Bifurcation Analysis of a Predator-Prey System with Ratio-Dependent Functional Response

In this paper, we are concerned with the structural stability of a density dependent predator. prey system with ratio-dependent functional response. Starting with the geometrical analysis of hyperbolic curves, we obtain that the system has one or two positive equilibria under various conditions. Inspired by the S-procedure and semi-definite programming, we use the sum of squares decomposition based method to ensure the global asymptotic stability of the positive equilibrium through the associated polynomial Lyapunov functions. By exploring the monotonic property of the trace of the Jacobianmatrix with respect to r under the given different conditions, we analytically verify that there is a corresponding unique r∗such that the trace is equal to zero and prove the existence of Hopf bifurcation, respectively.