Article
Mathematics, Applied
Yong Wang, Mengping Guo, Weihua Jiang
Summary: This paper studies the Hopf bifurcations and Turing bifurcations of the Gierer-Meinhardt activator-inhibitor model. The interesting and complex spatially periodic solutions and patterns induced by bifurcations are analyzed theoretically and numerically. The conditions for the existence of Hopf bifurcation and Turing bifurcation are established, and the Turing instability region caused by diffusion is obtained. Moreover, the dynamic behaviors near the Turing bifurcation are studied using weakly nonlinear analysis techniques, and the spatial pattern types are predicted by the amplitude equation. Numerical simulations are conducted to verify the results of the analysis.
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2023)
Review
Engineering, Mechanical
Luyao Guo, Xinli Shi, Jinde Cao
Summary: Researchers have investigated Turing patterns of the Gierer-Meinhardt model on complex networks, studying the influences of system parameters, network types, and average degree on pattern formations through numerical simulations. Additionally, an exponential decay of Turing patterns on complex networks was presented, providing a quantitative depiction of the influence of network topology on pattern formations and the possibility of predicting pattern formations.
NONLINEAR DYNAMICS
(2021)
Article
Engineering, Multidisciplinary
Shuangrui Zhao, Hongbin Wang
Summary: This paper explores the coexistence of multi-stable patterns and the superposition of patterns in the classical Gierer-Meinhardt system from the perspective of Turing-Turing bifurcation. The study reveals the existence of semi-stable patterns superimposed by two different spatial resonances and the coexistence of four stable steady states with different characteristic wavelengths. Numerical simulations are consistent with the theoretical analysis. The findings suggest that experimental patterns of vascular mesenchymal cells can be interpreted as the superposition of different spatial modal patterns.
APPLIED MATHEMATICAL MODELLING
(2022)
Article
Mathematics, Interdisciplinary Applications
Rasoul Asheghi
Summary: This paper investigates a reduction of the Gierer-Meinhardt Activator-Inhibitor model and analyzes its global dynamics and stability in both homogeneous and inhomogeneous systems. It shows the conditions and direction of a generalized Hopf bifurcation occurring in the inhomogeneous model.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Mathematics, Interdisciplinary Applications
Rasoul Asheghi
Summary: In this paper, we study the Hopf bifurcation in a Gierer-Meinhardt model with different sources and analyze the impact of diffusion rates on system stability. The normal form of this bifurcation is computed up to the third order, and the direction of Hopf bifurcation is determined using normal form theory. Numerical simulations are also provided to illustrate the analytical results.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics
Biao Liu, Ranchao Wu
Summary: This paper investigates the bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system using the couple map lattice model (CML) method. The linear stability and Turing instability conditions are analyzed, and numerical simulations are conducted.
Article
Mathematics, Interdisciplinary Applications
Ranchao Wu, Lingling Yang
Summary: This paper analyzes the bifurcation of the local Gierer-Meinhardt model and discovers the existence of a degenerate Bogdanov-Takens bifurcation of codimension 3 in the model, which has not been reported in the literature. The paper explores the existence, stability, bifurcation, and the resulting complex and interesting dynamics of equilibria using stability analysis, normal form method, and bifurcation theory. Numerical results are also provided to validate the theoretical findings.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Mathematics, Applied
Theodore Kolokolnikov, Frederic Paquin-Lefebvre, Michael J. Ward
Summary: Spatially localized 1D spike patterns occur in various two-component reaction-diffusion systems in the singular limit of a large diffusivity ratio. A linear instability, known as competition instability, plays a key role in initiating a coarsening process of 1D spike patterns. This instability is subcritical and leads to a symmetry-breaking bifurcation point where an unstable branch of asymmetric equilibria emerges.
