Stable Pest-eradication Periodic Solution Research Articles

Integrated Pest Management with a Mixed Birth Rate for Prey Species

X. Song and Z. Xiang [7] develop an impulsive differential equations model for a two-prey, one-predator model with stage structure for the predator. They demon-strate the conditions on the impulsive period for which a globally asymptotically stable pest-eradication periodic solution exists, as well as conditions on the im-pulsive period for which the prey species is permanently maintained under an economically acceptable threshold. We extend their model by including stage structure for both predator and prey and also by adding stochastic elements in the birth rate of the prey. As in [7], we find the conditions under which a globally asymptotically stable pest-eradication periodic solution exists.

Integrated Pest Management with a Mixed Birth Rate for Prey Species

X. Song and Z. Xiang [7] develop an impulsive differential equations model for a two-prey, one-predator model with stage structure for the predator. They demon-strate the conditions on the impulsive period for which a globally asymptotically stable pest-eradication periodic solution exists, as well as conditions on the im-pulsive period for which the prey species is permanently maintained under an economically acceptable threshold. We extend their model by including stage structure for both predator and prey and also by adding stochastic elements in the birth rate of the prey. As in [7], we find the conditions under which a globally asymptotically stable pest-eradication periodic solution exists.

Integrated pest management with stochastic birth rate for prey species

Song and Xiang (2006) developed an impulsive differential equations model for a two-prey one-predator model with stage structure for the predator. They demonstrate the conditions on the impulsive period for which a globally asymptotically stable pest-eradication periodic solution exists, as well as conditions on the impulsive period for which the prey species is permanently maintained under an economically acceptable threshold. We extend their model by including stage structure for both predator and prey as well as by adding stochastic elements in the birth rate of the prey. As in Song and Xiang (2006), we find the conditions under which a globally asymptotically stable pest eradication periodic solution exists. In addition, we numerically show the relationship between the stochastically varying birth rate of the prey and the necessary efficacy of the pesticide for which the probability of eradication of the prey species is above 90%. This is significant because the model recognizes varying environmental and climatic conditions which affect the resources needed for pest eradication.

Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

Impulsive Biological Pest Control Strategies of the Sugarcane Borer

We propose an impulsive biological pest control of the sugarcane borer (Diatraea saccharalis) by its egg parasitoid Trichogramma galloi based on a mathematical model in which the sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. By using the Floquet theory and the small amplitude perturbation method, we show that there exists a globally asymptotically stable pest‐eradication periodic solution when some conditions hold. The numerical simulations show that the impulsive release of parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

Analysis of a Stage-Structured Predator-Prey System Concerning Impulsive Control Strategy

Based on integrated pest management, a stage-structured predator-prey system with Holling type-II functional response concerning impulsive control strategy is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a globally stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further, sufficient conditions for the permanence of the system is established. Numerical simulations are carried out to illustrate the effect of impulses on the dynamics of the system.

Dynamical behavior of a pest management model with impulsive effect and nonlinear incidence rate

In this paper, we consider the pest management model with spraying microbial pesticide and releasing the infected pests, and the infected pests have the function similar to the microbial pesticide and can infect the healthy pests, further weaken or disable their prey function till death. By using the Floquet theory for impulsive differential equations, we show that there exists a globally asymptotically stable pest eradication periodic solution when the impulsive period τ τmax. Finally, by means of numerical simulation, we showthatwith the increaseof impulsive period, the system displays complicated behaviors.

The Viral Infection Model in Pest Management

In this paper,a pest management viral infection model with impulsive releases of viral particles and natural enemies is proposed and investigated.The sufficient conditions of globally asymptotic stability of pest-eradication periodic solution and the permanence of the model are obtained by using Floquet's theorem,small-amplitude perturbation skills and comparison theorem.The results provide effective tactic basis for the practical pest management.

Analysis of an impulsive pest management SEI model with nonlinear incidence rate

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with nonlinear incidence concerning impulsive strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically attractive pest-eradication periodic solution when the impulsive period satisfies A1. When the impulsive period satisfies A2, the stability of pest-eradication periodic solution is lost, the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes

In this paper, the dynamics of an impulsively controlled predator-prey model with modified Leslie-Gower and Holling type II schemes is analyzed. Choosing pest birth rate r 1 as control parameter, we show that there exists a globally asymptotically stable pest-eradication periodic solution when r 1 is less than some critical value \(r_{1}^{*}\) and the system is permanent when r 1 is larger than the critical value \(r_{1}^{*}.\) By use of standard techniques of bifurcation theory, we prove that this threshold there are periodic oscillations in pest and predator. Furthermore, some situations which leads to chaotic behavior of the system are investigated by means of numerical simulations.

