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Abstract ：
In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov-Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population.
Keyword ：
additive Allee effect additive Allee effect bifurcation bifurcation fear effect-type feedback control fear effect-type feedback control Logistic model Logistic model two limit cycles two limit cycles
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| GB/T 7714 | Zhu, Qun , Li, Zhong , Chen, Fengde . Bifurcation in a single-species logistic model with addition Allee effect and fear effect-type feedback control [J]. | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 . |
| MLA | Zhu, Qun 等. "Bifurcation in a single-species logistic model with addition Allee effect and fear effect-type feedback control" . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS (2024) . |
| APA | Zhu, Qun , Li, Zhong , Chen, Fengde . Bifurcation in a single-species logistic model with addition Allee effect and fear effect-type feedback control . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 . |
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Abstract ：
A Leslie-Gower predator-prey model with Smith growth and constant-yield harvesting is proposed in this paper. We show that the system admits at most two boundary equilibria, both of which are unstable. The degenerate positive equilibrium of the system is a cusp of codimension 2, and the system undergoes cusp-type Bogdanov-Takens bifurcation of codimension 2. Moreover, we prove that the system has a weak focus of order at most 3, and the system can undergo a degenerate Hopf bifurcation of codimension 3. Our results reveal that the constant-yield harvesting can lead to richer dynamic behaviors.
Keyword ：
Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation harvesting harvesting Hopf bifurcation Hopf bifurcation Leslie-Gower Leslie-Gower Smith growth Smith growth
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| GB/T 7714 | He, Mengxin , Li, Zhong . Dynamic behaviors of a Leslie-Gower predator-prey model with Smith growth and constant-yield harvesting [J]. | ELECTRONIC RESEARCH ARCHIVE , 2024 , 32 (11) : 6424-6442 . |
| MLA | He, Mengxin 等. "Dynamic behaviors of a Leslie-Gower predator-prey model with Smith growth and constant-yield harvesting" . | ELECTRONIC RESEARCH ARCHIVE 32 . 11 (2024) : 6424-6442 . |
| APA | He, Mengxin , Li, Zhong . Dynamic behaviors of a Leslie-Gower predator-prey model with Smith growth and constant-yield harvesting . | ELECTRONIC RESEARCH ARCHIVE , 2024 , 32 (11) , 6424-6442 . |
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Abstract ：
A Leslie-Gower predator-prey model with Allee effect on prey and hunting cooperation on predator is considered. We show the solution of model is positive and ultimately upper bounded, and prove the origin is an attractor by applying the blow-up method. The model has at most two positive equilibria, one is always a hyperbolic saddle and the other is a weak focus of multiplicity at least two. Moreover, we confirm that the degenerate equilibrium can be a cusp of codimension at most 3. A series of bifurcations can occur, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. Selecting Allee effect and hunting cooperation as bifurcation parameters, we investigate the influence of Allee effect and hunting cooperation on the dynamics of the model. Finally, through numerical simulations, we illustrate the Allee effects (or hunting cooperation) is detrimental to the coexistence of two species when the strength of the Allee parameter (or hunting cooperation) increases.
Keyword ：
Allee effect Allee effect Bifurcation Bifurcation Hunting cooperation Hunting cooperation Leslie-Gower Leslie-Gower
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| GB/T 7714 | Liu, Yingzi , Zhang, Zhiyang , Li, Zhong . The Impact of Allee Effect on a Leslie-Gower Predator-Prey Model with Hunting Cooperation [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (2) . |
| MLA | Liu, Yingzi 等. "The Impact of Allee Effect on a Leslie-Gower Predator-Prey Model with Hunting Cooperation" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 23 . 2 (2024) . |
| APA | Liu, Yingzi , Zhang, Zhiyang , Li, Zhong . The Impact of Allee Effect on a Leslie-Gower Predator-Prey Model with Hunting Cooperation . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (2) . |
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This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington-DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results.
