| Citation: |
Yingxian Zhu, Shuangfei Li, Yunxian Dai. STABILITY ANALYSIS OF A FRACTIONAL PREDATOR-PREY SYSTEM WITH TWO DELAYS AND INCOMMENSURATE ORDERS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 981-1006. doi: 10.11948/20220093
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STABILITY ANALYSIS OF A FRACTIONAL PREDATOR-PREY SYSTEM WITH TWO DELAYS AND INCOMMENSURATE ORDERS
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Abstract
In this paper, we consider a fractional predator-prey system with two delays and incommensurate orders. Firstly, the local stability of positive equilibrium of the system without delay is discussed. Secondly, we calculate the critical value of Hopf bifurcation by taking one delay as bifurcation parameter. Then, as two nonidentical delays change simultaneously, the stability switching curves, the directions of crossing and the existence of Hopf bifurcation are obtained. Finally, numerical simulations are presented to verify the given theoretical results.
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References
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Yingxian Zhu, Shuangfei Li, Yunxian Dai. STABILITY ANALYSIS OF A FRACTIONAL PREDATOR-PREY SYSTEM WITH TWO DELAYS AND INCOMMENSURATE ORDERS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 981-1006. doi: 10.11948/20220093
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Figure 1. The positive equilibrium
$ J^{*} $ of system$ (4.1) $ is locally asymptotically stable when$ \tau_{1}=0 $ and$ \tau_{2}=0 $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 2. The positive equilibrium
$ J^{*} $ of system$ (4.1) $ is locally asymptotically stable when$ \tau_{1}=0.8<\tau_{10} $ and$ \tau_{2}=0 $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 3. The system
$ (4.1) $ appears Hopf bifurcation when$ \tau_{1}=1>\tau_{10} $ and$ \tau_{2}=0 $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 4. The positive equilibrium
$ J^{*} $ of system$ (4.1) $ is locally asymptotically stable when$ \tau_{1}=0 $ with$ \tau_{2}=0.2<\tau_{20} $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 5. The system
$ (4.1) $ appears Hopf bifurcation when$ \tau_{1}=0 $ and$ \tau_{2}=0.35>\tau_{20} $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 6. Graph of F(
$ \omega $ ) when$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 7. Plot of the stability switching curves when
$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 8. The stable region of system
$ (4.1) $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 9. The positive equilibrium
$ J^{*} $ of system$ (4.1) $ is locally asymptotically stable when$ \tau_{1}=0.315 $ and$ \tau_{2}=0.24<0.2433 $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 10. The system
$ (4.1) $ appears Hopf bifurcation when$ \tau_{1}=0.315 $ and$ \tau_{2}=0.25>0.2433 $ with$ \gamma_{1}=0.9 $ and$ \gamma_{2}=0.95 $ . -
Figure 11. Plot of the stability switching curves when
$ \gamma_{1}= \gamma_{2}=1 $ . -
Figure 12. The stable region of system
$ (4.1) $ with$ \gamma_{1}=\gamma_{2}=1 $ .