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
Mathematics, Applied
Shanbing Li, Jianhua Wu
Summary: This article focuses on the stationary problem of a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, regulated by a Beddington-DeAngelis functional response. The global bifurcation structure of coexistence states is thoroughly described, and the parameter ranges for the existence of coexistence states are determined. Additionally, sufficient conditions for the nonexistence of coexistence states are established. The presence of prey-taxis/predator-taxis and the Beddington-DeAngelis functional response pose challenges to mathematical analysis, resulting in distinct phenomena.
ADVANCED NONLINEAR STUDIES
(2023)
Article
Mathematics
Julian Lopez-Gomez, Eduardo Munoz-Hernandez, Fabio Zanolin
Summary: This article discusses the existence, multiplicity, minimal complexity, and global structure of subharmonic solutions to planar Hamiltonian systems with periodic coefficients, using the classical predator-prey model as a key example. The paper employs a topological approach to determine their nature, multiplicity, minimal complexity, and global minimal structure based on the configuration of the coefficients. Additionally, the article introduces a dynamical system approach that allows for the detection of chaotic-type solutions in addition to subharmonic solutions.
Article
Mathematics, Applied
G. Tigan, C. Lazureanu, F. Munteanu, C. Sterbeti, A. Florea
Summary: This work focuses on a two-dimensional Kolmogorov system with two independent parameters, studying its local analytical properties near the origin and describing its behavior through bifurcation diagrams. Applications of such systems, particularly in modeling population dynamics in biology and ecology, are highlighted.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Ruizhi Yang, Fatao Wang, Dan Jin
Summary: In this study, the nonlocal competition in prey, additional food in predator, and time delay are incorporated into a predator-prey model. The local stability of the coexisting equilibrium is analyzed by studying the eigenvalue spectrum. The investigation also focuses on the time delay inducing Hopf bifurcation. The results show that nonlocal competition together with time delay can induce spatially inhomogeneous bifurcating periodic solutions in the diffusive predator-prey model.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Shilpa Garai, N. C. Pati, Nikhil Pal, G. C. Layek
Summary: We report the existence of periodic and shrimp-shaped structures in the bi-parameter space of a predator-prey model. Our analysis of stability behaviors, bifurcations, and Lyapunov exponent shows complex dynamical behaviors and the emergence of a new type of periodic structure. Additionally, we observe the coexistence of three heterogeneous attractors and the presence of basin boundaries, indicating the unpredictability of the model. Our findings highlight the dependence of predator-prey oscillations on initial densities in certain parameter regions.
CHAOS SOLITONS & FRACTALS
(2022)
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
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
Binhao Hong, Chunrui Zhang
Summary: In this paper, the dynamical behavior of a predator-prey model with discrete time is explored through theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed, with Flip bifurcation and Hopf bifurcation occurring at the unique positive equilibrium point. Chaotic cases are observed at some corresponding internal equilibria when small perturbations are applied to the bifurcation parameter. Numerical simulations using maximum Lyapunov exponent and phase diagrams reveal a complex dynamical behavior.
Article
Mathematics, Applied
Haokun Qi, Xinzhu Meng, Tasawar Hayat, Aatef Hobiny
Summary: This paper proposes a reaction-diffusion predator-prey model with fear effect under a predator-poisoned environment and analyzes its stability and bifurcation behavior. The study finds that the proper diffusion rate is beneficial for the survival of populations and changes in diffusion rates can cause steady state bifurcations. The validity of the theoretical analysis is verified through numerical simulations.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Seralan Vinoth, R. Vadivel, Nien-Tsu Hu, Chin-Sheng Chen, Nallappan Gunasekaran
Summary: This study investigates the impact of fear on prey populations and prey refuges in a predator-harvested Leslie-Gower model. The research focuses on analyzing the number and stability properties of all positive equilibria and uses numerical simulation to evaluate the stability. Additionally, sensitivity investigations are performed on model solutions in relation to fear impact, prey refuges, and harvesting.
