| Citation: |
Qianqian Li, Fengde Chen, Lijuan Chen, Zhong Li. DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2089-2123. doi: 10.11948/20240399
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DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT
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Abstract
This paper proposes a discrete amensalism with the Beddington-DeAngelis functional response response and fear effect on the first species. By comparison and study of different bifurcations, the introduction of the Beddington-DeAngelis functional response not only increased the dynamical behaviour of the system, including the emergence of pitchfork bifurcation and fold bifurcation, but also reduced the rate of extinction of the first species. Furthermore, we analyze the influence of the fear effect on the system, specifically focusing on the boundary equilibrium $E_2$ and the positive equilibrium $E_1^*$. Our findings reveal that when the second species is in a chaotic state, due to the persistence of the second species, the fear effect may have increased the stability of the first species or accelerated the extinction of the first species; when the second species is stable, the fear effect plays an essential part in maintaining the stability of the first species. Moreover, an appropriate fear effect promotes the coexistence of the first and second species. However, if the fear effect becomes excessively large, it directly results in the extinction of the first species. The discovery further enhances the understanding of the influence generated by amensalism through the fear effect.
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Keywords:
- Amensalism /
- Beddington-DeAngelis functional response /
- fear effect /
- bifurcation /
- Chaos control
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References
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Qianqian Li, Fengde Chen, Lijuan Chen, Zhong Li. DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2089-2123. doi: 10.11948/20240399
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Figure 1.
Flip bifurcation image of system (1.7) at
$ E_2(0, \gamma) $ $ \alpha=6 $ $ c=1 $ $ \delta=1 $ $ \beta=1 $ $ k=1$ -
Figure 2.
The transcritical bifurcation diagram of system (1.7) at
$ E_2 $ $ \alpha=4 $ $ \delta=1 $ $ k=1 $ $ \gamma=1 $ $ c=2 $ $ \beta=0.25 $ $ \alpha=4 $ $ \delta=1 $ $ k=1 $ $ \gamma=1 $ $ c=2 $ $ \beta=1.5 $ -
Figure 3.
The pitchfork bifurcation diagram of system (1.7) at
$ E_2 $ $ \alpha=4 $ $ \delta=1 $ $ k=1 $ $ \gamma=1 $ $ c=2 $ $ \beta=1 $ $ \alpha=4 $ $ \delta=1 $ $ k=1 $ $ \gamma=1 $ $ c=2 $ $ \beta=2 $ -
Figure 4.
The fold bifurcation diagram of system (1.7) at
$ E_3^*(x_3^*, \gamma) $ $ c=c^*=5.37 $ $ \alpha=4 $ $ \beta=1 $ $ \delta=1 $ $ \gamma=1 $ $ k=0.1 $ -
Figure 5.
Flip bifurcation diagram of system (1.7) at
$ E_1^*(x_1^*, \gamma) $ $ \alpha=5 $ $ c=1 $ $ k=0.1 $ $ \delta=1 $ $ \gamma=1 $ -
Figure 6.
Diagrams of bifurcation for controlled system (5.2).
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Figure 7.
(a) Global stability of
$ E_1 $ $ \alpha=2.1 $ $ \delta $ $ c=11 $ $ \beta=1 $ $ \gamma=0.5 $ $ E_1 $ $ k=1 $ $ k=5 $ -
Figure 8.
(a) Global stability of
$ E_1^* $ $ \alpha=2 $ $ \delta $ $ c=0.5 $ $ \beta=1 $ $ \gamma=1 $ $ x $ $ k=0.5 $ $ k=2 $ $ k=4 $ -
Figure 9.
The bifurcation plot of
$ x $ $ k $ -
Figure 10.
(a) represent bifurcation plot of system (1.4) with
$ e_1=4.3 $ $ e_2=0.3 $ $ a_2=1 $ $ c=0.2 $ $ \alpha=5 $ $ \beta=0.5 $ $ c=1 $ $ \delta=1 $ $ \gamma=1 $