| Citation: |
Ming Kang, Fengjie Geng, Ming Zhao. DYNAMICAL BEHAVIORS OF A STOCHASTIC PREDATOR-PREY MODEL WITH ANTI-PREDATOR BEHAVIOR[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1209-1224. doi: 10.11948/20210497
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DYNAMICAL BEHAVIORS OF A STOCHASTIC PREDATOR-PREY MODEL WITH ANTI-PREDATOR BEHAVIOR
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Abstract
In this paper, a stochastic predator-prey model is proposed and studied, where the model has anti-predator behavior. By constructing a suitable Lyapunov function, combined with the Itô's formula and the stochastic comparison theorem, the existence and uniqueness of the global positive solution of the system are proved. Then the stochastic boundedness of the system is established, and we discussed the asymptotic behavior of the solution which fluctuates around the equilibrium point of the deterministic model. Moreover, we provide sufficient conditions for the persistence and extinction of the predator and prey. Finally, the results obtained in this paper are verified by numerical simulation, and the anti-predator behavior and stochastic perturbation are analyzed as well.
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Keywords:
- Stochastic predator-prey model /
- anti-predator behavior /
- boundedness /
- persistence /
- extinction
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References
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Ming Kang, Fengjie Geng, Ming Zhao. DYNAMICAL BEHAVIORS OF A STOCHASTIC PREDATOR-PREY MODEL WITH ANTI-PREDATOR BEHAVIOR[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1209-1224. doi: 10.11948/20210497
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Figure 1.
the paths of the populations
$ x(t) $ $ y(t) $ $ \sigma_{1}=0.01, \sigma_{2}=0.01 $ -
Figure 2.
the paths of the populations
$ x(t) $ $ y(t) $ $ \sigma_{1}=0.9, \sigma_{2}=0.01 $ -
Figure 3.
the paths of the populations
$ x(t) $ $ y(t) $ $ \sigma_{1}=0.1, \sigma_{2}=1.2 $ -
Figure 4.
the paths of the populations
$ x(t) $ $ y(t) $ $ \sigma_{1}=0.8, \sigma_{2}=1.1 $ -
Figure 5.
the paths of the populations
$ x(t) $ $ y(t) $ $ \sigma_{1}=0.01, \sigma_{2}=0.01, \eta=0.01 $ -
Figure 6.
the paths of the populations
$ x(t) $ $ y(t) $ $ \eta=0.8, \sigma_{1}=0.01, \sigma_{2}=0.01 $ -
Figure 7.
the paths of the populations
$ x(t) $ $ y(t) $ $ \eta=0.8, \sigma_{1}=0.1, \sigma_{2}=0.3 $