Article
Mathematics, Interdisciplinary Applications
Fatao Wang, Ruizhi Yang
Summary: In this paper, we investigate a cross-diffusion predator-prey system with Holling type functional response. We analyze the local stability, Turing instability, spatial pattern formation, Hopf and Turing-Hopf bifurcation of the equilibrium. Numerical simulation reveals that the system experiences cross-diffusion-driven instability and exhibits various patterns such as spots, stripe-spot mixtures, and labyrinthine patterns. The study also shows that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are crucial factors for the stability of the predator-prey system.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics
Ruizhi Yang, Qiannan Song, Yong An
Summary: This paper considers a diffusive predator-prey system with a functional response that increases in both predator and prey densities. The Turing instability and Hopf bifurcation are studied by analyzing the characteristic roots of the system. By calculating the normal form of the Turing-Hopf bifurcation and conducting numerical simulations, the dynamic properties of different types of solutions in each parameter region of the phase diagram are found to be extremely rich.
Article
Physics, Multidisciplinary
Youwei Yang, Daiyong Wu, Chuansheng Shen, Fengping Lu
Summary: Nonlocal competition and Allee effect are studied in a predator-prey system, where the prey faces nonlocal competition and the predator is subject to Allee effect. The effects of predation on the spatial distribution of prey are investigated. The conditions for stable coexistence equilibrium, spatially inhomogeneous Hopf bifurcation, and Turing bifurcation are studied. Numerical simulations are carried out to illustrate the theoretical results, showing that nonlocal prey competition can destabilize the coexistence equilibrium point and drive spatially inhomogeneous bifurcations. The results also indicate that a larger habitat domain requires a larger prey diffusion coefficient for coexistence in the spatially homogeneous form.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2023)
Article
Engineering, Mechanical
Qing Hu, Jianwei Shen
Summary: This paper investigates the pattern dynamics of a prey-predator network system with diffusion and delay. The effect of delay and diffusion on the network system is obtained by linear stability analysis, including stability, Hopf bifurcation, and Turing pattern. The numerical simulation verifies the results.
NONLINEAR DYNAMICS
(2022)
Article
Engineering, Mechanical
Xuebing Zhang, Qi An, Ling Wang
Summary: In this study, a delayed diffusive predator-prey model with fear effect is considered due to the delay in the impact of fear on the growth rate of prey. The existence of equilibria, occurrence of Turing, Hopf and Turing-Hopf bifurcation, and global asymptotic stability of the positive equilibrium are analyzed, with various spatiotemporal patterns induced by delay confirmed through numerical simulations.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics
Yining Xie, Jing Zhao, Ruizhi Yang
Summary: This paper proposes a diffusive predator-prey model with a strong Allee effect and nonlocal competition in the prey and a fear effect and gestation delay in the predator. The study mainly focuses on the local stability of the coexisting equilibrium and the existence and properties of Hopf bifurcation. Bifurcation diagrams with the fear effect parameter (s) and the Allee effect parameter (a) are provided, showing that the stable region of the coexisting equilibrium increases (or decreases) with an increase in the fear effect parameter (s) (or the Allee effect parameter (a)). The results demonstrate that the fear effect parameter (s), the Allee effect parameter (a), and gestation delay (t) can be utilized to control the growth of prey and predator populations.
Article
Mathematics
Liangying Miao, Zhiqian He
Summary: In this article, the authors investigate the Hopf bifurcation and Turing instability of a predator-prey model with hunting cooperation. The study analyzes the stability of the equilibrium and determines the conditions for the direction and stability of the bifurcating periodic solution. The results demonstrate the significant role of hunting cooperation in the model's dynamics, leading to beneficial effects on the predator population and increased Turing instability. Numerical simulations are utilized to visualize the complex dynamic behavior.