Global Dynamics Behaviors of Viral Infection Model for Pest Management

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

Complex Dynamics of an Impulsive Control System in which Predator Species Share a Common Prey

In an ecosystem, multiple predator species often share a common prey and the interactions between the predators are neutral. In view of this fact, we propose a three-species prey-predator system with the functional responses and impulsive controls to model the process of pest management. It is proved that the system has a locally stable pest-eradication periodic solution under the assumption that the impulsive period is less than some critical value. In particular, two single control strategies (biological control alone or chemical control alone) are proposed. Finally, we compare three pest control strategies and find that if we choose narrow-spectrum pesticides that are targeted to a specific pest’s life cycle to kill the pest, then the combined strategy is preferable. Numerical results show that our system has complex dynamics including period-doubling bifurcation, quasi-periodic oscillation, chaos, intermittency and crises.

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.

A FOOD CHAIN SYSTEM WITH HOLLING IV FUNCTIONAL RESPONSES AND IMPULSIVE EFFECT

In the paper, according to biological and chemical control strategy for pest control, our main purpose is to construct a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive effect concerning integrated pest management (IPM), and investigate the dynamic behaviors of this system. By using the Floquet theory and comparison theorem of impulsive differential equation and analytic method, we prove that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, condition for permanence of the system is established. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level.

Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with saturation incidence concerning impulsive control strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. When the impulsive period is larger than some critical value, the stability of the pest-eradication periodic solution is lost; the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. 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 period-doubling cascade, symmetry-breaking pitchfork bifurcation, quasi-periodic oscillate, chaos, and non-unique dynamics.

Bifurcation and chaos of a pest-control food chain model with impulsive effects

According to biological and chemical control strategy for pest control, we investigate the dynamics of a predator–prey food chain with impulsive effect, periodic releasing natural enemies and spraying pesticide at different fixed times, by using impulsive differential equation. Choose pest birth rate r 2 as control parameter, we show that there exists a stable pest-eradication periodic solution when r 2 is less than some critical value r 2 ∗ and the system is permanence when r 2 is larger than the critical value r 2 ∗ . By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in prey, middle-predator and top predator. Furthermore, bifurcation diagrams have shown that there exists complexity for the pulsed system including periodic doubling cascade.

Modelling and analysis of an impulsive SI model with Monod-Haldane functional response

An impulsive SI model with Monod-Haldane functional response for pest control is proposed and investigated. First, we have proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the above system can be permanent. Then, influences of impulsive perturbation including impulse period, the time of spraying pesticide and the quantity of releasing infective pests on the above system have been studied. Moreover, numerical simulations show that the system has rich dynamical behaviors. Finally, it is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying is more effective than the classical one if the chemical control is adopted rationally.

IMPULSIVE CONTROL STRATEGIES FOR PEST MANAGEMENT

In this paper, we propose two impulsive differential systems concerning biological and, respectively, integrated pest management strategies. In each case, 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 a certain 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 non-trivial periodic solution is also studied by means of numerical simulations. In the case in which a single control is used, one can only use the amount of infective pests which are periodically released in order to control pests at desirable low levels, while in the case in which integrated management is used, one can use the proportion of pests removed by means of spraying chemical pesticides together with the amount of infective pests which are periodically released to control pests at desirable low levels.

Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control

By introducing impulsive biological control strategy, the dynamic behaviors of the two-prey one-predator model with defensive ability and Holling type-II functional response are investigated. By using Floquet’s Theorem and the small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical minimum value, and permanence conditions (that is, the impulsive period is greater than some critical maximum value) are established via the method of comparison involving multiple Liapunov functions. It is shown that our impulsive control strategy is more effective than the classical one. Furthermore, the effect of impulsive perturbations on the unforced continuous system is studied. From simulations, we find that the system has more complex dynamic behaviors and is dominated by periodic, quasi-periodic, and chaotic solutions.

The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects

In this paper, we consider the prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects. By applying the Floquet theory of linear periodic impulsive equation, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, that is, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, and thus, we further prove that the system is uniformly permanent if the impulsive period is larger than some critical value, and meanwhile we also give the conditions for the extinction of one of the two preys and permanence of the remaining species. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population's intrinsic growth rate. The resulting bifurcation diagrams show that with the varying of parameters, the system experiences process of cycles, periodic windows, periodic-doubling cascade, symmetry breaking bifurcation as well as chaos.