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学者姓名：陈凤德
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Abstract ：
The stability of the ecosystem is essential for the sustainable development of the earth. Studying the population model's dynamic behavior, including the Allee and anxiety effects, can better represent the ecosystem's working mechanism, which is crucial for preserving ecological balance. In light of this, the objective of this paper is to build a Leslie-Gower model that incorporates Allee effect on the birth rate and the saturated fear effect on the predator, then analyze its dynamic behavior and the impact of the saturated fear effect on population density. In the process of analysis, the existence and stability of boundary and positive equilibria are established, demonstrating that the origin is an attractor using the blow-up method. By varying the saturated fear effect parameter, the corresponding system will undergo supercritical, sub-critical, and even degenerate Hopf bifurcations. The existence of Bogdanov-Takens bifurcation of codimension-2 (or codimension-3) is demonstrated near the unique positive equilibrium. In light of these bifurcation phenomena, the validity of the theoretical results is confirmed through graphical representations via numerical simulations. The results show that while the fear effect on the predator favorably contributes to the ecological stability, high levels of either the Allee effect or the saturated fear effect pose a hazard to the stability of the ecosystem.
Keyword ：
Allee effect Allee effect Bogdanov-Takens bifurcation Bogdanov-Takens bifurcation Hopf bifurcation Hopf bifurcation Saturated fear effect Saturated fear effect
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| GB/T 7714 | Zhang, Yilin , Chen, Lijuan , Xu, Junyan et al. Analysis of a Leslie-Gower Model with Allee Effect on Birth Rate and Saturated Fear Effect [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2025 , 35 (07) . |
| MLA | Zhang, Yilin et al. "Analysis of a Leslie-Gower Model with Allee Effect on Birth Rate and Saturated Fear Effect" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 35 . 07 (2025) . |
| APA | Zhang, Yilin , Chen, Lijuan , Xu, Junyan , Chen, Fengde . Analysis of a Leslie-Gower Model with Allee Effect on Birth Rate and Saturated Fear Effect . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2025 , 35 (07) . |
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Abstract ：
Predator-prey interactions are among the most common and crucial ecological phenomena in nature. Over the course of long-term evolution, prey populations have developed various anti-predation strategies to cope with the threat of predators, with population dispersal being one of the most common strategies. In traditional ecological models, the prey population is typically constrained by direct predation. However, an increasing body of empirical evidence suggests that the fear effect from the predator significantly alters the physiological behavior of prey, leading to a decrease in reproduction rate and an increase in mortality rate. In this paper, we investigate a predator-prey system incorporating asymmetric dispersal and the fear effect, which influences the birth and death rates of the prey species. We rigorously establish the existence and local stability of equilibrium points, derive sufficient conditions for global stability, and prove the occurrence of a transcritical bifurcation at the boundary equilibrium. Our analysis reveals an optimal dispersal rate that maximizes prey population density; beyond this threshold, increased dispersal drives both populations to extinction. Furthermore, the fear effect and its maximum cost exhibit significant negative impacts on predator abundance, though they do not alter the equilibrium stability or existence. These findings provide critical insights for designing habitat corridors in endangered species conservation and underscore the pivotal role of prey dispersal in shaping population dynamics.
Keyword ：
bifurcation bifurcation dispersal dispersal fear effect fear effect global stability global stability predator-prey predator-prey
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| GB/T 7714 | Meng, Xinyu , Chen, Lijuan , Chen, Fengde . Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect [J]. | SYMMETRY-BASEL , 2025 , 17 (3) . |
| MLA | Meng, Xinyu et al. "Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect" . | SYMMETRY-BASEL 17 . 3 (2025) . |
| APA | Meng, Xinyu , Chen, Lijuan , Chen, Fengde . Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect . | SYMMETRY-BASEL , 2025 , 17 (3) . |
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Abstract ：
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 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 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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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 paper, a patchy model in which the migration is induced by the fear effect on the predator was investigated. By applying dynamical theory, the complete study on persistence of the system and the local/global stability of equilibria were discussed. Choosing the diffusion coefficient D1 as the bifurcation parameter, transcritical bifurcation occurring near the trivial equilibrium was demonstrated. We concluded that low dispersal is favorable for the coexistence of both species, but large dispersal leads to the extinction of species. There is an optimal diffusion coefficient to make the density of the prey population reach its maximum. In addition, the level of fear effect k and the maximum fear cost η are beneficial to the total population density of prey. © 2025 the Author(s)
Keyword ：
Abiotic Abiotic Invertebrates Invertebrates Predator prey systems Predator prey systems
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| GB/T 7714 | Zhong, Jin , Xia, Yue , Chen, Lijuan et al. Dynamical analysis of a predator-prey system with fear-induced dispersal between patches [J]. | Mathematical Biosciences and Engineering , 2025 , 22 (5) : 1159-1184 . |
| MLA | Zhong, Jin et al. "Dynamical analysis of a predator-prey system with fear-induced dispersal between patches" . | Mathematical Biosciences and Engineering 22 . 5 (2025) : 1159-1184 . |
| APA | Zhong, Jin , Xia, Yue , Chen, Lijuan , Chen, Fengde . Dynamical analysis of a predator-prey system with fear-induced dispersal between patches . | Mathematical Biosciences and Engineering , 2025 , 22 (5) , 1159-1184 . |
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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, 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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