system are obtained. We also prove that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. Our results provide reliable tactic basis for the practical pest management.     

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.     

Eco-epidemiology model with age structure and prey-dependent consumption for pest management

In this paper, a prey-dependent consumption predator–prey (natural enemy-pest) model with age structure for the predators and infectious disease in the prey, is considered. Infectious pests, immature natural enemies and mature natural enemies are released impulsively. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. The results provide a reliable theoretical tactics for pest management.     

Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

In this paper, we study a predator–prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey population’s intrinsic growth rate and find the conditions for the stability and for the permanence of this system.     

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.     

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.     

Two profitless delays for an SEIRS epidemic disease model with vertical transmission and pulse vaccination

Since the investigation of impulsive delay differential equations is beginning, the literature on delay epidemic models with pulse vaccination is not extensive. In this paper, we propose a new SEIRS epidemic disease model with two profitless delays and vertical transmission, and analyze the dynamics behaviors of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using a new modeling method, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and vertical transmission can bring different effects on the dynamics behaviors of the model by numerical analysis. Our results also show the delays are “profitless”. In this paper, the main feature is to introduce two discrete time delays, vertical transmission and impulse into SEIRS epidemic model and to give pulse vaccination strategies.     

The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators

In this paper, we introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amount μ₁ and μ₂ are lager than some critical value. Further, to obtain a more effective pest control strategy, we give the conditions (involving the estimate of μ₁ and μ₂) in which the model is uniformly permanent and the pest population is under the economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.     

Existence and stability of periodic solution of a Lotka–Volterra predator–prey model with state dependent impulsive effects

According to biological and chemical control strategy for pest, we investigate the dynamic behavior of a Lotka–Volterra predator–prey state-dependent impulsive system by releasing natural enemies and spraying pesticide at different thresholds. By using Poincaré map and the properties of the Lambert W$W$ function, we prove that the sufficient conditions for the existence and stability of semi-trivial solution and positive periodic solution. Numerical simulations are carried out to illustrate the feasibility of our main results.     

A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey

In this paper, we formulate a robust prey-dependent consumption predator-prey model with a delay of digestion and impulsive perturbation on the prey. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-eradication’ periodic solution and show that the ‘predator-eradication’ periodic solution is globally attractive when harvesting for the prey is over certain value. Using a new qualitative analysis method for impulsive and delay differential equations, we prove the system is uniformly persistent when harvesting for the prey is under certain value. Further, we show the delay of digestion is a “profitless” time delay. Moreover, we show our theoretical results by numerical simulation. In this paper, the main feature is that we introduce a delay of digestion and impulsive effects into the predator-prey model and exhibit a new mathematical method which is applied to investigate the system which is governed by both impulsive and delay differential equations.     

An impulsive ratio-dependent predator–prey system with diffusion

We investigate the predator–prey system with diffusion, when biological and environmental parameters are assumed to change in periodical manner over time. The system is affected by impulses which can be considered as a control. Conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solutions are obtained.     

具有Holling Ⅲ功能反应和阶段结构的Gompertz生态系统的持久性

以生态学与微分方程的理论和方法为基础,建立了一类具有HollingⅢ功能反应和阶段结构的生态Gompertz模型.利用频闪映射,获得了捕食者灭绝周期解,分析了此周期解的全局吸引性.在对食饵进行脉冲收获和捕食者具有成长期时滞条件下,运用脉冲微分方程比较定理和小振幅扰动技巧,获得了系统一致持续生存的条件.     

食饵具有流行病的阶段结构捕食模型

本文考虑了一类食饵具有流行病和阶段结构的脉冲时滞捕食模型．利用脉冲时滞微分方程的相关理论和方法，获得易感害虫根除周期解全局吸引的充分条件以及当脉冲周期在一定范围内时，天敌与易感害虫可以共存且易感害虫的密度可以控制在经济危害水平E（EIL）之下．我们的结论为现实的害虫管理提供了可靠的策略依据．     

Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population

In this paper, a stage-structured predator–prey model is proposed and analyzed to study how the type of refuges used by prey population influences the dynamic behavior of the model. Two types of refuges: those that protect a fixed number of prey and those that protect a constant proportion of prey are considered. Mathematical analyses with regard to positivity, boundedness, equilibria and their stabilities, and bifurcation are carried out. Persistence condition which brings out the useful relationship between prey refuge parameter and maturation time delay is established. Comparing the conclusions obtained from analyzing properties of two types of refuges using by prey, we observe that value of maturation time at which the prey population and hence predator population go extinct is greater in case of refuges which protect a constant proportion of prey.     

Periodicity in a generalized semi-ratio-dependent predator–prey system with time delays and impulses

With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi-ratio-dependent predator–prey system with time delays and impulses, which covers many models appeared in the literature. When the results reduce to the semi-ratio-dependent predator–prey system without impulses, they generalize and improve some known ones.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.     

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.     

具有饱和反应率和阶段时滞结构的害虫生物防治模型

考虑了一个害虫和天敌都有阶段结构及具有饱和反应率的阶段时滞脉冲捕食者-食饵模型,利用人工周期定量地投放有病的害虫和天敌去治理害虫.借助脉冲时滞微分方程的相关理论和方法获得易感害虫根除周期解全局吸引的充分条件以及天敌与易感害虫可以共存且易感害虫的密度可以控制在经济危害水平之下的充分条件.我们的结论为现实的害虫管理提供了可靠的策略依据.