Abstract
We consider a delayed Pest-predator model under insecticide use. First, the paper considers the stability and local Hopf bifurcation for a modified Pest-predator model with time delay. In succession, using the normal form theory and center manifold argument, we obtain some explicit results which determine the stability, direction and other properties of bifurcation periodic solutions. I. INTRODUCTION With the rapid development of chemistry, many pesticides are applied in the world. However, insecticide pollution is also recognized as a major health hazard to human beings and to natural enemies. Thus it is required that we should combine pesticide efficacy tests with biological control research, so that the effects on the pest and the natural enemies are considered as a unified whole. Many researchers have been devoting to study the Pest-predator model. In this paper, our research is based on the Pest-predator models under insecticide use. However, here we consider the model with time delay. Let x(t), y(t) denote the density of the pest and predator (natural enemies) at the time t, respectively. We could have the Pest-predator model as followings
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