Article
Mathematics, Interdisciplinary Applications
Rui Yang, Xiao-Qing Yu
Summary: In this paper, the delayed reaction-diffusion Gierer-Meinhardt system with Neumann boundary condition is studied. Necessary and sufficient conditions for Turing instability, Hopf bifurcation and Turing-Hopf bifurcation are obtained through linear stability analysis and root distribution of the characteristic equation.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Erika Hausenblas, Akash Ashirbad Panda
Summary: This article investigates the behavior of the stochastic reaction-diffusion Gierer-Meinhardt system in one or two-dimensional space. By introducing perturbations with an infinite-dimensional Wiener process, the existence and pathwise uniqueness of solutions are studied.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
G. K. Duong, N. I. Kavallaris, H. Zaag
Summary: This paper thoroughly investigates the blowing up behavior induced through diffusion of the solution to a non-local problem, providing constructions of solutions that blow up in finite time at interior points under certain conditions. The final asymptotic profile at the blowup point and the form of Turing patterns occurring in that case are also described. The technique used for constructing the blowing up solutions mainly relies on previously developed approaches.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Yuta Ishii
Summary: This paper investigates the Gierer-Meinhardt model with heterogeneity in both activator and inhibitor on a Y-shaped compact metric graph. A one-peak stationary solution, concentrated at a suitable point, is constructed using the Lyapunov-Schmidt reduction method. It is shown that the location of concentration point is determined by the interaction of the activator's heterogeneity function with the geometry of the domain represented by the associated Green's function. Additionally, the precise location of concentration point for the non-heterogeneity case is determined, and the effect of heterogeneity is demonstrated using a concrete example.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics
G. K. Duong, T. E. Ghoul, N. I. Kavallaris, H. Zaag
Summary: In this paper, we consider a nonlocal parabolic PDE and study the construction and asymptotic behavior of the blow-up solution in the critical parameter regime. Using a formal and rigorous approach, we find an approximate solution, linearize the equation, and reduce the problem to a finite-dimensional one. By applying index theory, we solve the finite-dimensional problem and obtain the exact solution to the full problem.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Mengxin Chen, Ranchao Wu, Yancong Xu
Summary: The paper introduces and investigates a depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and homogeneous Neumann boundary conditions. It provides theoretical analysis and numerical simulations to explore stability, existence, and bifurcation properties in the model. The results confirm the theoretical findings and provide additional insights through numerical simulations.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics, Interdisciplinary Applications
Haicheng Liu, Bin Ge
Summary: In this paper, the Gierer-Meinhardt model with cross-diffusion is established, and the Turing instability of its periodic solutions is studied. The conditions of Turing instability are derived and verified through theoretical analysis and numerical simulations, showing that the Turing instability of periodic solutions is induced by cross-diffusion.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Chemistry, Multidisciplinary
Mengxin Chen, Tian Wang
Summary: In this paper, we investigate the dynamical behaviors of a generalized Lengyel-Epstein model with zero-flux boundary conditions. Through theoretical analysis and numerical experiments, we obtain conclusions about the attraction region, stability, and steady states of the equation. The results suggest that the diffusion rates of the substance play a significant role in the dynamical behaviors of the model.
JOURNAL OF MATHEMATICAL CHEMISTRY
(2023)
Article
Mathematics, Applied
Mengxin Chen, Ranchao Wu
Summary: In this paper, a diffusive predator-prey system with network connection and harvesting policy is analyzed. The stability and dynamical behaviors of the system are explored, showing that the networked system exhibits distinct characteristics compared to the model without network structure.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Mengxin Chen, Ranchao Wu, Xiaohui Wang
Summary: This paper investigates the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. The existence and non-existence of steady states, temporal bifurcation, and the boundedness of global positive solutions are explored. Results on the upper and lower bounds of positive solutions for the associated elliptic system are provided. The impact of the ratio-dependent Holling III functional response and strong Allee effect on the dynamical behaviors of the species interaction systems is illustrated through numerical simulations.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Physics, Multidisciplinary
Mengxin Chen, Qianqian Zheng
Summary: In this paper, the authors investigate the pattern dynamics of a population model with chemotaxis. They study the existence of solutions and the global stability of the coexistence equilibrium. The authors also explore the steady state bifurcation induced by chemotaxis in the spatiotemporal model, with multiple thresholds depending on different assumptions. They provide an amplitude equation to determine the direction of the steady state bifurcation and validate the theoretical analysis with numerical results.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Ranchao Wu
Summary: In this study, the existence of steady states, bifurcations, and spatiotemporal patterns are investigated for a diffusive predator-prey model. The nonexistence and existence of nonconstant steady states are justified using priori estimates, Poincare inequalities, and Leray-Schauder degree, respectively. The weakly nonlinear analysis is employed to establish the amplitude equations and various complex pattern solutions are identified from these equations.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Hari Mohan Srivastava
