| Citation: |
Dan Jin, Ruizhi Yang. HOPF BIFURCATION IN A PREDATOR-PREY MODEL WITH MEMORY EFFECT AND INTRA-SPECIES COMPETITION IN PREDATOR[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1321-1335. doi: 10.11948/20220127
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HOPF BIFURCATION IN A PREDATOR-PREY MODEL WITH MEMORY EFFECT AND INTRA-SPECIES COMPETITION IN PREDATOR
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Abstract
This paper investigates the spatiotemporal dynamics of a reaction diffusion predator-prey model that incorporates memory delay and intra-species competition in predator. We provide rigorous results of the model including the local stability of positive equilibrium, the existence and the property of Hopf bifurcation. We show that increasing the intra-species competition is not beneficial to the stability of the positive equilibrium. Moreover, we obtain that the stable region of the positive equilibrium will decrease with the increase of memory-based diffusion coefficient when it larger than the critical value. In addition, the memory delay may also affect the stability of the positive equilibrium. When the memory delay crosses the critical value, the stable positive equilibrium becomes unstable, and the stably inhomogeneous periodic solutions appears. These results indicate that the memory delay and intra-species competition play an important role in the spatiotemporal dynamics of predator-prey model.
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References
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Dan Jin, Ruizhi Yang. HOPF BIFURCATION IN A PREDATOR-PREY MODEL WITH MEMORY EFFECT AND INTRA-SPECIES COMPETITION IN PREDATOR[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1321-1335. doi: 10.11948/20220127
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Figure 1.
The population densities of prey and predator with parameter
$ e $ -
Figure 2.
Bifurcation diagram of system
$ (1.2) $ $ e $ $ d=1.5 $ -
Figure 3.
Bifurcation diagram of system
$ (1.2) $ $ d $ $ e=0.1 $ -
Figure 4.
The numerical simulations of system
$ (1.2) $ $ e=0.1 $ $ \tau=10 $ $ (u_*, v_*) $ -
Figure 5.
The numerical simulations of system
$ (1.2) $ $ e=0.1 $ $ \tau=12 $ $ (u_*, v_*) $ -
Figure 6.
The numerical simulations of system
$ (1.2) $ $ e=0.3 $ $ \tau=6.5 $ $ (u_*, v_*) $ -
Figure 7.
The numerical simulations of system
$ (1.2) $ $ e=0.3 $ $ \tau=7.2 $ $ (u_*, v_*) $