| Citation: |
Özlem AK GÜMÜŞ. A STUDY ON STABILITY, BIFURCATION ANALYSIS AND CHAOS CONTROL OF A DISCRETE-TIME PREY-PREDATOR SYSTEM INVOLVING ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3166-3194. doi: 10.11948/20220532
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A STUDY ON STABILITY, BIFURCATION ANALYSIS AND CHAOS CONTROL OF A DISCRETE-TIME PREY-PREDATOR SYSTEM INVOLVING ALLEE EFFECT
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Abstract
This paper examines the stability and bifurcation of a discrete-time prey-predator system that is modified by the Allee effect on the prey population. The system undergoes flip and Neimark-Sacker bifurcations in a small neighborhood of the unique positive fixed point depending on the densities of prey-predator. The OGY method and hybrid control method are used to control the chaotic behavior that results from Neimark-Sacker bifurcation. In addition, numerical simulations are performed to illustrate the theoretical results. To keep the ecosystem stable, it is crucial to research how populations of prey and predator interact. The Allee effect is a significant evolutionary force that alters population size by affecting both prey and predator behavior. It would be more realistic to look into population behavior in light of this effect, which results from population density (number of individuals per unit area). The increase in the density of predator in the model with the Allee effect pushes the prey to extinction. When the density of predator is suppressed, the stability continues for a certain time before undergoing bifurcation.
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References
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Özlem AK GÜMÜŞ. A STUDY ON STABILITY, BIFURCATION ANALYSIS AND CHAOS CONTROL OF A DISCRETE-TIME PREY-PREDATOR SYSTEM INVOLVING ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3166-3194. doi: 10.11948/20220532
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Figure 1.
(a) The trajectories of prey and predator densities in the system (2.2) when
$ a=0.9 $ $ b=0.2 $ $ c=0.5 $ $ d=3.5 $ $ a=0.9 $ $ b=0.2 $ $ c=0.5 $ $ d=3.5 $ -
Figure 2.
(a) The trajectories of prey and predator densities in the system (2.4) when
$ a=1.39 $ $ b=0.2 $ $ c=0.5 $ $ d=3.5 $ $ a=1.39 $ $ b=0.2 $ $ c=0.5 $ $ d=3.5 $ $ a=3.6 $ $ b=0.2 $ $ c=0.5 $ $ d=1.4 $ $ a=3.6 $ $ b=0.2 $ $ c=0.5 $ $ d=1.4 $ -
Figure 3.
(a) The trajectories of prey and predator densities in the system (2.7) when
$ a=2.33 $ $ b=0.2 $ $ c=0.031 $ $ d=3.5 $ $ a=2.33 $ $ b=0.2 $ $ c=0.031 $ $ d=3.5 $ $ a=2.53 $ $ b=0.2 $ $ c=0.033 $ $ d=3.5 $ $ a=2.53 $ $ b=0.2 $ $ c=0.033 $ $ d=3.5 $ -
Figure 4.
Bifurcations diagram of the prey-predator system (4.1) with the parameter values
$ a\in (5.3, 5.8) $ $ b=0.2 $ $ d=3.5 $ $ c=0, 5 $ -
Figure 5.
(a) Bifurcations diagram of the prey-predator system (4.2) with the parameter values
$ a\in(2, 4) $ $ b=0.2 $ $ d=3.5 $ $ c=0, 05 $ $ a=2.5 $ $ b=0.2 $ $ d=3.5 $ $ c=0, 05 $ $ a=2.64151 $ $ b=0.2 $ $ d=3.5 $ $ c=0, 05 $ $ a=2.7 $ $ b=0.2 $ $ d=3.5 $ $ c=0, 05 $ -
Figure 6.
Stability region of the controlled system (4.3) in
$ k_{1}k_{2} $ -
Figure 7.
(a) The trajectories of the controlled system (4.4) when
$ b=0.2 $ $ d=3.5 $ $ c=0.05 $ $ a=2.64151 $ $ \gamma=0.97 $ $ b=0.2 $ $ d=3.5 $ $ c=0.05 $ $ a=2.64151 $ $ \gamma=0.97 $ -
Figure 8.
(a) The trajectories of prey-predator densities when
$ a=2.9 $ $ b=0.2 $ $ c=0.5 $ $ d=0.9 $ $ a=2.9 $ $ b=0.2 $ $ c=0.5 $ $ d=0.9 $ $ a=3.1 $ $ b=0.2 $ $ c=0.5 $ $ d=0.9 $ $ a=3.1 $ $ b=0.2 $ $ c=0.5 $ $ d=0.9 $ $ a=2.9 $ $ b=0.2 $ $ c=0.5 $ $ d=1.4 $ $ a=2.9 $ $ b=0.2 $ $ c=0.5 $ $ d=1.4 $ -
Figure 9.
(a) The trajectories of prey-predator densities with the initial conditions
$ x_{0}=0.5 $ $ y_{0}=8.1 $ $ a=2.53 $ $ b=0.2 $ $ c=0.30 $ $ d=3.5 $ $ a=2.53 $ $ b=0.2 $ $ c=0.30 $ $ d=3.5 $ $ x_{0}=0.5 $ $ y_{0}=11.1 $ $ a=3.381 $ $ b=0.2 $ $ c=0.30 $ $ d=3 $ $ a=3.381, b=0.2 $ $ c=0.30 $ $ d=3 $ $ x_{0}=0.5 $ $ y_{0}=6.1 $ $ a=3.381 $ $ b=0.2 $ $ c=0.033 $ $ d=2.99 $ $ a=3.381 $ $ b=0.2 $ $ c=0.033 $ $ d=2.99 $ -
Figure 10.
Neimark-Sacker bifurcations of system (1.1) when
$ a \in (1, 1.4) $ $ b=0.0001 $ $ d=9.5 $ $ c=0.0001 $