Summary: In this paper, we investigate the role of prey-taxis in an ecological model. The local stability of the positive equilibrium and the occurrence conditions of the steady state bifurcation are given. By treating the prey-taxis constant e as the bifurcation parameter, we confirm the model possesses the steady state bifurcation at e =ekS for k & ISIN; N0/{0}. Numerical experiments show the stable bifurcating solution.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Ranchao Wu
Summary: This study introduces prey-taxis, which describes the directed movement of predator species, into the Previte-Hoffman model and investigates steady-state bifurcation with no-flux boundary conditions and prey-taxis. The stability analysis of the positive equilibrium, the existence of Hopf bifurcation, and steady-state bifurcation are presented. The Crandall-Rabinowitz local bifurcation theory is employed to determine the existence and stability of nonconstant steady-state bifurcation. Results show that only repulsive prey-taxis can induce steady-state bifurcation in the Previte-Hoffman model, leading to the occurrence of spatiotemporal patterns demonstrated through numerical simulations.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Mathematics, Applied
Mengxin Chen, Ranchao Wu, Biao Liu, Liping Chen
Summary: This paper investigates a general reaction-diffusion Brusselator model under homogeneous Neumann boundary conditions. The stability of the unique positive equilibrium is studied, and the existence of the Hopf bifurcation is identified. Occurrence conditions of the Turing instability, Turing-Turing, and Turing-Hopf bifurcations are established. The spatiotemporal solutions resulting from the bifurcation are explored, and the validity of the theoretical results is verified through numerical simulations.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2023)
Article
Chemistry, Physical
Mengxin Chen, Ranchao Wu
Summary: In this paper, the stability and bifurcations of the nutrient-microorganism model with chemotaxis under no-flux boundary conditions are analyzed. It is found that the presence of chemotaxis can lead to steady state bifurcation, namely Hopf-Turing bifurcation, in the model. The model exhibits induced spatially homogeneous periodic solution, non-constant steady state, and spatially inhomogeneous periodic solution. These results suggest that the inclusion of chemotaxis in the model can give rise to diverse spatiotemporal dynamical behaviors.
ZEITSCHRIFT FUR NATURFORSCHUNG SECTION A-A JOURNAL OF PHYSICAL SCIENCES
(2023)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Seokjun Ham, Yongho Choi, Hyundong Kim, Junseok Kim
Summary: This paper investigates the pattern dynamics of a harvested predator-prey model with no-flux boundary conditions. The positive equilibrium types and the direction of Hopf bifurcation are analyzed for the local temporal model. The conditions for the existence of Turing instability and the selection of different patterns are presented using amplitude equations with the assistance of weakly nonlinear analysis. Spot patterns and mixed patterns are displayed in 2D space, on spherical and torus surfaces, demonstrating the strong influence of prey population diffusion rate on pattern structures.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mengxin Chen, Qianqian Zheng
Summary: In this paper, the pattern dynamics of the predator-prey model with interval biological coefficients and no-flux boundary conditions was studied. Boundedness of the solutions was shown by comparison principle and constructing an invariant rectangle domain with different interval variable values. The stable and unstable intervals of the positive equilibrium were discussed by treating the diffusion coefficient of the predator as the critical parameter. Amplitude equations around the threshold of the Turing instability were deduced using weakly nonlinear analysis method to classify the existence and stability of various pattern solutions. Non-symmetrical and symmetrical spatial patterns were displayed in 2D space through numerical simulation results with different interval biological coefficients.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Engineering, Mechanical
Mengxin Chen
Summary: This paper investigates the spatiotemporal inhomogeneous pattern phenomenon of a predator-prey model with chemotaxis and time delay. The study provides sufficient conditions to ensure the existence of Turing instability by adjusting the control parameters of time delay and chemotaxis. It also determines the occurrence conditions of Turing-Hopf bifurcation using time delay control parameter and chemotaxis sensitivity coefficient. The research shows that both time delay and chemotaxis can affect the formation of spatiotemporal inhomogeneous patterns.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Mengxin Chen, Hari Mohan Srivastava
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematical & Computational Biology
Mengxin Chen, Xuezhi Li, Ranchao Wu
Summary: In this paper, the predator-prey model with strong Allee and fear effects is analyzed. The paper establishes the existence and stability of the equilibria and explores the degenerate point and different types of bifurcation. The nonexistence and existence of nonconstant steady states are presented using energy estimates and the Leray-Schauder degree.
INTERNATIONAL JOURNAL OF BIOMATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Mengxin Chen, Zhenyong Hu, Qianqian Zheng, Hari Mohan Srivastava
Summary: This article investigates an SI model with saturated treatment, non-monotonic incidence rate, logistic growth, and homogeneous Neumann boundary conditions. The global existence and uniform boundedness of the parabolic system are analyzed. The global stability of the disease-free and endemic equilibria are studied separately. Additionally, a priori estimates and propositions about nonconstant steady states for the elliptic system are provided. Furthermore, it is discovered that the diffusion rates of susceptible and infected populations can affect the nonexistence of nonconstant steady states. An interesting finding is that the absence of disease-free equilibrium and basic reproduction number occurs when the intrinsic growth rate of susceptible individuals is lower than the rate of vaccination.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
