Identifying weak focus of order 3 in a Leslie–Gower prey–predator model with prey harvesting

Abstract

In this paper, we identify a weak focus with order up to 3 for a Leslie–Gower prey–predator model with three parameters. The known work computed the first Lyapunov quantity and discussed Hopf bifurcations of this system, but the identification of weak focus and its maximal order was not completed yet. In this paper, we decompose the varieties of the first three Lyapunov quantities of this system by resultant, realroot isolation, and pseudo-division so as to prove that the center-type equilibrium is a weak focus with order up to 3. Moreover, we give the parameter conditions of each order. Finally, numerical simulations are employed to illustrate the results obtained.