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学者姓名：陈凤德
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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 等. "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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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 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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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 ：
bifurcation bifurcation dispersal dispersal fear effect fear effect global stability global stability predator-prey predator-prey
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| GB/T 7714 | Zhong, J. , Xia, Y. , Chen, L. 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, J. 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, J. , Xia, Y. , Chen, L. , Chen, F. . 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 ：
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 ：
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 . |
| 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 (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 . |
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Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.
Keyword ：
Allee effect Allee effect bifurcation bifurcation Competition model Competition model fear effect fear effect reaction-diffusion system reaction-diffusion system stability stability
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| GB/T 7714 | Chen, Shangming , Chen, Fengde , Srivastava, Vaibhava et al. Dynamical analysis of a Lotka-Volterra competition model with both Allee and fear effects [J]. | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 , 17 (08) . |
| MLA | Chen, Shangming et al. "Dynamical analysis of a Lotka-Volterra competition model with both Allee and fear effects" . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS 17 . 08 (2024) . |
| APA | Chen, Shangming , Chen, Fengde , Srivastava, Vaibhava , Parshad, Rana D. . Dynamical analysis of a Lotka-Volterra competition model with both Allee and fear effects . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 , 17 (08) . |
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One of the most important factors influencing animal growth is non-genetics, which includes factors like nutrition, management and environmental conditions. By consuming their prey, predators can directly affect their ecology and evolution, but they can also have an indirect impact by affecting their prey's nutrition and reproduction. Preys used to change their habitats, their foraging and vigilance habits as anti-predator responses. Cooperation during hunting by the predators develops significant fear in their prey which indirectly affects their nutrition. In this work, we propose a two-species stage-structured predator-prey system where the prey are classified into juvenile and mature prey. We assume that the conversion of juvenile prey to matured prey is affected by the fear of predation risk. Non-negativity and boundedness of the solutions are demonstrated theoretically. All the biologically feasible equilibrium states are determined, and their stabilities are analyzed. The role of various important factors, e.g. hunting cooperation rate, predation rate and rate of fear, on the system dynamics is discussed. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MatCont 7.3. Finally, the proposed model is extended into a harvesting model under quadratic harvesting strategy and the associated control problem has been analyzed for optimal harvesting.
Keyword ：
bifurcation analysis bifurcation analysis Hunting cooperation Hunting cooperation numerical simulation numerical simulation stability analysis stability analysis stage structure stage structure
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| GB/T 7714 | Kashyap, Ankur Jyoti , Doley, Dhanesh , Chen, Fengde et al. A stage-structured prey-predator interaction model with the impact of fear and hunting cooperation [J]. | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 . |
| MLA | Kashyap, Ankur Jyoti et al. "A stage-structured prey-predator interaction model with the impact of fear and hunting cooperation" . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS (2024) . |
| APA | Kashyap, Ankur Jyoti , Doley, Dhanesh , Chen, Fengde , Bordoloi, Arnab Jyoti . A stage-structured prey-predator interaction model with the impact of fear and hunting cooperation . | INTERNATIONAL JOURNAL OF BIOMATHEMATICS , 2024 . |
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This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity.
Keyword ：
Allelopathy Allelopathy Competition Competition Global stability Global stability Pitchfork bifurcation Pitchfork bifurcation Reaction-diffusion system Reaction-diffusion system Saddle-node bifurcation Saddle-node bifurcation Transcritical bifurcation Transcritical bifurcation
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| GB/T 7714 | Chen, Shangming , Chen, Fengde , Srivastava, Vaibhava et al. Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (4) . |
| MLA | Chen, Shangming et al. "Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 23 . 4 (2024) . |
| APA | Chen, Shangming , Chen, Fengde , Srivastava, Vaibhava , Parshad, Rana D. . Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (4) . |
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In the study of continuous amensalism systems, it has been widely accepted that Michaelis-Menten type harvesting has a significant impact on the survival and extinction of species. However, scholars have not yet studied discrete amenslism models that include Michaelis-Menten type harvesting. To characterize such dynamics, a discrete amensalism system with Michaelis-Menten type harvesting for the second species is investigated. Firstly, we study the existence and stability of all possible equilibrium points. Under different parameters, there are two stable equilibria, which means that the model is not always globally stable. Then, the conditions of various types of bifurcations likely: pitchfork bifurcations, transcritical bifurcations, fold bifurcations, and flip bifurcations have been established. In addition, a global dynamics analysis of the model is also conducted. Finally, the significance of Michaelis-Menten type harvesting in species relationships is shown by numerical simulations. Although proper harvesting reduces the density of the second species, it favors the stable coexistence of both species and excessive harvesting leads directly to the extinction of the second species. Therefore, the results of this paper can provide a reference for research on how to maximize harvesting without destroying the ecological balance of the species.
Keyword ：
Amensalism Amensalism Bifurcation Bifurcation Michaelis-Menten type harvesting Michaelis-Menten type harvesting Stability Stability
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| GB/T 7714 | Li, Qianqian , Chen, Fengde , Chen, Lijuan et al. Dynamical Analysis of a Discrete Amensalism System with Michaelis-Menten Type Harvesting for the Second Species [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (SUPPL 1) . |
| MLA | Li, Qianqian et al. "Dynamical Analysis of a Discrete Amensalism System with Michaelis-Menten Type Harvesting for the Second Species" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 23 . SUPPL 1 (2024) . |
| APA | Li, Qianqian , Chen, Fengde , Chen, Lijuan , Li, Zhong . Dynamical Analysis of a Discrete Amensalism System with Michaelis-Menten Type Harvesting for the Second Species . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2024 , 23 (SUPPL 1) . |
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