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学者姓名：李忠
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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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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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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 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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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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Abstract ：
Predator-prey interaction is considered a natural phenomenon in the ecological system. Empirical research conducted on vertebrates has demonstrated that the presence of predators can have a significant impact on the survival rates and reproductive capabilities of prey populations. Recently, there has been research on mathematical models of predator-prey systems that include different predator functional responses and fear effects. These studies have overlooked the influence of fear on the death rates of species that are hunted. From the given findings, we present a mathematical model of predator-prey systems that includes fear costs impacting the rates of reproduction and death in prey population. By reducing the discrete model into different normal forms, we prove that there exists a set of codimension-1 and codimension-2 bifurcations, which include transcritical, flip, Neimark-Sacker bifurcations, 1:2 and 1:4 strong resonance bifurcations. These findings indicate that, compared with the system without fear effect, the increase of the fear effect parameter k1 that affects the birth rate of prey and the fear effect parameter k2 that affects the death rate of prey will strengthen the oscillation of prey population and reduce the oscillation of predator population. In addition, the increases of k1 and k2 have no effect on the density of the prey population but reduce the density of the predator population. When the fear effect k1 and other parameter values remain the same, the system generates an expanding limit circle as k2 increases, indicating that the effect of fear effects on the death rate enhances the stability of the system.
Keyword ：
fear effect fear effect Neimark-Sacker bifurcation Neimark-Sacker bifurcation Predator-prey Predator-prey strong resonance bifurcation strong resonance bifurcation
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| GB/T 7714 | Li, Qianqian , Chen, Fengde , Li, Zhong et al. Impact of Fear on Death Rate of Prey Species: Codimension-1 Bifurcations and Strong Resonances in a Discrete Predator-Prey Model [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2025 . |
| MLA | Li, Qianqian et al. "Impact of Fear on Death Rate of Prey Species: Codimension-1 Bifurcations and Strong Resonances in a Discrete Predator-Prey Model" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS (2025) . |
| APA | Li, Qianqian , Chen, Fengde , Li, Zhong , Chen, Lijuan . Impact of Fear on Death Rate of Prey Species: Codimension-1 Bifurcations and Strong Resonances in a Discrete Predator-Prey Model . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 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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In ecosystems, hunting cooperation of the predator will intensify the survival pressure faced by the prey, and then promote anti-predator behavior in the prey population. Therefore, it is of biological significance to study the prey's anti-predator behavior and the predator's hunting cooperation simultaneously. In this paper, a mathematical model that encompasses both aspects is developed. Through rigorous and comprehensive analysis of equilibria and bifurcation phenomena, this study reveals how these two behaviors influence population dynamics and ecosystem stability. In detail, as the hunting cooperation parameter a increases, the density of the prey population decreases. When a is extremely large or small, it will hinder the persistence of predator and prey populations, and only moderate hunting cooperation can maintain the ecosystem balance. Furthermore, moderate anti-predator behavior of the prey can suppress predators' hunting efficiency, while once the anti-predator behavior exceeds a critical threshold, it will disrupt the coexistence of predator and prey populations. The findings in this research can be viewed as an improvement and a complement to those with anti-predator behavior or hunting cooperation.
Keyword ：
Anti-predator Anti-predator Bogdanov-Takensbifurcation Bogdanov-Takensbifurcation Hopf bifurcation Hopf bifurcation Hunting cooperation Hunting cooperation
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| GB/T 7714 | Zhang, Yilin , Chen, Lijuan , Chen, Fengde et al. Interplay between preys' anti-predator behavior and predators' cooperative hunting: A mathematical approach [J]. | CHAOS SOLITONS & FRACTALS , 2025 , 199 . |
| MLA | Zhang, Yilin et al. "Interplay between preys' anti-predator behavior and predators' cooperative hunting: A mathematical approach" . | CHAOS SOLITONS & FRACTALS 199 (2025) . |
| APA | Zhang, Yilin , Chen, Lijuan , Chen, Fengde , Li, Zhong . Interplay between preys' anti-predator behavior and predators' cooperative hunting: A mathematical approach . | CHAOS SOLITONS & FRACTALS , 2025 , 199 . |
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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 et al. "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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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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