| Citation: |
Chuanying Zhang, Ranchao Wu, Mengxin Chen. HOPF BIFURCATION IN A DELAYED PREDATOR-PREY SYSTEM WITH GENERAL GROUP DEFENCE FOR PREY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 810-840. doi: 10.11948/20200011
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HOPF BIFURCATION IN A DELAYED PREDATOR-PREY SYSTEM WITH GENERAL GROUP DEFENCE FOR PREY
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Abstract
In this paper, a delayed predator-prey system with group defence for prey is investigated. Firstly, in the absence of spatial diffusion and time delay, the stability of positive equilibrium and existence of the Hopf bifurcation are investigated, as well as the direction of the Hopf bifurcation, which is determined by applying the first Lyapunov number. Then, the occurrence of the Hopf bifurcation in the diffusion-driven delayed system is further explored. By using the center manifold reduction and the normal form theorem, the conditions ensuring the occurrence of Hopf bifurcation, its direction and its stability are formulated in terms of different parameters. Finally, some numerical simulations are carried out to verify the theoretical results and the existence of the homogeneous periodic solution is exhibited by setting different values of parameters. Moreover, stable temporal periodic solutions and spatially inhomogeneous periodic solutions are identified from the numerical simulations. The obtained results are also explained and discussed from the practical point of view.
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Keywords:
- Stability /
- hopf bifurcation /
- time delay /
- group defense
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References
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Chuanying Zhang, Ranchao Wu, Mengxin Chen. HOPF BIFURCATION IN A DELAYED PREDATOR-PREY SYSTEM WITH GENERAL GROUP DEFENCE FOR PREY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 810-840. doi: 10.11948/20200011
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Figure 1. The equilibrium
$ E_* $ is locally asymptotically stable. - Figure 2. The periodic solution bifurcated from the Hopf bifurcation is stable
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Figure 3. Solutions
$ u(x, t) $ and$ v(x, t) $ with diffusion and different aggregation efficiency$ \alpha $ . The Hopf bifurcation could exist in the following situations: (i)$ \alpha = 0.2 $ (top panel), (ii)$ \alpha = 0.5 $ (middle panel), (iii)$ \alpha = 0.75 $ (bottom panel), other parameters could refer to (i-iii) in Remark 2.2. Here choose the initial values$ u(x, 0) = u_*-0.001\cos2x, \; v(x, 0) = u_*-0.001\cos2x $ . -
Figure 4.
$ E_* = (0.5905, 0.2687) $ is locally asymptotically stable, where$ \tau = 1.34 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.5905+0.5\cos(2x), 0.2687+0.4\cos(2x)) $ . -
Figure 5. When
$ \alpha = 0.2, $ the bifurcating periodic solutions are stable, where$ \tau = 1.48 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.5905+0.02\cos(2x), 0.2687+0.02\cos(2x)) $ . -
Figure 6. When
$ \alpha = 0.5, $ the bifurcating periodic solutions are stable, where$ \tau = 1.29 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.4624+0.5\cos(2x), 0.3656+0.5\cos(2x)) $ . -
Figure 7. When
$ \alpha = 0.7, $ the bifurcating periodic solutions are stable, where$ \tau = 1.5 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.663-0.02\cos(2x), 0.2979-0.02\cos(2x)) $ . -
Figure 8. For system (1.4), there exist quasi periodic solution, where
$ \alpha = 0.2, \; \tau = 1.2942 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.4624+0.005\cos(5x), 0.3656+0.05\cos(5x)) $ . -
Figure 9. For system (1.4), the spatially inhomogeneous bifurcating periodic solution are stable, where
$ \alpha = 0.93, \; \tau = 2.08 $ and initial condition is$ (u(x, 0), v(x, 0)) = (0.5601-0.2\cos(2x), 0.4224-0.2\cos(2x)) $ .