Article
Mathematics, Interdisciplinary Applications
Zhimin Bi, Shutang Liu, Miao Ouyang
Summary: This paper investigates the spatial dynamics of a class of spatial fractional predator-prey systems with time delay and Allee effect. The conditions for Hopf bifurcation and Turing bifurcation are obtained, and the abundant dynamic behaviors of the system are demonstrated through numerical simulation. The numerical results show that time delay, Allee effect, and fractional diffusion can affect the formation of 3D Turing patterns and the constitution of 3D spiral wave patterns.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics
Yujia Xiang, Yuqi Jiao, Xin Wang, Ruizhi Yang
Summary: In this paper, a delayed diffusive predator-prey model incorporating the Allee effect, nonlocal competition in prey, and hunting cooperation in predators is proposed. The local stability of coexisting equilibrium and the occurrence of Hopf bifurcation are studied by analyzing the eigenvalue spectrum. The properties of Hopf bifurcation are further investigated using the center manifold theorem and normal form method. The numerical simulations verify the analysis results and demonstrate the influence of various parameters on the model, including the Allee effect parameter, hunting cooperation parameter, and time delay.
ELECTRONIC RESEARCH ARCHIVE
(2023)
Article
Mathematics, Applied
Yudan Ma, Ming Zhao, Yunfei Du
Summary: In this study, a predator-prey model with strong Allee effect and Holling type II functional response is proposed and investigated. Through dynamical analysis, the existence of equilibria and bifurcations of the system are derived. It is found that the strong Allee effect plays a crucial role in the dynamics of the system.
Article
Mathematics, Interdisciplinary Applications
Xuan Tian, Shangjiang Guo
Summary: The study explores a diffusive predator-prey model with Allee effect and constant stocking rate, finding that the Allee effect is the driving factor for the formation of Turing patterns. Turing patterns emerge only when the prey's diffusion rate is faster than that of the predator, contrasting with the classical predator-prey system. The research also delves into Hopf and steady-state bifurcations using Lyapunov-Schmidt reduction, shedding light on the mechanisms behind generating spatiotemporal patterns.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Mathematics, Applied
Kimun Ryu, Wonlyul Ko
Summary: This study provides a qualitative analysis of a diffusive predator-prey system with a hunting-cooperation functional response under homogeneous Neumann boundary conditions. The study investigates the global attractor for nonnegative time-dependent solutions, the nonpersistence leading to predator extinction, the local stability at all feasible nonnegative constant steady states, the occurrence of Hopf bifurcation, the existence and nonexistence of nonconstant positive steady states, and the limiting behavior of positive steady states according to diffusion rate. Interesting results, such as bistability, predator extinction induced by a large consumption rate, and stationary patterns in the system with hunting cooperation in predators, are observed.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics
Ruizhi Yang, Xiao Zhao, Yong An
Summary: In this study, a delayed predator-prey model with diffusion and anti-predator behavior is investigated. The stability of the positive equilibrium is analyzed, and the existence of Hopf bifurcation is discussed based on the Hopf bifurcation theory. The properties of Hopf bifurcation are derived using the theory of center manifold and normal form method. Finally, the impact of time delay on the model is examined through numerical simulations.
Article
Mathematics, Applied
Qiannan Song, Ruizhi Yang, Chunrui Zhang, Lei Wang
Summary: This paper investigates the Turing instability and Hopf bifurcation of a diffusive predator-prey model with Beddington-DeAngelis functional response. Bifurcation parameters m, d(2) are used to study the Turing-Hopf bifurcation, and the normal form for this bifurcation is computed. Complex spatiotemporal dynamics near the Turing-Hopf bifurcation point are identified, and numerical simulations are provided to illustrate the theoretical results.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Heping Jiang
Summary: In this paper, complex dynamical behaviors of a diffusive Leslie-Gower predator-prey model with a ratio-dependent Holling type III functional response and nonlinear prey harvesting under homogeneous Neumann boundary conditions are studied. The existence and stability of extinction and coexistence equilibrium states are determined by analyzing the distribution of eigenvalues, and the bifurcations of the system are investigated. Additionally, Turing-Hopf bifurcation points induced by harvesting rate and delay are derived based on theoretical analysis and numerical simulation. Our results indicate that delay and nonlinear prey harvesting rates can generate spatially inhomogeneous periodic solutions.
