| Citation: |
Fu Feng, Jianping Shi, Hui Fang. HOPF BIFURCATION OF A FRACTIONAL-ORDER PREY-PREDATOR-SCAVENGER SYSTEM WITH HUNTING DELAY AND COMPETITION DELAY[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1234-1258. doi: 10.11948/20220253
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HOPF BIFURCATION OF A FRACTIONAL-ORDER PREY-PREDATOR-SCAVENGER SYSTEM WITH HUNTING DELAY AND COMPETITION DELAY
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Abstract
This paper deals with Hopf bifurcation of a fractional-order prey-predator-scavenger system (FPSS in short) with hunting delay and two-predator competition delay. We introduce the notion of Hopf bifurcation of fractional-order system with double delays. The characteristic equation of the linearized system of FPSS is obtained by using the method of linearization and Laplace transform. Choosing the hunting delay and the competition delay as bifurcation parameters, respectively, we obtain the stability switch conditions and the critical delay values of emergence of Hopf bifurcation by analyzing the characteristic equation of the linearized system around a coexistence equilibrium. Especially, the delay bifurcation curve of emergence of Hopf bifurcation for FPSS with nonzero double delays is determined. Numerical simulations are performed to further illustrate our theoretical results.
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References
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Fu Feng, Jianping Shi, Hui Fang. HOPF BIFURCATION OF A FRACTIONAL-ORDER PREY-PREDATOR-SCAVENGER SYSTEM WITH HUNTING DELAY AND COMPETITION DELAY[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1234-1258. doi: 10.11948/20220253
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Figure 1. Waveform plots of Case Ⅱ with
$ \tau_1=\tau_2=12.5<\tau_0=12.7408 $ . The coexistence equilibrium point$ E_7 $ of system (4.1) is asymptotically stable. -
Figure 2. Waveform plots of Case Ⅱ with
$ \tau_1=\tau_2=12.8>\tau_0=12.7408 $ . System (4.1) undergoes periodic oscillation. -
Figure 3. Waveform plots of Case Ⅴ with
$ \tau_1=0, \tau_2=93.5<\bar{\tau}_2=94.22664 $ . The coexistence equilibrium point$ E_7 $ of (4.1) is asymptotically stable. -
Figure 4. Waveform plots of Case Ⅴ with
$ \tau_1=0, \tau_2=94.6>\bar{\tau}_2=94.22664 $ . System (4.1) undergoes periodic oscillation. -
Figure 5. Waveform plots of Case Ⅵ with
$ \tau_1=13.4<13.84928, \tau_2=12 $ . The coexistence equilibrium point$ E_7 $ of system (4.1) is asymptotically stable. -
Figure 6. Waveform plots of Case Ⅵ with
$ \tau_1=14>13.84928, \tau_2=12 $ . System (4.1) undergoes periodic oscillation. -
Figure 7. The delay bifurcation curve
$ S_1 $ on the$ (\tau_2, \tau_1) $ -plane in Case Ⅵ. -
Figure 8. Waveform plots of Case Ⅲ with
$ \tau_1=9.1<\bar{\tau}_1=9.52225, \tau_2=0 $ . The coexistence equilibrium point$ E_7 $ of system (4.2) is asymptotically stable. -
Figure 9. Waveform plots of Case Ⅲ with
$ \tau_1=9.7>\bar{\tau}_1=9.52225, \tau_2=0 $ . System (4.2) undergoes periodic oscillation. -
Figure 10. Waveform plots of Case Ⅳ with
$ \tau_1=6, \tau_2=8<\bar{\tau}_{2}=8.33174 $ . The coexistence equilibrium point$ E_7 $ of system (4.2) is asymptotically stable. -
Figure 11. Waveform plots of Case Ⅵ with
$ \tau_1=6, \tau_2=8.5>\bar{\tau}_{2}=8.33174 $ . System (4.2) undergoes periodic oscillation. -
Figure 12. The delay bifurcation curve
$ S_2 $ on the$ (\tau_1, \tau_2) $ -plane in Case Ⅳ.