A Holling II functional response food chain model with impulsive perturbations

In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.     

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.     

Dynamic complexities of a food chain model with impulsive perturbations and Beddington–DeAngelis functional response

In this paper, the dynamic complexities of a three trophic level food chain system with impulsive perturbations and Beddington–DeAngelis functional responses is studied. Conditions for extinction of predator are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics in the positive octant.     

Mathematics analysis and chaos in an ecological model with an impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model with an impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest-level prey and mid-level predator eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems.     

Extinction and permanence of a two-prey two-predator system with impulsive on the predator

In this paper, the dynamic behaviors of a two-prey two-predator system with impulsive effect on the predator of fixed moment are investigated. By applying the Floquet theory of liner periodic impulsive equation, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is large than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining three species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.     

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.     

The dynamical behavior of a predator–prey system with Gompertz growth function and impulsive dispersal of prey between two patches

In this paper, we investigate a predator–prey model with Gompertz growth function and impulsive dispersal of prey between two patches. Using the dynamical properties of single‐species model with impulsive dispersal in two patches and comparison principle of impulsive differential equations, necessary and sufficient criteria on global attractivity of predator‐extinction periodic solution and permanence are established. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations

The effect of periodic forcing and impulsive perturbations on predator–prey model with Holling type IV functional response is investigated. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of the prey. The impulsive perturbations are affected by introducing periodic constant impulsive immigration of predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can easily give rise to complex dynamics, including (1) quasi-periodic oscillating, (2) period doubling cascade, (3) chaos, (4) period halfing cascade.     

A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect

In this paper, the predator–prey system with the Beddington–DeAngelis functional response is developed, by introducing a proportional periodic impulsive catching or poisoning for the prey populations and a constant periodic releasing for the predator. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. This model has the potential to protect predator from extinction, but under some conditions may also lead to extinction of the prey. That is, the system exists a locally stable prey-eradication periodic solution when the impulsive period satisfies an inequality. The condition for permanence is established via the method of comparison involving multiple Liapunov̀ functions. Further, by numerical simulation method the influences of the impulsive perturbations and mutual interference by predators on the inherent oscillation are investigated. With the increasing of releasing for the predator, the system appears a series of complex phenomenon, which include (1) period-doubling, (2) chaos attractor, (3) period-halfing. (4) non-unique dynamics (meaning that several attractors coexist).     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

具有收获和阶段结构的时滞脉冲的捕食-食饵模型

研究了食饵分布在不同斑块,捕食者具有阶段结构和收获的时滞脉冲的捕食-食饵模型.利用离散动力系统的频闪映射,得到了捕食者灭绝周期解的存在性和它的精确表达式.使用比较原理,得到了捕食者灭绝周期解全局渐近稳定的充分条件和系统的持久性.最后,用Matlab软件进行数值仿真验证了获得的结果.     

An impulsive periodic predator–prey system with Holling type III functional response and diffusion

This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.     

Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy

According to biological and chemical control strategy for pest control, we investigate the dynamic behavior of a Holling II functional response predator–prey system concerning impulsive control strategy-periodic releasing natural enemies and spraying pesticide at different fixed times. By using Floquet theorem and small amplitude perturbation method, we prove that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, the condition for the permanence of the system is also given. 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 periodic, quasiperiodic and chaotic solutions, which implies that the presence of pulses makes the dynamic behavior more complex. Finally, we conclude that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently.     

Survival analysis for a periodic predator–prey model with prey impulsively unilateral diffusion in two patches

In this paper, we study a periodic predator–prey system with prey impulsively unilateral diffusion in two patches. Firstly, based on the results in [41], sufficient conditions on the existence, uniqueness and globally attractiveness of periodic solution for predator-free and prey-free systems are presented. Secondly, by using comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the permanence and extinction of prey species x with predator have other food source are established. Finally, the theoretical results both for non-autonomous system and corresponding autonomous system are confirmed by numerical simulations, from which we can see some interesting phenomena happen.     

The study of predator–prey system with defensive ability of prey and impulsive perturbations on the predator

Predator–prey system with non-monotonic functional response and impulsive perturbations on the predator is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than the critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. Further, using numerical simulation method the influences of the impulsive perturbations on the inherent oscillation are investigated. With the increasing of the impulsive value, the system displays a series of complex phenomena, which include (1) quasi-periodic oscillating, (2) period-doubling, (3) period-halfing, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis and (6) chaotic bands with periodic windows.     

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.     

Modeling and analysis of a modified Leslie–Gower type three species food chain model with an impulsive control strategy

This paper describes a modified Leslie–Gower type three species food chain model with harvesting. We have incorporated impulsive control strategy to the system. Theories of impulsive differential equations, small amplitude perturbation skills and comparison technique are used to study dynamical behavior of the system. Sufficient conditions are derived to ensure global stability of the lowest-level prey and mid-level predator eradication periodic solution. Sufficient conditions are also derived to examine the permanence of the system. Numerical simulations are carried out to verify the analytical results, and the system is analyzed through graphical illustrations. It is observed that the stability of the system exhibits several states, ranging from stable situation to cyclic oscillatory behavior, under different favorable conditions. These results are useful to study the dynamic complexity of ecological systems. The computation of the largest Lyapunov exponent demonstrates the chaotic dynamic nature of the system. The qualitative nature of strange attractor is examined. It is to be noted that the harvesting effort can cause a stable equilibrium to become unstable and even a switching of stabilities.     

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.     

Dynamics of cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response

In this paper, cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response is studied. By using comparison theorem and some analysis techniques as well as the coincidence degree theory, sufficient conditions are obtained for the permanence, extinction and the existence of positive periodic solution.     

Dynamic analysis of pest control model with population dispersal in two patches and impulsive effect

In this paper, we investigate the pest control model with population dispersal in two patches and impulsive effect. By exploiting the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we can obtain that the susceptible pest eradication periodic solution is globally asymptotically stable if the impulsive periodic τ is less than the critical value τ₀ . Further, we also prove that the system is permanent when the impulsive periodic τ is larger than the critical value τ₀. Hence, in order to drive the susceptible pest to extinction, we can take impulsive control strategy such that τ < τ₀ according to the effect of the viruses on the environment and the cost of the releasing pest infected in a laboratory. Finally, numerical simulations validate the obtained theoretical results for the pest control model with population dispersal in two patches and impulsive effect.