摘要：We study the local dynamics and bifurcations of a two-dimensional discrete-time predator–prey model in the closed first quadrant R + 2 $\mathbb{R}_{+}^,$ . It is proved that the model has two boundary equilibria: O ( 0 , 0 ) $O(0,0)$ , A ( α 1 − 1 α 1 , 0 ) $A (\frac{\alpha _)-1}{\alpha _)},0 )$ and a unique positive equilibrium B ( 1 α 2 , α 1 α 2 − α 1 − α 2 α 2 ) $B (\frac){\alpha _,},\frac{ \alpha _)\alpha _,-\alpha _)-\alpha _,}{\alpha _,} )$ 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 ( α 1 − 1 α 1 , 0 ) $A (\frac{\alpha _)-1}{\alpha _)},0 )$ and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium B ( 1 α 2 , α 1 α 2 − α 1 − α 2 α 2 ) $B (\frac){\alpha _,},\frac{\alpha _) \alpha _,-\alpha _)-\alpha _,}{\alpha _,} )$ . It is also proved that the model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium B ( 1 α 2 , α 1 α 2 − α 1 − α 2 α 2 ) $B (\frac){ \alpha _,},\frac{\alpha _)\alpha _,-\alpha _)-\alpha _,}{\alpha _,} )$ 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.
关键词：Discrete-time predator–prey model ; Stability and bifurcations ; Center manifold theorem ; Fractal dimension ; Chaos control ; Numerical simulation
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