Keyword ：
Amensalism model Amensalism model Beddington-DeAngelis functional response Beddington-DeAngelis functional response bifurcation bifurcation fear effect fear effect global dynamics global dynamics stability stability
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| GB/T 7714 | Zhu, Qun , Chen, Fengde , Li, Zhong et al. Global Dynamics of Two-Species Amensalism Model with Beddington-DeAngelis Functional Response and Fear Effect [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2024 , 34 (06) . |
| MLA | Zhu, Qun et al. "Global Dynamics of Two-Species Amensalism Model with Beddington-DeAngelis Functional Response and Fear Effect" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 34 . 06 (2024) . |
| APA | Zhu, Qun , Chen, Fengde , Li, Zhong , Chen, Lijuan . Global Dynamics of Two-Species Amensalism Model with Beddington-DeAngelis Functional Response and Fear Effect . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2024 , 34 (06) . |
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Abstract ：
In this paper, we introduce constant-yield prey harvesting into the Holling-Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov-Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model.
Keyword ：
Constant-yield harvesting Constant-yield harvesting degenerate Bogdanov-Takens bifurcation of codimension 4 degenerate Bogdanov-Takens bifurcation of codimension 4 degenerate Hopf bifurcation degenerate Hopf bifurcation generalist predator generalist predator
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| GB/T 7714 | Wu, Hongqiuxue , Li, Zhong , He, Mengxin . Bifurcation Analysis of a Holling-Tanner Model with Generalist Predator and Constant-Yield Harvesting [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2024 , 34 (06) . |
| MLA | Wu, Hongqiuxue et al. "Bifurcation Analysis of a Holling-Tanner Model with Generalist Predator and Constant-Yield Harvesting" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 34 . 06 (2024) . |
| APA | Wu, Hongqiuxue , Li, Zhong , He, Mengxin . Bifurcation Analysis of a Holling-Tanner Model with Generalist Predator and Constant-Yield Harvesting . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2024 , 34 (06) . |
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Abstract ：
In this paper, we investigate the dynamic behavior of a modified Leslie-Gower predator-prey model with the Allee effect on both prey and predator. It is shown that the model has at most two positive equilibria, where one is always a hyperbolic saddle and the other is a weak focus with multiplicity of at least three by concrete example. In addition, we analyze the bifurcations of the system, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The results show that the model has a cusp of codimension three and undergoes a Bogdanov-Takens bifurcation of codimension two. The system undergoes a degenerate Hopf bifurcation and has two limit cycles (the inner one is stable and the outer one is unstable). These enrich the dynamics of the modified Leslie-Gower predator-prey model with the double Allee effects.
Keyword ：
Allee effect Allee effect Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation Hopf bifurcation Hopf bifurcation Leslie-Gower Leslie-Gower limit cycle limit cycle
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| GB/T 7714 | Xing, Mengyun , He, Mengxin , Li, Zhong . Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects [J]. | MATHEMATICAL BIOSCIENCES AND ENGINEERING , 2024 , 21 (1) : 792-831 . |
| MLA | Xing, Mengyun et al. "Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects" . | MATHEMATICAL BIOSCIENCES AND ENGINEERING 21 . 1 (2024) : 792-831 . |
| APA | Xing, Mengyun , He, Mengxin , Li, Zhong . Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects . | MATHEMATICAL BIOSCIENCES AND ENGINEERING , 2024 , 21 (1) , 792-831 . |
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Abstract ：
A Leslie–Gower predator–prey model with square root response function and generalist predator is considered, and the existence and stability of equilibria of the system are discussed. It is shown that the system undergoes a degenerate Hopf bifurcation of codimension exactly two, where there exist two limit cycles. In addition, we find that the system has a cusp of codimension two and exhibits a Bogdanov–Takens bifurcation of codimension two. Our results reveal richer dynamics than the system with no generalist predator. © 2024 Elsevier Ltd
Keyword ：
Bogdanov–Takens bifurcation Bogdanov–Takens bifurcation Generalist predator Generalist predator Hopf bifurcation Hopf bifurcation Leslie–Gower Leslie–Gower Square root response function Square root response function
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| GB/T 7714 | He, M. , Li, Z. . Dynamics of a Lesile–Gower predator–prey model with square root response function and generalist predator [J]. | Applied Mathematics Letters , 2024 , 157 . |
| MLA | He, M. et al. "Dynamics of a Lesile–Gower predator–prey model with square root response function and generalist predator" . | Applied Mathematics Letters 157 (2024) . |
| APA | He, M. , Li, Z. . Dynamics of a Lesile–Gower predator–prey model with square root response function and generalist predator . | Applied Mathematics Letters , 2024 , 157 . |
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Abstract ：
A Leslie-Gower predator-prey model with nonlinear harvesting and a generalist predator is considered in this paper. It is shown that the degenerate positive equilibrium of the system is a cusp of codimension up to 4, and the system admits the cusp-type degenerate Bogdanov-Takens bifurcation of codimension 4. Moreover, the system has a weak focus of at least order 3 and can undergo degenerate Hopf bifurcation of codimension 3. We verify, through numerical simulations, that the system admits three different stable states, such as a stable fixed point and three limit cycles (the middle one is unstable), or two stable fixed points and two limit cycles. Our results reveal that nonlinear harvesting and a generalist predator can lead to richer dynamics and bifurcations (such as three limit cycles or tristability); specifically, harvesting can cause the extinction of prey, but a generalist predator provides some protection for the predator in the absence of prey.