Article
Mathematics, Applied
Daiyong Wu, Hongyong Zhao
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
(2017)
Article
Mathematics, Applied
Daiyong Wu, Hongyong Zhao, Yuzhen Bai
APPLIED MATHEMATICS LETTERS
(2018)
Article
Mathematics, Applied
Daiyong Wu, Hongyong Zhao
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2018)
Article
Mathematics, Applied
Daiyong Wu, Min Zhao, Hai Zhang
ADVANCES IN DIFFERENCE EQUATIONS
(2018)
Article
Mathematics, Applied
Daiyong Wu, Hongyong Zhao, Yuan Yuan
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2019)
Article
Mathematics, Applied
Daiyong Wu
ADVANCES IN DIFFERENCE EQUATIONS
(2013)
Article
Physics, Multidisciplinary
Daiyong Wu, Min Zhao
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2019)
Article
Mathematics, Applied
Yuxin Dong, Daiyong Wu, Chuansheng Shen, Luhong Ye
Summary: Experiments show that the fear of predators reduces the birth rate of the prey population, but it does not lead to their extinction. The sensitivity of the prey to predators affects the population density of both prey and predators. The introduction of saturated fear cost and predator-taxis sensitivity into the predator-prey interactions model proves to be feasible and provides insights into the stability of the model.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics, Applied
Youwei Yang, Daiyong Wu, Chuansheng Shen, Jian Gao, Fengping Lu
Summary: The study introduces a nonlocal prey competition model to investigate the relationship between prey competition and spatial position. The research reveals that the critical delay threshold of prey competition increases with the fear level or intra-prey competition coefficient. Nonlocal prey competition can induce Hopf bifurcation for spatially inhomogeneous form and generate periodic solutions. Furthermore, the nonlocal effect can increase the risk of extinction for the species.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Interdisciplinary Applications
Daiyong Wu, Fengping Lu, Chuansheng Shen, Jian Gao
Summary: Recently, spatial memory and Allee effect have been investigated independently in population models. This paper combines these two aspects in a predator-prey system and examines their interaction. The Allee effect leads to bistability, with the predator-free steady-state always locally stable. Prey-taxis plays a stabilizing role in positive constant steady-state, while spatial memory delay generates inhomogeneous Hopf bifurcation and stability switching. In the absence of spatial memory delay, a stronger Allee effect on the predator requires a larger prey-taxis coefficient to maintain stable coexistence in a homogeneous spatial form. With the same prey-taxis coefficient, the critical threshold of spatial memory delay for a large predator diffusion coefficient is significantly greater than that for a small predator diffusion coefficient. Moreover, the amplitudes of spatial patterns oscillate as spatial memory delay varies.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Computer Science, Interdisciplinary Applications
Daiyong Wu, Youwei Yang, Peng Wu
Summary: This paper investigates the spatiotemporal predator-prey system with prey-taxis and nonconstant mortality. The global stability of spatially homogeneous steady states is obtained by constructing a Lyapunov functional. The minimum mortality of predators determines their survival. In addition, the parameter ranges for the stability or instability of spatially homogeneous steady states are identified, and steady state bifurcation is discussed by choosing the prey-taxis coefficient. It is found that diffusion-induced instability can occur under nonconstant mortality, and prey-taxis can eliminate the spatial patterns induced by diffusion. Numerical simulations show that prey-taxis and nonconstant mortality can lead to various pattern formations.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Physics, Multidisciplinary
Youwei Yang, Daiyong Wu, Chuansheng Shen, Fengping Lu
Summary: Nonlocal competition and Allee effect are studied in a predator-prey system, where the prey faces nonlocal competition and the predator is subject to Allee effect. The effects of predation on the spatial distribution of prey are investigated. The conditions for stable coexistence equilibrium, spatially inhomogeneous Hopf bifurcation, and Turing bifurcation are studied. Numerical simulations are carried out to illustrate the theoretical results, showing that nonlocal prey competition can destabilize the coexistence equilibrium point and drive spatially inhomogeneous bifurcations. The results also indicate that a larger habitat domain requires a larger prey diffusion coefficient for coexistence in the spatially homogeneous form.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2023)
