Relaxation Oscillations In A Class Of Predator-Prey Systems With Holling Ⅳ Function

This paper consider a class of three-dimensional, singularly perturbedpredator-prey systems with Holling IV response function, having twopredator competing exploitatively for the same prey in a constantenvironment. The model have the form as below:After the preface that introduce the subject and the knowledgenecessary to prepare in the chapter2, we first consider the system’stwo-dimensional dynamics on the boundary plane to obtain saddle-nodebifurcation and the existence and the uniqueness of the limit cycle, whichmeans persistence and extinction in the system. Then, by using dynamicalsystems techniques and the geometric singular perturbation theory, we giveprecise conditions which guarantee the existence of stable relaxationoscillations for systems within the class. Such result shows the coexistenceof the predators and the prey with quite diversified time response whichtypically happens when the prey population grows much faster than thoseof predators.