| Citation: |
Yunxian Dai, Ping Yang, Zhiliang Luo, Yiping Lin. BOGDANOV-TAKENS BIFURCATION IN A DELAYED MICHAELIS-MENTEN TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1333-1346. doi: 10.11948/2156-907X.20180238
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BOGDANOV-TAKENS BIFURCATION IN A DELAYED MICHAELIS-MENTEN TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH PREY HARVESTING
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Abstract
In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated two-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytical results. -
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References
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Yunxian Dai, Ping Yang, Zhiliang Luo, Yiping Lin. BOGDANOV-TAKENS BIFURCATION IN A DELAYED MICHAELIS-MENTEN TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1333-1346. doi: 10.11948/2156-907X.20180238
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Figure 1. The B-T bifurcation diagram and phase portraits of system (3.1) in the
$\mu_1-\mu_2$ plane for$\mu_1 < 0, \mu_2 < 0 $ . -
Figure 2. Numerical simulations for system (1.3) near the Bogdanov-Takens bifurcation point
$(h_*, ~c_*)$ for$(\mu_1, \mu_2)=(-0.001, -0.4), ~(\mu_1, \mu_2)\in ①$ , where the positive equilibrium$E_1$ is stable. -
Figure 3. Numerical simulations for system (1.3) near the Bogdanov-Takens bifurcation point
$(h_*, ~c_*)$ for$(\mu_1, \mu_2)=(-0.001, -0.145), ~(\mu_1, \mu_2)\in ②$ sufficiently close to the line$H$ .$E_1$ is unstable and a stable bifurcating periodic solution occurs. -
Figure 4.
$E_1$ is unstable for$(\mu_1, \mu_2)=(-0.001, -0.035), ~(\mu_1, \mu_2)\in ②$ sufficiently closing to the line$HL$ . The orbit of the periodic solution sufficiently closes to the homoclinic orbit.