Keyword ：
Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation generalist predator generalist predator harvesting harvesting Hopf bifurcation Hopf bifurcation
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| GB/T 7714 | He, Mengxin , Li, Zhong . Bifurcation of a Leslie-Gower Predator-Prey Model with Nonlinear Harvesting and a Generalist Predator [J]. | AXIOMS , 2024 , 13 (10) . |
| MLA | He, Mengxin et al. "Bifurcation of a Leslie-Gower Predator-Prey Model with Nonlinear Harvesting and a Generalist Predator" . | AXIOMS 13 . 10 (2024) . |
| APA | He, Mengxin , Li, Zhong . Bifurcation of a Leslie-Gower Predator-Prey Model with Nonlinear Harvesting and a Generalist Predator . | AXIOMS , 2024 , 13 (10) . |
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Abstract ：
In this paper, a Leslie-Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie-Gower model with Allee effect and fear effect.
Keyword ：
Allee effect Allee effect Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation Fear effect Fear effect Hopf bifurcation Hopf bifurcation
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| GB/T 7714 | Liu, Tingting , Chen, Lijuan , Chen, Fengde et al. Dynamics of a Leslie-Gower Model with Weak Allee Effect on Prey and Fear Effect on Predator [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2023 , 33 (01) . |
| MLA | Liu, Tingting et al. "Dynamics of a Leslie-Gower Model with Weak Allee Effect on Prey and Fear Effect on Predator" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 33 . 01 (2023) . |
| APA | Liu, Tingting , Chen, Lijuan , Chen, Fengde , Li, Zhong . Dynamics of a Leslie-Gower Model with Weak Allee Effect on Prey and Fear Effect on Predator . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2023 , 33 (01) . |
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This paper analyzes an allelopathic phytoplankton competition model, which was proposed by Bandyopadhyay [Dynamical analysis of a allelopathic phytoplankton model, J Biol Syst 14(02):205-217, 2006]. Our study refines the previous results and finds at most three positive equilibria for the system. The existence conditions of all positive equilibria and the corresponding stability cases are given in the paper. Interesting dynamical phenomena such as bistability, saddle-node bifurcation, and cusp bifurcation are found. It is shown that the rate of toxin releases heavily influences the positive equilibria of the system under certain conditions. Numerical simulations verify the feasibility of the theoretical results.
Keyword ：
Allelopathy Allelopathy Competition Competition Cusp Bifurcation Cusp Bifurcation Global Stability Global Stability Saddle-Node Bifurcation Saddle-Node Bifurcation
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| GB/T 7714 | Chen, Shangming , Chen, Fengde , Li, Zhong et al. BIFURCATION ANALYSIS OF AN ALLELOPATHIC PHYTOPLANKTON MODEL [J]. | JOURNAL OF BIOLOGICAL SYSTEMS , 2023 . |
| MLA | Chen, Shangming et al. "BIFURCATION ANALYSIS OF AN ALLELOPATHIC PHYTOPLANKTON MODEL" . | JOURNAL OF BIOLOGICAL SYSTEMS (2023) . |
| APA | Chen, Shangming , Chen, Fengde , Li, Zhong , Chen, Lijuan . BIFURCATION ANALYSIS OF AN ALLELOPATHIC PHYTOPLANKTON MODEL . | JOURNAL OF BIOLOGICAL SYSTEMS , 2023 . |
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