Dynamics of a Leslie–Gower type predator–prey system with herd behavior and constant harvesting in prey

In this paper, the dynamics of a Leslie–Gower type predator–prey system with herd behavior and constant harvesting in prey are investigated. Earlier work has shown that the herd behavior in prey merely induces a supercritical Hopf bifurcation in the classic Leslie–Gower predator–prey system in the absence of harvesting. However, the work in this paper shows that the presence of herd behavior and constant harvesting in prey can give rise to numerous kinds of bifurcation at the non-hyperbolic equilibria in the classic Leslie–Gower predator–prey system such as two saddle–node bifurcations and one Bogdanov–Takens bifurcation of codimension two at the degenerate equilibria and one degenerate Hopf bifurcation of codimension three at the weak focus. Some numerical simulations are also provided to verify the theoretical results and evaluate their biological implications such as the changes of phase diagram near the degenerate equilibrium due to the Bogdanov–Takens bifurcation and the coexistence of multiple limit cycles arising from the degenerate Hopf bifurcation. Hence, the research results reveal that the herd behavior and constant harvesting in prey have a strong influence on the dynamics and also contribute to promoting the ecological diversity and maintaining the long-term economic benefits.