Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator–prey model with linear harvesting rate: Analysis and computation

In this paper, we propose a ratio-dependent predator–prey model with linear harvesting rate, which can describe the real predator–prey system better than the previously published model with constant harvesting in [D. Xiao, L.S. Jennings, Bifurcation of a ratio-dependent predator–prey system with constant rate harvesting, SIAM J. Appl. Math. 65 (3) (2005) 737–753]. We show the different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of “conditional coexistence” depending on the initial values. By choosing the modulus of linear harvesting rate as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator–prey models, is that the origin is a non-hyperbolic equilibrium, whose characteristics crucially determine the main properties of the model. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. Experimental motivation is made to summarize the effect of harvesting on coexistence and extinction.