an:07020831
Zbl 07020831
Khan, Abdul Qadeer
Bifurcations of a two-dimensional discrete-time predator-prey model
EN
Adv. Difference Equ. 2019, Paper No. 56, 23 p. (2019).
00427324
2019
j
39A10 40A05 92D25 70K50 35B35
discrete-time predator-prey model; stability and bifurcations; center manifold theorem; fractal dimension; chaos control; numerical simulation
Summary: We study the local dynamics and bifurcations of a two-dimensional discrete-time predator-prey model in the closed first quadrant \(\mathbb{R}_{+}^{2}\). It is proved that the model has two boundary equilibria: \(O(0,0)\), \(A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )\) and a unique positive equilibrium \(B (\frac{1}{\alpha _{2}},\frac{ \alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )\) 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: \(O(0,0)\), \(A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )\) and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium \(B (\frac{1}{\alpha _{2}},\frac{\alpha _{1} \alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )\). It is also proved that the model undergoes a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium \(B (\frac{1}{ \alpha _{2}},\frac{\alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )\) 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.