| Citation: |
Guodong Zhang, Huangyu Guo, Jing Han. HOPF BIFURCATION AND CONTROL FOR THE DELAYED PREDATOR-PREY MODEL WITH NONLINEAR PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2954-2976. doi: 10.11948/20240013
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HOPF BIFURCATION AND CONTROL FOR THE DELAYED PREDATOR-PREY MODEL WITH NONLINEAR PREY HARVESTING
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Abstract
In our study, we focused on investigating a delayed differential-algebraic system. The system incorporates a square root functional response and non-linear prey harvesting. Employing the normal form of differential algebraic systems and the central manifold theory, we conducted a detailed analysis of the system's stability and bifurcation phenomena, with time delay identified as a critical bifurcation parameter. When the time delay reached a critical value, the system's equilibrium points underwent the Hopf bifurcation, resulting in system instability. To achieve stability, we introduced a feedback controller, successfully transitioning the system from an unstable to a stable state. Through subsequent numerical simulations, we validated the accuracy and correctness of our research conclusions.
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Keywords:
- Stability /
- predator-prey system /
- time delay /
- Hopf bifurcation /
- periodic solution
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References
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Guodong Zhang, Huangyu Guo, Jing Han. HOPF BIFURCATION AND CONTROL FOR THE DELAYED PREDATOR-PREY MODEL WITH NONLINEAR PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2954-2976. doi: 10.11948/20240013
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Figure 1.
Under the condition
$ \tau = 0.45 < \tau_{0} $ $ Y_{0} $ $ x_{0} = 3.9 $ $ y_{0} = 5.4 $ $ E_{0} = 0.49 $ -
Figure 2.
At
$ \tau = 0.595 < \tau_{0} $ $ Y_{0} $ $ x_{0} = 3.9 $ $ y_{0} = 5.4 $ $ E_{0} = 0.49 $ -
Figure 3.
When
$ k_{1}=1.3 > 0.98 $ $ Y_{0} $ $ x_{0} = 3.9 $ $ y_{0} = 5.4 $ $ E_{0} = 0.49 $