| Citation: |
Qimei Zhou, Yuming Chen, Shangming Chen, Fengde Chen. DYNAMIC ANALYSIS OF A DISCRETE AMENSALISM MODEL WITH ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2416-2432. doi: 10.11948/20220332
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DYNAMIC ANALYSIS OF A DISCRETE AMENSALISM MODEL WITH ALLEE EFFECT
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Abstract
This paper concerns with a discretization of a continuous-time amensalism model with Allee effect on the first species. Compared with the continuous analog, the discrete system has different and quite rich dynamical behavior. First, we obtain the existence of fixed points and their local stabilities. Then we confirm the occurrence of fold bifurcation and period doubling bifurcation by using the center manifold theorem and bifurcation theory. Followed is a hybrid control strategy to control the period-doubling bifurcation and stabilize unstable periodic orbits embedded in the complex attractor. Numerical simulations indicate that Allee effect is beneficial to the stability of the first species to a certain extent. Moreover, when the first species is affected by Allee effect, solutions can quickly approach the corresponding fixed point.
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References
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Qimei Zhou, Yuming Chen, Shangming Chen, Fengde Chen. DYNAMIC ANALYSIS OF A DISCRETE AMENSALISM MODEL WITH ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2416-2432. doi: 10.11948/20220332
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Figure 1.
When
$ (\alpha, m, \beta)=(2.2, 0, 0.8) $ $ E^*(1.4, 1) $ $ (x_0, y_0)=(0.5, 0.6) $ -
Figure 2.
The fold bifurcation diagram of system (1.5) at the fixed point
$ E_3^*(x_3^*, 1) $ $ \alpha=\alpha_2=1.8 $ $ m=0.4, \beta=0.8 $ $ \gamma=0.5 $ -
Figure 3.
The period-doubling bifurcation diagram of (1.5) at
$ E_2^*(2.25, 1) $ $ \alpha $ $ (\beta, m, \gamma)=(0.8, 0.5, 0.4) $ $ \alpha=\alpha^{**}\approx 3.32 $ -
Figure 4.
Phase portraits for various values of
$ \alpha $ -
Figure 5.
The bifurcation diagram with
$ m $ -
Figure 6.
Time series of solutions of system (1.5) with different values of
$ m $ -
Figure 7.
Trajectories of the densities of the first and second species with
$ (x_0, y_0) = (0.5, 0.6) $ -
Figure 8.
Time series of the first species for the controlled system (5.1) with different
$ \varpi $ $ \alpha=4.2 $ $ \beta=0.8 $ $ m=0.5 $ $ \gamma=0.4 $ $ (x_0, y_0)=(0.5, 0.6) $