Bifurcations in a Leslie–Gower predator–prey model with strong Allee effects and constant prey refuges

In this paper, we study a Leslie–Gower predator–prey model with strong Allee effects and constant prey refuges. It is shown that the model can undergo a cusp type degenerate Bogdanov–Takens bifurcation of codimension 4, focus and elliptic types degenerate Bogdanov–Takens bifurcations of codimension 3, and degenerate Hopf bifurcation of codimension 3 as the parameters vary. The model can exhibit the coexistence of multiple positive steady states, multiple limit cycles, and homoclinic loops. Our results indicate that a larger prey refuge contributes to the coexistence of both species. Numerical simulations, including three limit cycles, quadristability, a large-amplitude limit cycle enclosing three positive steady states and a homoclinic loop, two large-amplitude limit cycles enclosing three positive steady states, are presented to illustrate the theoretical results.