Critical Condition Of The General Natural Enemies Of Pests Control System

Controlling insect pests has been becoming an increasing important and com-plicate issue over the past decade years. How to reduce harm caused by insect pests on important plant, animal and human, which has been a great concern of entomologists and social. Integrated pest management(IPM) uses a combination of biological, cultural, and chemical tactics that reduce pests to tolerable level, with little cost to minimal effect on the environment. Chemical control is one of the IPM strategy, and it is very useful because it can kill significant proportion of an insect population quickly. However, the pesticide application may have adverse effects on the foodstuffs, human illness and environment. As a result, the biological control plays key roles in suppressing insect pests for preserving the environment. Biological control is an important component of IPM strategy. It is defined as the reduction of pest populations by natural enemies and typically involves an active human role and achieve a better economic benefit. It has been proving that IPM is more ef-fect than the class methods (such as biological control or chemical control) both of experimentally and theoretically points of view. These indicate that different pest control techniques should work together rather than against each other.Recently, many continuous and discrete predator-prey models concerning IPM strategy have been investigated in home and abroad. In those works, there are two assumptions, one of which is that a single biological control strategy is applied; Another assumption is that all control tactics including biological and chemical controls are implemented at the same time, which means that the application of pesticides can kill the released natural enemies instantly. In view of the above two problems, S. Nundloll and its partners put forward the generalized predator-prey model with different harvesting and releasing period in 2008. It has been proved that there exists a positive pest-free periodic solution and globally asymptotically stable threshold condition. However, it has not completely avoided the effect of natural enemies killed because of the harvesting.According to a practical point of view, sanyi Tang propose a simple impulsive pest-natural enemy model with different frequencies of spraying pesticides and re-leasing natural enemies in 2010. In the case of completely avoiding the effect of natural enemies because of spraying peseticide, it proved the exist of pest-free pe- riodic solution and globally asymptotically stable threshold condition, and studied the optimum rate for spraying and releasing. This result may help people to decide optimum timing for IPM.Comprehensive above two research, we construct and investigated the dynamics of a quite generalized impulsive predator-prey model according with the different periodic biological and chemical control we consider two different cases in terms of the timing of IPM applications.(i) spray-ing pesticides more frequently than releasing natural enemies; (ii) spraying pesticides less frequently than releasing natural enemies. In section 2, it is proved that there is a locally stable prey-free periodic solution by using Floquet theory, and this periodic solution is global attractivity by using an analytic method for case 1. For case 2, we show that there exists a global stable pest-free periodic solution in section 3. The most important problems in IPM are how many pest populations should be killed and natural enemies should be released to avoid economic loss, and how the control tactics impact on the threshold conditions. Therefore, in section 4, we investigate how the control tactics impact on the threshold conditions by the Holling Type II functional response curve. Furthermore, the existence of multiple attractors at which pests and natural enemies can coexist for a wide range of parameters. Finally, the effects of random perturbation of the constant killing rate of pesticides on pests and on natural enemies, releasing rates and releasing constant on the switch-like transitions can prove that varying dosages and frequencies of pesticide application and the number of natural enemies released are crucial for pest control.