Dynamic complexity of microbial pesticide model

In this paper, a mathematical model for the entomopathogenic nematode attacking the pest is investigated. This novel theoretical framework could result in an objective criterion on how to release the entomopathogenic nematode in order to control the pest population under the economic threshold (ET) which indicates the maximally admissible pest densities. Firstly, continuous release of the entomopathogenic nematode is taken. By using qualitative analysis method, the sufficient condition of the global stability of the positive equilibria and the existence and uniqueness of limit cycle of the system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. Using the Floquet's theorem and small-amplitude perturbation, we obtain that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.