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学者姓名：李忠
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Abstract ：
In this article, we analyze the bifurcation of a modified LeslieGower system with Holling type II functional response and fear effect. We discuss the existence and stability of equilibria. The system admits at most two positive equilibria, where one is always a saddle and the other is an anti-saddle, and a unique degenerate equilibrium which is a cusp of codimension three. In addition, with the change of parameters, the system undergoes saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and cusp type degenerate Bogdanov-Takens bifurcation of codimension three. We show that the system has two limit cycles (i.e., the inner one is unstable and the outer one is stable), and then undergoes the bistable phenomena. Finally, the existence of bifurcations are verified by numerical simulations.
Keyword ：
Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation cusp cusp Fear effect Fear effect Hopf bifurcation Hopf bifurcation modified Leslie-Gower modified Leslie-Gower
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| GB/T 7714 | Zhang, Mengxin , Li, Zhong . BIFURCATION ANALYSIS OF A MODIFIED LESLIE-GOWER PREDATOR-PREY SYSTEM WITH FEAR EFFECT ON PREY [J]. | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B , 2025 , 30 (10) : 3730-3760 . |
| MLA | Zhang, Mengxin 等. "BIFURCATION ANALYSIS OF A MODIFIED LESLIE-GOWER PREDATOR-PREY SYSTEM WITH FEAR EFFECT ON PREY" . | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 30 . 10 (2025) : 3730-3760 . |
| APA | Zhang, Mengxin , Li, Zhong . BIFURCATION ANALYSIS OF A MODIFIED LESLIE-GOWER PREDATOR-PREY SYSTEM WITH FEAR EFFECT ON PREY . | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B , 2025 , 30 (10) , 3730-3760 . |
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Abstract ：
This paper proposes a discrete amensalism with the BeddingtonDeAngelis 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 E2 and the positive equilibrium E1 & lowast;. 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.
Keyword ：
Beddington-DeAngelis functional response Beddington-DeAngelis functional response bifurcation bifurcation Chaos control. Chaos control. fear ef- fear ef- fect fect Keywords Amensalism Keywords Amensalism
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| GB/T 7714 | Li, Qianqian , Chen, Fengde , Chen, Lijuan et al. DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT [J]. | JOURNAL OF APPLIED ANALYSIS AND COMPUTATION , 2025 , 15 (4) : 2089-2123 . |
| MLA | Li, Qianqian et al. "DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT" . | JOURNAL OF APPLIED ANALYSIS AND COMPUTATION 15 . 4 (2025) : 2089-2123 . |
| APA | Li, Qianqian , Chen, Fengde , Chen, Lijuan , Li, Zhong . DYNAMICAL ANALYSIS OF A DISCRETE AMENSALISM SYSTEM WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND FEAR EFFECT . | JOURNAL OF APPLIED ANALYSIS AND COMPUTATION , 2025 , 15 (4) , 2089-2123 . |
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Abstract ：
This paper investigates the effect of fear effect and constant-type harvesting on the dynamic of a Leslie-Gower predator-prey model. Initially, an analysis is carried out to identify all potential equilibria and evaluate their stability. Furthermore, the dynamic behavior at these points is examined, revealing various bifurcations such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation. In particular, the model undergoes a degenerate Hopf bifurcation, which leads to the existence of two limit cycles. Additionally, we demonstrate that the Bogdanov-Takens bifurcation of codimension 2 occurs in this model. Ultimately, these findings are validated through numerical simulations, demonstrating that continuous harvesting or the significant fear effect is not conducive to either predator or prey surviving.
Keyword ：
bifurcation. bifurcation. fear effect fear effect harvesting harvesting predator-prey predator-prey
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| GB/T 7714 | Huangfu, Chenyang , Li, Zhong . Bifurcation analysis of a Leslie-Gower predator-prey system with fear effect and constant-type harvesting [J]. | NONLINEAR ANALYSIS-MODELLING AND CONTROL , 2025 , 30 (3) : 439-459 . |
| MLA | Huangfu, Chenyang et al. "Bifurcation analysis of a Leslie-Gower predator-prey system with fear effect and constant-type harvesting" . | NONLINEAR ANALYSIS-MODELLING AND CONTROL 30 . 3 (2025) : 439-459 . |
| APA | Huangfu, Chenyang , Li, Zhong . Bifurcation analysis of a Leslie-Gower predator-prey system with fear effect and constant-type harvesting . | NONLINEAR ANALYSIS-MODELLING AND CONTROL , 2025 , 30 (3) , 439-459 . |
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In this paper, we study a Leslie-Gower predator-prey model with strong Allee effects and constant prey refuges. It is shown that the model can undergo a cusp type degenerate Bogdanov-Takens bifurcation of codimension 4, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension 3, and degenerate Hopf bifurcation of codimension 3 as the parameters vary. The model can exhibit the coexistence of multiple positive steady states, multiple limit cycles, and homoclinic loops. Our results indicate that a larger prey refuge contributes to the coexistence of both species. Numerical simulations, including three limit cycles, quadristability, a large-amplitude limit cycle enclosing three positive steady states and a homoclinic loop, two large-amplitude limit cycles enclosing three positive steady states, are presented to illustrate the theoretical results.
