LAUSR

We study the local dynamics and bifurcations of a two-dimensional discrete-time predator–prey model in the closed first quadrant R+2$\mathbb{R}_{+}^{2}$. It is proved that the model has two boundary equilibria: O(0,0)$O(0,0)$, A(α1−1α1,0)$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$ and a unique positive equilibrium B(1α2,α1α2−α1−α2α2)$B (\frac{1}{\alpha _{2}},\frac{ \alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$ under some restriction to the parameter. We study the local dynamics along their topological types by imposing the method of linearization. It is proved that a fold bifurcation occurs about the boundary equilibria: O(0,0)$O(0,0)$, A(α1−1α1,0)$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$ and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium B(1α2,α1α2−α1−α2α2)$B (\frac{1}{\alpha _{2}},\frac{\alpha _{1} \alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$. It is also proved that the model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium B(1α2,α1α2−α1−α2α2)$B (\frac{1}{ \alpha _{2}},\frac{\alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$ and meanwhile a stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between predator and prey populations. Numerical simulations are presented to verify not only the theoretical results but also to exhibit the complex dynamical behavior such as the period-2, -4, -11, -13, -15 and -22 orbits. Further, we compute the maximum Lyapunov exponents and the fractal dimension numerically to justify the chaotic behaviors of the discrete-time model. Finally, the feedback control method is applied to stabilize chaos existing in the discrete-time model.