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Article
Mathematics
Chuang Xiang et al.
Summary: This paper investigates the complex dynamics and bifurcations in the Holling-Tanner model with a generalist predator and prey refuge. The model exhibits various types of bifurcations and can demonstrate multiple coexistent periodic orbits and homoclinic loops. The constant prey refuge prevents prey extinction and induces global coexistence. A significant finding is that the refuge can induce a stable, large-amplitude limit cycle enclosing one or three positive steady states. Numerical simulations are provided to illustrate and complement the theoretical results.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Zhen Wei et al.
Summary: This paper investigates a Holling type II predator-prey system with prey refuge, Allee effect, fear effect, and time delay. The existence and stability of the system's equilibria are studied. The system undergoes a Hopf bifurcation at the positive equilibrium when the delay exceeds certain critical values. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solution are analyzed using the center manifold theorem and normal form theory. The results show that the fear effect and Allee effect have negative impacts, while the prey refuge has a positive impact. The birth rate plays a significant role in the equilibrium's stability. Numerical simulations are provided to validate the main results.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Mathematical & Computational Biology
Yuanfu Shao
Summary: In this paper, a delayed predator-prey model is established, taking into account the impacts of predator fear, anti-predation response, refuge for prey, additional food supplement for predator, and the delayed fear induced by the predator. The persistence and extinction of species, as well as the existence and uniqueness of a coexistence fixed point, are analyzed. The local asymptotic stability of the equilibrium is investigated using the characteristic equation theory of a variational matrix. By applying the Hopf bifurcation theorem, the bifurcation thresholds of various parameters are obtained, including fear, refuge coefficient, quality and quantity of additional food, and anti-predation delayed response produced by prey. Examples are given to verify the theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal the important role of these parameters in the long-term behaviors of species and emphasize the need for their correct application to preserve the continuous development of species.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2023)
Article
Mathematics, Interdisciplinary Applications
Jiao-Guo Wang et al.
Summary: In this paper, a Beddington-DeAngelis prey-predator model with fear effect, refuge and harvesting is investigated. The positivity of solutions and boundedness of the system are given. The existence and local stability of equilibria are obtained. Different codimension-one bifurcations and Bogdanov-Takens bifurcation of codimension-two occur as predicted by the center manifold theorem and bifurcation theory. Numerical simulations are carried out to confirm the theoretical results.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Physics, Multidisciplinary
Renji Han et al.
Summary: This study investigates a generalist predator-prey model with nonlinear cross-diffusion, prey refuge and ratio-dependent functional response. The refuge parameter plays a key role in the dynamics of the model, influencing uniform persistence, stability of equilibria, Turing instability space and more. The study also compares the stabilizing effects of refuge depending on prey and refuge depending on both species. Numerical simulations reveal the growth of spatiotemporal patterns controlled by prey refuge and self- and cross-diffusion, including spots, stripes, mixtures and more. This proposed model contributes to the field of ecology with its richness and complexity.
EUROPEAN PHYSICAL JOURNAL PLUS
(2022)
Article
Mathematics, Interdisciplinary Applications
Ashraf Adnan Thirthar et al.
Summary: This study explores the shape of interaction between three species and proposes a new predator-prey model. The fear of predators in prey species and the sheltering behavior of predators for super predators are found to play critical roles in system stability.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Computer Science, Interdisciplinary Applications
Udai Kumar et al.
Summary: In this work, the authors investigate the impact of the Allee effect on the dynamics of predator-prey interaction. By incorporating the Allee effect, they modify the predator-prey model to account for the reproductive characteristics of the predator. The study reveals that the Allee effect enhances the stability behavior of the model and leads to additional equilibrium points and various bifurcations.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mechanics
Renji Han et al.
Summary: This research focuses on the combined effect of cooperative hunting and prey refuge in a predator-prey model. The study confirms that the competition between hunting cooperation and prey refuge can determine the dynamics of the system. The findings demonstrate that when the hunting cooperation factor exceeds the prey refuge coefficient, both Hopf and Turing bifurcations occur. Additionally, a distinct Turing instability mechanism emerges when the prey diffusivity exceeds that of the predator.
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
(2022)
Article
Ecology
Koushik Garain et al.
Summary: This study introduces a continuous time predator-prey model with component Allee effect and density dependent mortality of predators. The effects of density independent mortality and Allee effect on the system are highlighted, leading to observations of multiple hydra effects and the phenomenon of bubbling. Bifurcation diagrams and possible bifurcations of the system are presented, showing the possibility of tristability in the system with two stable co-existence fixed points.
ECOLOGICAL COMPLEXITY
(2021)
Article
Mathematics, Applied
Renji Han et al.
Summary: In this study, a predator-prey interacting model with prey refuge in proportion to both the species and Beddington-DeAngelis functional response is proposed and examined. The coefficient of refuge plays a significant role in modifying the system dynamics and mediating the population permanence, stability of coexisting equilibrium, and Turing instability parameter space. Numerical simulations show complex dynamics including prey refugia, self-diffusion controlling spatiotemporal pattern growth, and various spatial patterns like spots, stripes, mixtures, and rings.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Biology
Renji Han et al.
JOURNAL OF BIOLOGICAL SYSTEMS
(2020)
Article
Mathematics, Applied
Shilpa Samaddar et al.
Article
Engineering, Mechanical
K. Durga Prasad et al.
NONLINEAR DYNAMICS
(2019)
Article
Biology
Mickael Teixeira Alves et al.
JOURNAL OF THEORETICAL BIOLOGY
(2017)
Article
Mathematics, Interdisciplinary Applications
Joydev Ghosh et al.
CHAOS SOLITONS & FRACTALS
(2017)
Article
Biology
Xiaoying Wang et al.
JOURNAL OF MATHEMATICAL BIOLOGY
(2016)
Article
Engineering, Multidisciplinary
Lakshmi Narayan Guin et al.
APPLIED MATHEMATICAL MODELLING
(2014)
Article
Engineering, Multidisciplinary
Md. Samsuzzoha et al.
APPLIED MATHEMATICAL MODELLING
(2013)
Article
Biology
Eduardo Liz et al.
JOURNAL OF MATHEMATICAL BIOLOGY
(2012)
Article
Biology
P. D. N. Srinivasu et al.
BULLETIN OF MATHEMATICAL BIOLOGY
(2011)
Article
Biology
Nuno M. Oliveira et al.
BULLETIN OF MATHEMATICAL BIOLOGY
(2010)
Article
Biology
P. D. N. Srinivasu et al.
JOURNAL OF MATHEMATICAL BIOLOGY
(2010)
Review
Biology
Zhihui Ma et al.
MATHEMATICAL BIOSCIENCES
(2009)
Article
Biology
Nakul Chitnis et al.
BULLETIN OF MATHEMATICAL BIOLOGY
(2008)
Article
Ecology
P. D. N. Srinivasu et al.
THEORETICAL POPULATION BIOLOGY
(2007)
