Hopf Point Analysis for Ratio-Dependent Food Chain Models

In this paper periodic and quasi-periodic behavior of a food chain model with three trophic levels are studied. Michaelis-Menten type ratio-dependent functional response is considered. There are two equilibrium points of the system. It is found out that at most one of these equilibrium points is stable at a time. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. However for the second equilibrium point the computation is more tedious and bifurcation points can only be found by numerical experiments. It has been found that around these points there are periodic solutions and when this point is unstable, the solution is an enlarging spiral from inside and approaches to a limit cycle.