| Citation: |
Heping Jiang, Xiaosong Tang. HOPF BIFURCATION IN A DIFFUSIVE PREDATOR-PREY MODEL WITH HERD BEHAVIOR AND PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 671-690. doi: 10.11948/2156-907X.20180142
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HOPF BIFURCATION IN A DIFFUSIVE PREDATOR-PREY MODEL WITH HERD BEHAVIOR AND PREY HARVESTING
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Abstract
In this paper, the dynamics of a diffusive delayed predator-prey model with herd behavior and prey harvesting subject to the homogeneous Neumann boundary condition is considered. Firstly, choosing the harvesting term as a bifurcation parameter, then we obtain the existence and the stability of the equilibrium by analyzing the distribution of the roots of associated characteristic equation. Secondly, time delay is regarding as a bifurcation parameter, and the use of the normal form theory and center manifold theorem, the existence, stability and direction of bifurcating periodic solutions are all demonstrated detailly. Finally, summarizing some numerical simulations to illustrate the theoretical analysis.-
Keywords:
- Hopf bifurcation /
- predator-prey model /
- herd behavior /
- prey harvesting /
- delay
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References
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Heping Jiang, Xiaosong Tang. HOPF BIFURCATION IN A DIFFUSIVE PREDATOR-PREY MODEL WITH HERD BEHAVIOR AND PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 671-690. doi: 10.11948/2156-907X.20180142
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Figure 1. The stability region in the
$\beta-h$ plane, (A):$\eta=0.5 < 1$ ; (B):$\eta=1.2>1$ . -
Figure 2. The trajectory graphs and phase portrait of (2.1), which shows the positive equilibrium
$\left(u^*, v^*\right)$ is asymptotically stable when$h=0.38>0.3016$ . Here we set some parameter values$\gamma=2, \eta=0.5, \beta=0.55$ , and the initial value is$u_0=0, 6, v_0=0.4$ . -
Figure 3. The trajectory graphs and phase portrait of (2.1), there exists a stable periodic orbit that bifurcating from the positive equilibrium
$\left(u^*, v^*\right)$ when the parameter$h=0.28 < 0.3016$ . Here we set these parameter values$\gamma=2, \eta=0.5, \beta=0.55$ , and the initial value is$u_0=0, 6, v_0=0.4$ . -
Figure 4. The positive equilibrium
$\left(u^*, v^*\right)$ of (1.3) is asymptotically stable when the value of the parameteris$h=0.38>0.3016$ . Here we set these parameter values$d_1=0.02, d_2=1, \gamma=2, \eta=0.5, \beta=0.55$ , and the initial values is$u(x, 0)=u^*+0.05\cos x, v(x, 0)=v^*+0.05\cos x$ . -
Figure 5. There exists stable spatially homogenous periodic solutions that bifurcating from the positive equilibrium
$\left(u^*, v^*\right)$ of (1.3) when$h=0.28 < 0.3016$ . Here we set parameter values$d_1=0.02, d_2=1, \gamma=2, \eta=0.5, \beta=0.55$ , and the initial values is$u(x, 0)=u^*+0.05\cos x, v(x, 0)=v^*+0.05\cos x$ . -
Figure 6. The trajectory graphs and phase portrait of (1.4) without diffusion. The positive equilibrium
$\left(u^*, v^*\right)$ is asymptotically stable when the parameter$\tau=1.68 < 1.7348$ . Here we set parameter values$\gamma=2, \eta=0.5, \beta=0.7, h=0.3$ and the initial value is$u_0=0.3, v_0=0.2$ . -
Figure 7. The trajectory graphs and phase portrait of (1.4) without diffusion. There exists a stable periodic orbit that bifurcating from the positive equilibrium
$\left(u^*, v^*\right)$ when the parameter$\tau=1.8>1.7348$ . Here we set parameter values$\gamma=2, \eta=0.5, \beta=0.7, h=0.3$ and the initial value is$u_0=0.3, v_0=0.2$ . -
Figure 8. The positive equilibrium
$\left(u^*, v^*\right)$ of (1.4) is asymptotically stable when$\tau=1.68 < 1.7348$ . Here we choose parameter values$d_1=0.02, d_2=1, \gamma=2, \eta=0.5, \beta=0.7, h=0.3$ and set the initial values$u(x, 0)=u^*+0.01\cos x, v(x, 0)=v^*+0.01\cos x$ . -
Figure 9. There exists stable spatially homogenous periodic solutions which are bifurcating from the positive equilibrium
$\left(u^*, v^*\right)$ of (1.4) when$\tau=1.8>1.7348$ . Here we choose parameter values$d_1=0.02, d_2=1, \gamma=2, \eta=0.5, \beta=0.7, h=0.3$ and set the initial values is$u(x, 0)=u^*+0.01\cos x, v(x, 0)=v^*+0.01\cos x$ .