| Citation: |
Haicheng Liu, Bin Ge, Qiyuan Liang, Jiaqi Chen. A DELAYED SEMILINEAR PARABOLIC PREDATOR-PREY SYSTEM WITH HABITAT COMPLEXITY AND HARVESTING EFFECTS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2561-2582. doi: 10.11948/20210015
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A DELAYED SEMILINEAR PARABOLIC PREDATOR-PREY SYSTEM WITH HABITAT COMPLEXITY AND HARVESTING EFFECTS
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Abstract
In this paper, we propose a delayed reaction-diffusive system with habitat complexity and harvesting effects, and study dynamic behaviors of the system. Firstly, for the system without time delay, the stability of equilibria is studied. It is found that when habitat complexity reaches a certain critical value, the positive equilibrium will change from unstable to locally asymptotically stable. Secondly, time delay effect on the dynamic behaviors of diffusion system is studied. The existence conditions of Hopf bifurcation are given, and the properties of bifurcating periodic solutions are studied by using the center manifold and normal form theories, including the direction of Hopf bifurcation, the stability of bifurcating periodic solutions and the period. Finally, the corresponding numerical simulations and biological interpretation are made to verify the results of theoretical analysis.
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Keywords:
- Predator-prey system /
- habitat complexity effect /
- harvesting effect /
- time delay /
- diffusion term
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References
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Haicheng Liu, Bin Ge, Qiyuan Liang, Jiaqi Chen. A DELAYED SEMILINEAR PARABOLIC PREDATOR-PREY SYSTEM WITH HABITAT COMPLEXITY AND HARVESTING EFFECTS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2561-2582. doi: 10.11948/20210015
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Figure 1.
$ {{P}^{*}} = (99.568, 4.825) $ is locally asymptotically stable, and the initial value is$ (99.5, 4.8) $ . -
Figure 2. The system produces periodic solutions, and the initial value is
$ (82.9, 4.4) $ . -
Figure 3. The system produces spatially homogeneous periodic solutions, and the initial value is
$ (62.2, 4.2) $ . -
Figure 4.
$ \tau = 1<{{\tau }_{0}} $ , the system is locally asymptotically stable at$ {{P}^{*}} = ({{u}_{0}}, {{v}_{0}}) $ . -
Figure 5.
$ \tau = 1.2>{{\tau }_{0}} $ , the system produces periodic solutions at$ {{P}^{*}} = ({{u}_{0}}, {{v}_{0}}) $ .