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
Engineering, Mechanical
Ruizhi Yang, Chenxuan Nie, Dan Jin
Summary: This paper investigates a delayed diffusive predator-prey model with nonlocal competition and habitat complexity. The local stability of coexisting equilibrium is studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is explored using time delay as a bifurcation parameter. Conditions for determining the bifurcation direction and stability of the bifurcating periodic solution are derived using the normal form method and center manifold theorem. The results suggest that only the combination of nonlocal competition and diffusion can induce stably spatial inhomogeneous bifurcating periodic solutions.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
N. Mukherjee, V. Volpert
Summary: This study explores a prey-predator model with sexual reproduction in prey population and nonlocal consumption of resources, focusing on Turing patterns induced by nonlocal interaction in two spatial dimensions. Analytical derivation of Turing bifurcation conditions for the nonlocal model is conducted, studying the bifurcation scenario of stationary hotspot pattern generated from the homogeneous steady-state. The study also investigates the transformation of periodic and aperiodic solutions exhibited by the local model into stationary Turing patterns due to the effects of nonlocal interaction terms.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Soufiane Bentout, Salih Djilali, Abdon Atangana
Summary: In this study, an age-structured prey-predator model with infection was proposed to examine the effect of predator maturation age on the interaction between predator and prey, as well as the spread of infectious disease. It was found that the minimal maturation duration can impact the behavior of the solution, potentially leading to periodic solutions generated by Hopf bifurcation for three different equilibrium states. The mathematical results were numerically validated using graphical illustrations.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
J. Lopez-Gomez, P. H. Rabinowitz
JOURNAL OF DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics
Julian Lopez-Gomez, Pierpaolo Omari
JOURNAL OF DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
D. Aleja, I Anton, J. Lopez-Gomez
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2020)
Article
Mathematics, Applied
Julian Lopez-Gomez, Pierpaolo Omari
ADVANCED NONLINEAR STUDIES
(2020)
Biographical-Item
Mathematics, Applied
Julian Lopez-Gomez, Patrizia Pucci
ADVANCED NONLINEAR STUDIES
(2020)
Article
Mathematics, Applied
D. Aleja, I. Anton, J. Lopez-Gomez
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
M. Fencl, J. Lopez-Gomez
Summary: This paper analyzes the structure of nodal solutions of a class of one-dimensional superlinear indefinite boundary value problems with indefinite weight functions, finding that high-order eigenvalues may not be concave, leading to bifurcations of nodal solutions from multiple points. The combination of analytical and numerical tools demonstrates how mathematical analysis aids numerical study and vice versa.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics, Applied
Julian Lopez-Gomez, Eduardo Munoz-Hernandez, Fabio Zanolin
Summary: This paper investigates the existence and multiplicity of periodic solutions to a planar Hamiltonian system under degenerate conditions, showing that depending on certain geometric configurations, a large number of periodic solutions can be guaranteed. The proof is based on the Poincare-Birkhoff twist theorem and applications are made to Volterra's predator-prey model with seasonal effects.
ADVANCED NONLINEAR STUDIES
(2021)
Article
Mathematics, Applied
Alberto Boscaggin, Walter Dambrosio, Eduardo Munoz-Hernandez
Summary: This article provides a Maupertuis-type principle for a system of ODEs related to special relativity, and proves the existence of multiple periodic solutions with prescribed energy for a relativistic N -centre type problem in the plane.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Alberto Boscaggin, Eduardo Munoz-Hernandez
Summary: This paper examines planar Hamiltonian systems and proves that the above nonlinear system has subharmonic solutions of any order k large enough, by revisiting the index theory and using the Poincare-Birkhoff fixed point theorem. The existence of subharmonic solutions is determined by the difference in rotation numbers of the linearizations of the system at zero and at infinity. Applications to planar Hamiltonian systems arising from second order scalar ODEs are also discussed.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
Julian Lopez-Gomez, Eduardo Munoz-Hernandez, Fabio Zanolin
Summary: This paper studies the global structure of nodal solutions of a generalized Sturm-Liouville boundary value problem associated with a quasilinear equation. It is the first study to address the general case when lambda is a real number and a > 0. The semilinear case with a < 0 has been recently treated by López-Gómez and Rabinowitz.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Julian Lopez-Gomez, Eduardo Munoz-Hernandez
Summary: This paper introduces a spatially heterogeneous diffusive predator-prey model that unifies classical Lotka-Volterra and Holling-Tanner models through a spatially heterogeneous prey saturation coefficient. By studying general mixed boundary conditions of non-classical type, the paper employs recent technical devices to obtain some of its main results.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
Article
Mathematics, Applied
Julian Lopez-Gomez, Eduardo Munoz-Hernandez, Fabio Zanolin
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2020)
