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
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, Applied
Yan Li, Zhiyi Lv, Xiuzhen Fan
Summary: This paper focuses on a diffusive predator-prey model with prey-taxis and prey-stage structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator-prey model is determined. Hopf bifurcation and steady-state bifurcation are also investigated.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Interdisciplinary Applications
Lei Kong, Fengjiao Lu
Summary: The influence of indirect prey-taxis on the dynamics of a predator-prey system with predator functional response is studied. The study analyzes the stability and bifurcations of the system, deriving critical values of the indirect prey-taxis coefficient. The research finds that attractive indirect prey-taxis can destabilize the system and induce the emergence of spatially inhomogeneous periodic solutions. The secretion level of chemoattractant by the prey plays a role in determining the likelihood of spatial patterns.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Qingyan Shi, Yongli Song
Summary: This paper studies the effect of time delay on the dynamics of a diffusive predator-prey model with predator-taxis under Neumann boundary condition. It is found that the joint effect of predator-taxis and delay can lead to spatially nonhomogeneous periodic patterns via spatially nonhomogeneous Hopf bifurcations. Moreover, double Hopf bifurcations are observed due to the interaction either between homogeneous and nonhomogeneous Hopf bifurcations or between nonhomogeneous Hopf bifurcations with different modes, which cannot occur when considering only delay or predator-taxis diffusion in the system.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Automation & Control Systems
Zhichao Jiang, Yan Zhao, Xueli Bai, Zexian Zhang
Summary: This paper introduces a delayed feedback controller with a delay-dependent coefficient into a multiple delay phytoplankton-zooplankton system. By choosing delays as bifurcation parameters, it is proved that the onset of Hopf bifurcation can be delayed and the stability domain can be extended under this control mechanism. The influence of the decay rate on the system dynamics cannot be ignored, and numerical simulations verify the effectiveness of the delayed feedback controller in bifurcation control.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2021)
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, 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
Sainan Wu
Summary: This paper considers a reaction-diffusion predator-prey model with indirect prey-taxis and predator-taxis. The model obtains globally bounded solutions under different parameters and conditions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
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
Inkyung Ahn, Changwook Yoon
Summary: This paper analyzes the prey-predator models with indirect predator-taxis, where the chemical secreted by the predator triggers the repellent behavior of prey. The global existence and uniform boundedness of classical solutions up to two spatial dimensions are proved under the assumption of quadratic decay of predator. The linear stability analysis shows that large chemosensitivity gives rise to pattern formations, and global stability results for the nontrivial constant steady states are obtained through proper Lyapunov functionals.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2021)
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
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
Mathematical & Computational Biology
Yan Li, Zhiyi Lv, Fengrong Zhang, Hui Hao
Summary: In this paper, a diffusive predator-prey model with hyperbolic mortality and prey-taxis under homogeneous Neumann boundary condition is studied. The influence of prey-taxis on the local stability of constant equilibria is analyzed. Prey-taxis is found to affect the stability of the unique positive constant equilibrium, but has no influence on the stability of the trivial equilibrium and the semi-trivial equilibrium. Hopf bifurcation and steady state bifurcation related to prey-taxis are then derived, indicating the important role of prey-taxis in the dynamics.
INTERNATIONAL JOURNAL OF BIOMATHEMATICS
(2024)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Ranchao Wu, Hongxia Liu, Xiaoxue Fu
Summary: This paper investigates the Leslie-Gower type predator-prey system with the ratio-dependent Holling III functional response and Neumann boundary conditions. The existence of the codimension-two Turing-Hopf point is identified, and amplitude equations are derived using weakly nonlinear analysis to explore the spatiotemporal dynamics near the C2THP. The temporal patterns, hexagonal patterns, and plane wave patterns can be presented through amplitude equations, along with the sufficient conditions of their existence and stability.
CHAOS SOLITONS & FRACTALS
(2021)