Keyword ：
Constant prey refuge Constant prey refuge Hopf bifurcation Hopf bifurcation Leslie-Gower predator-prey model Leslie-Gower predator-prey model Strong Allee effect Strong Allee effect
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| GB/T 7714 | Chen, Fengde , Li, Zhong , Pan, Qin et al. Bifurcations in a Leslie-Gower predator-prey model with strong Allee effects and constant prey refuges [J]. | CHAOS SOLITONS & FRACTALS , 2025 , 192 . |
| MLA | Chen, Fengde et al. "Bifurcations in a Leslie-Gower predator-prey model with strong Allee effects and constant prey refuges" . | CHAOS SOLITONS & FRACTALS 192 (2025) . |
| APA | Chen, Fengde , Li, Zhong , Pan, Qin , Zhu, Qun . Bifurcations in a Leslie-Gower predator-prey model with strong Allee effects and constant prey refuges . | CHAOS SOLITONS & FRACTALS , 2025 , 192 . |
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In this paper, a Leslie-Gower predator-prey model with simplified Holling type IV functional response and Allee effect on predator is studied. We discuss the existence and stability of equilibria, and show that the system has at most three positive equilibria. Under sufficient conditions, the system can have a weak focus of order 3, a cusp of codimension 3, a nilpotent focus or an elliptic equilibrium of codimension 3, or a nilpotent elliptic equilibrium of codimension at least 4. Also, we prove that the degenerate Hopf bifurcation of codimension 3 occurs at the anti-saddle, cusp type degenerate Bogdanov-Takens bifurcation of codimension 3 occurs at the double equilibrium, and focus or elliptic type degenerate Bogdanov-Takens bifurcation of codimension 3 occurs at the triple equilibrium. Finally, some numerical simulations show that the system can exhibit two limit cycles, a semi-stable limit cycle or a limit cycle containing a homoclinic loop. Our results imply that the predator and prey can arrive a stable state of coexistence if Allee effect is small, or the predator will be extinct if Allee effect is large.
Keyword ：
Allee effect Allee effect Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation Hopf bifurcation Hopf bifurcation Simplified Holling type IV Simplified Holling type IV
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| GB/T 7714 | Zhang, Mengxin , Li, Zhong , Chen, Fengde et al. Bifurcation Analysis of a Leslie-Gower Predator-Prey Model with Allee Effect on Predator and Simplified Holling Type IV Functional Response [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2025 , 24 (3) . |
| MLA | Zhang, Mengxin et al. "Bifurcation Analysis of a Leslie-Gower Predator-Prey Model with Allee Effect on Predator and Simplified Holling Type IV Functional Response" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 24 . 3 (2025) . |
| APA | Zhang, Mengxin , Li, Zhong , Chen, Fengde , Chen, Lijuan . Bifurcation Analysis of a Leslie-Gower Predator-Prey Model with Allee Effect on Predator and Simplified Holling Type IV Functional Response . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2025 , 24 (3) . |
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Abstract ：
This paper investigates a Leslie-Gower predator-prey model with simplified Holling type IV functional response and constant-yield prey harvesting. We analyze conditions for the existence of positive equilibria and prove that the system has at most four positive equilibria. The results show that the double positive equilibrium is a cusp of codimension at most 4, the triple positive equilibrium is a degenerate saddle or nilpotent focus of codimension-3, and the quadruple positive equilibrium is a nilpotent cusp of codimension-5. In addition, as the parameters vary, the system can undergo a cusp-type (or focus-type) degenerate Bogdanov-Takens bifurcation of codimension-4 (or codimension-3). Furthermore, the positive equilibrium is a weak focus of order at most 4, and the model can undergo a degenerate Hopf bifurcation of codimension-4. Finally, our main results are verified by some numerical simulations, which also reveal that there exist three limit cycles containing one positive equilibrium, or one (or two) limit cycles containing three positive equilibria, or a limit cycle as well as a homoclinic loop.
Keyword ：
bifurcation bifurcation constant-yield harvesting constant-yield harvesting Leslie-Gower Leslie-Gower simplified Holling type IV simplified Holling type IV
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| GB/T 7714 | Huangfu, Chenyang , Li, Zhong , Chen, Fengde et al. Dynamics of a Harvested Leslie-Gower Predator-Prey Model with Simplified Holling Type IV Functional Response [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2025 , 35 (02) . |
| MLA | Huangfu, Chenyang et al. "Dynamics of a Harvested Leslie-Gower Predator-Prey Model with Simplified Holling Type IV Functional Response" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 35 . 02 (2025) . |
| APA | Huangfu, Chenyang , Li, Zhong , Chen, Fengde , Chen, Lijuan , He, Mengxin . Dynamics of a Harvested Leslie-Gower Predator-Prey Model with Simplified Holling Type IV Functional Response . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2025 , 35 (02) . |
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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 et al. "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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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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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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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 et al. "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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