| Citation: |
Ping Yang, Yiping Lin. STABILITY SWITCHING CURVES AND HOPF BIFURCATION ON A THREE SPECIES FOOD CHAIN WITH TWO DELAYS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1062-1076. doi: 10.11948/20220118
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STABILITY SWITCHING CURVES AND HOPF BIFURCATION ON A THREE SPECIES FOOD CHAIN WITH TWO DELAYS
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Abstract
A three species food chain with two time delays and double Holling type-Ⅱ functional responses is investigated. The conditions for the existence of positive equilibrium and Hopf bifurcation are presented. The stability area of positive equilibrium is surrounded by coordinate axis and stability switching curves. By using the theory of Hassard, Hopf bifurcation directions are determined analytically. Numerical simulations are presented on the frontier of stability to explain and support the analytic results.
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References
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Ping Yang, Yiping Lin. STABILITY SWITCHING CURVES AND HOPF BIFURCATION ON A THREE SPECIES FOOD CHAIN WITH TWO DELAYS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1062-1076. doi: 10.11948/20220118
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Figure 1. For
$ a_1=5, b_1=2, d_1=0.6, a_2=0.2, b_2=0.3, d_2=0.02, $ $ F1, F2, F3 $ are all less then$ 0 $ if$ \omega \in (0.0738, 0.2371) $ . -
Figure 2. For
$ a_1=5, b_1=2, d_1=0.6, a_2=0.2, b_2=0.3, d_2=0.02, $ $ \mathit{T} $ is a switching curve on$ \tau_1-\tau_2 $ plane with$ \omega \in (0.0738, 0.2371) $ .$ E_* $ is stable in the region surrounded by$ \tau_1=0, \tau_2=0 $ and curve$ T $ . -
Figure 3. When
$ \tau_1=2.91, \tau_2=3.44, $ $ E_* $ is asymptotically stable. -
Figure 4. When
$ \tau_1=3.51, \tau_2=3.44, $ $ E_* $ is unstable, and there is a stable periodic solution surrounding$ E_* $ . -
Figure 5. When
$ \tau_1=2.91, \tau_2=6.68, $ $ E_* $ is unstable and there is a stable periodic solution surrounding$ E_* $ .