| Citation: |
Chunxia Liu, Shumin Li, Yan Yan. HOPF BIFURCATION ANALYSIS OF A DENSITY PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE AND TWO TIME DELAYS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1589-1605. doi: 10.11948/2156-907X.20190029
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HOPF BIFURCATION ANALYSIS OF A DENSITY PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE AND TWO TIME DELAYS
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Abstract
In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.-
Keywords:
- Predator-prey system /
- density dependence /
- Hopf bifurcations /
- time-delay
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References
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Chunxia Liu, Shumin Li, Yan Yan. HOPF BIFURCATION ANALYSIS OF A DENSITY PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE AND TWO TIME DELAYS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1589-1605. doi: 10.11948/2156-907X.20190029
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Figure 1. When
$ \tau_{1} = 0, E^* $ is asymptotically stable for$ \tau_{2} = 2.75<\tau_{20} = 6.7756 $ . -
Figure 2. When
$ \tau_{1}=0, E^* $ undergoes a Hopf bifurcation for$ \tau_{2}=3.8<\tau_{20}=6.7756 $ . -
Figure 3. When
$ \tau_{2} = 0, E^* $ is asymptotically stable for$ \tau_{1} = 0.98<\tau_{10} = 1.4340 $ . -
Figure 4. When
$ \tau_{2} = 0, E^* $ undergoes a Hopf bifurcation for$ \tau_{1} = 1.01657<\tau_{10} = 1.4340 $ . -
Figure 5.
$ E^* $ is asymptotically stable for$ \tau_{1} = 0.10, \tau_{2} = 1.9<\tau_{20} = 6.7756 $ . -
Figure 6.
$ E^* $ undergoes a Hopf bifurcation for$ \tau_{1} = 0.1, \tau_{2} = 7.0>\tau_{20} = 6.7756 $ .