| Citation: |
Yahong Peng, Yujing Li. STABILITY AND HOPF BIFURCATION OF A DELAYED PREDATOR-PREY SYSTEM WITH NONLOCAL COMPETITION AND HERD BEHAVIOUR[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1932-1958. doi: 10.11948/20220422
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STABILITY AND HOPF BIFURCATION OF A DELAYED PREDATOR-PREY SYSTEM WITH NONLOCAL COMPETITION AND HERD BEHAVIOUR
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Abstract
In this paper, we investigate the stability and Hopf bifurcation of a diffusive predator-prey system with herd behaviour. The model is described by introducing both time delay and nonlocal prey intraspecific competition. Compared to the model without time delay, or without nonlocal competition, thanks to the together action of time delay and nonlocal competition, we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous. We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves. Furthermore, by the computation of normal forms for the system near equilibria, we investigate the stability and direction of Hopf bifurcation. Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.
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Keywords:
- Predator-prey model /
- time delay /
- nonlocal prey competition /
- Hopf bifurcation
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References
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Yahong Peng, Yujing Li. STABILITY AND HOPF BIFURCATION OF A DELAYED PREDATOR-PREY SYSTEM WITH NONLOCAL COMPETITION AND HERD BEHAVIOUR[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1932-1958. doi: 10.11948/20220422
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Figure 1.
The distribution of the first Hopf bifurcation value
$ \tau_{*} $ $ d_{1}-d_{2} $ $ d_2=d_{2}^{*} $ $ d_2=d_{2}^{c} $ $ \tau_{*}=\tau_{10} $ $ R_{10} $ $ \tau_{*}=\tau_{00} $ $ R_{0} $ -
Figure 2.
Bifurcation curves diagram for the system (1.5). Parameter values are
$ d_1=1,\; \beta=0.8,\; r=3,\; l=1.5 $ -
Figure 3.
The simulations for species
$ u $ $ d_1=1, \beta=0.8, r=3, l=1.5 $ $ d_2=0.38, \tau=1.1 $ $ P_{1} $ $ u(x,t)=0.64-0.5cos(0.5x), v(x,t)=0.2304-0.1cos(0.5x) $ $ t\in[-\tau,0] $ $ d_2=0.5, \tau=1.15 $ $ P_{2} $ $ u(x,t)=0.64+0.01cos(0.5x), v(x,t)=0.2304+0.01cos(0.5x) $ $ t\in[-\tau,0] $ $ d_2=0.4, \tau=1.2 $ $ P_{3} $ $ u(x,t)=0.64+0.05cos(0.5x), v(x,t)=0.2304+0,01cos(0.5x) $ $ t\in[-\tau,0]. $ $ x=0.785, $ $ x=3.925, $ -
Figure 4.
Bifurcation curves diagram for the system (1.5). Parameter values are
$ d_1=3,\; \beta=0.8,\; r=3,\; l=1.5 $ -
Figure 5.
The simulations for species
$ u $ $ d_1=3, \beta=0.8, r=3, l=1.5 $ $ d_2=0.2,\; \tau=0.6 $ $ P_{4} $ $ u(x,t)=0.64-0.5cos(0.5x), v(x,t)=0.2304-0.1cos(0.5x) $ $ t\in[-\tau,0] $ $ d_2=0.2, \tau=1.16 $ $ P_{5} $ $ u(x,t)=0.64+0.01cos(0.5x), v(x,t)=0.2304+0.01cos(0.5x) $ $ t\in[-\tau,0]; $ $ d_2=0.2, \tau=1.63 $ $ P_{6} $ $ u(x,t)=0.64-0.1cos(x), v(x,t)=0.2304-0.5cos(x) $ $ t\in[-\tau,0] $