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Supercritical Neimark-Sacker bifurcation of a discrete-time Nicholson-Bailey model. (English) Zbl 1394.39013
Summary: We study the local dynamics and supercritical Neimark-Sacker bifurcation of a discrete-time Nicholson-Bailey host-parasitoid model in the interior of \(\mathbb R^2_+\). It is proved that if \(\alpha>1\), then the model has a unique positive equilibrium point \(P\bigg(\frac{\alpha}{\alpha-1}(\mathrm{ln}(\alpha))^2,(\mathrm{ln}(\alpha))^2\bigg)\), which is locally asymptotically focus, unstable focus and nonhyperbolic under certain parametric condition. Furthermore, it is proved that the model undergoes a supercritical Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium point \(P\bigg(\frac{\alpha}{\alpha-1}(\mathrm{ln}(\alpha))^2,(\mathrm{ln}(\alpha))^2\bigg)\), and meanwhile, the stable closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasiperiodic oscillations between host and parasitoid populations. Some numerical simulations are presented to verify theoretical results.

MSC:
39A28 Bifurcation theory for difference equations
39A30 Stability theory for difference equations
39A10 Additive difference equations
40A05 Convergence and divergence of series and sequences
92D25 Population dynamics (general)
70K50 Bifurcations and instability for nonlinear problems in mechanics
35B35 Stability in context of PDEs
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