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学者姓名：李忠
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Abstract ：
Fear effects, as non-consumptive interactions in ecological systems, have emerged as a critical factor influencing species coexistence and population dynamics. Understanding the impact of fear effects on competing ecosystems helps humans better protect species and maintain ecological balance. In this paper, we propose and study a special Lotka-Volterra competition system, where one competitor experiences a reduction in birth rates and an increase in mortality rates due to fear of its competitor, and the effect of fear has saturated cost parameters beta and eta. The results show that using beta and eta as bifurcation parameters can induce transcritical and saddle-node bifurcations in the system. Furthermore, we find that the introduced saturated fear cost parameters beta and eta have interesting effects on the dynamical behavior of the Lotka-Volterra system. When the saturated fear cost parameters beta and eta satisfy certain conditions, the fear parameter k does not affect the global stability of equilibria. Even under extreme fear, the two species are still able to coexist stably, which aligns with certain ecological phenomena. All theoretical results are verified through numerical simulations.
Keyword ：
bifurcation bifurcation Competition model Competition model saturated fear effects saturated fear effects stability stability
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| GB/T 7714 | He, Guangwen , Chen, Fengde , Li, Zhong et al. The influence of saturated fear effects on species dynamics: A Lotka-Volterra competition model analysis [J]. | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2025 . |
| MLA | He, Guangwen et al. "The influence of saturated fear effects on species dynamics: A Lotka-Volterra competition model analysis" . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS (2025) . |
| APA | He, Guangwen , Chen, Fengde , Li, Zhong , Chen, Lijuan . The influence of saturated fear effects on species dynamics: A Lotka-Volterra competition model analysis . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2025 . |
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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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Abstract ：
Understanding how the fear affects biological populations will help us develop more effective ecological conservation measures. In this paper, we consider a stage-structured competition model in which one competitor fears the other, leading to an impact on reducing the birth rate of the former. First, we prove the non-negativity and boundedness of the solutions of this system and also analyze the existence and stability of the equilibria of the system. Our study shows that the fear effect greatly enriches the dynamical behavior of the originally stage-structured system. The fear parameter k makes it possible for the system to have two positive equilibria and induces transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. Interestingly, we find that the stage-structured competition system is more sensitive to the fear effect. Under the same conditions, a smaller value of k in system with stage structure can lead to alterations in the system's dynamic behaviors. Finally, numerical simulations are conducted to support and illustrate the theoretical results.
Keyword ：
Bifurcation Bifurcation Competition model Competition model Fear effect Fear effect Stability Stability Stage-structured Stage-structured
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| GB/T 7714 | He, Guangwen , Chen, Fengde , Li, Zhong et al. Impact of Fear on a Stage-Structured Lotka-Volterra Competition Model [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2025 , 24 (4) . |
| MLA | He, Guangwen et al. "Impact of Fear on a Stage-Structured Lotka-Volterra Competition Model" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 24 . 4 (2025) . |
| APA | He, Guangwen , Chen, Fengde , Li, Zhong , Chen, Lijuan . Impact of Fear on a Stage-Structured Lotka-Volterra Competition Model . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2025 , 24 (4) . |
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This paper investigates a model of amensalism, in which the first species is influenced by the combined effects of refuge and fear, while the second species exhibits an additive Allee effect. The paper analyzes the existence and stability of the equilibria of the system and derives the conditions for various bifurcations. In the global structure analysis, the stability at infinity is examined, and the phenomena of global stability and bistability in the system are analyzed. Additionally, a sensitivity analysis is employed to evaluate the impact of system parameters on populations. The study reveals that refuge has a significant positive effect on the first population, and refuge's effect becomes more pronounced as the fear level increases. Under the strong Allee effect, when the initial density of the second species is low, the second species may eventually become extinct; when the initial density is high, if the refuge parameter is below a certain threshold, increasing the refuge parameter slows down the extinction of the first species, whereas, when the refuge parameter exceeds this threshold, the two species can coexist. Under the weak Allee effect, when the refuge parameter surpasses a certain threshold, the two species can achieve long-term, stable coexistence, and the threshold for the weak Allee effect is higher than that for the strong Allee effect.
Keyword ：
additive Allee effect additive Allee effect amensalism model amensalism model bifurcation bifurcation fear effect fear effect global dynamics global dynamics refuge refuge
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| GB/T 7714 | Huang, Yuting , Chen, Fengde , Chen, Lijuan et al. Dynamic Analysis of an Amensalism Model Driven by Multiple Factors: The Interwoven Impacts of Refuge, the Fear Effect, and the Allee Effect [J]. | AXIOMS , 2025 , 14 (8) . |
| MLA | Huang, Yuting et al. "Dynamic Analysis of an Amensalism Model Driven by Multiple Factors: The Interwoven Impacts of Refuge, the Fear Effect, and the Allee Effect" . | AXIOMS 14 . 8 (2025) . |
| APA | Huang, Yuting , Chen, Fengde , Chen, Lijuan , Li, Zhong . Dynamic Analysis of an Amensalism Model Driven by Multiple Factors: The Interwoven Impacts of Refuge, the Fear Effect, and the Allee Effect . | AXIOMS , 2025 , 14 (8) . |
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Abstract ：
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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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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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 et al. "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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This study proposes a Lotka-Volterra competition model incorporating a novel fear-based feedback control mechanism. Through theoretical analysis and numerical simulations, several important conclusions are obtained. The system admits at most two positive equilibria; while the feedback control parameter does not affect their existence, it plays a crucial role in determining their stability. A sufficiently large control intensity can enhance the stability of the positive equilibrium, thereby promoting species coexistence. The system exhibits rich bifurcation phenomena, including transcritical, pitchfork, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Notably, a stable limit cycle may arise through a Hopf bifurcation. Under weak competition, the system admits at most one positive equilibrium, which is always stable regardless of the intensity of fear effects. In contrast, under strong competition, the system displays more complex dynamics, including the possible occurrence of multiple positive equilibria and intricate bifurcation structures. These results highlight the crucial role of fear-based feedback control in facilitating stable coexistence in competitive ecological systems.
Keyword ：
Bifurcation Bifurcation Competition model Competition model Fear effects Fear effects Feedback control Feedback control Limit cycle Limit cycle Stability Stability
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| GB/T 7714 | He, Guangwen , Chen, Fengde , Li, Zhong et al. Dynamical regulation of lotka-volterra competition systems: a novel fear-effect-based feedback control mechanism [J]. | NONLINEAR DYNAMICS , 2025 . |
| MLA | He, Guangwen et al. "Dynamical regulation of lotka-volterra competition systems: a novel fear-effect-based feedback control mechanism" . | NONLINEAR DYNAMICS (2025) . |
| APA | He, Guangwen , Chen, Fengde , Li, Zhong , Chen, Lijuan . Dynamical regulation of lotka-volterra competition systems: a novel fear-effect-based feedback control mechanism . | NONLINEAR DYNAMICS , 2025 . |
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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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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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