Article
Mathematics, Interdisciplinary Applications
Malay Banerjee, Swadesh Pal, Pranali Roy Chowdhury
Summary: This paper investigates the spatio-temporal pattern formation in a complex habitat. The results show that the shape and size of the habitat play a significant role in determining the spatio-temporal patterns.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Interdisciplinary Applications
Hantaek Bae
Summary: This paper investigates two models for pattern formation in active systems on a d-dimensional torus. The first model focuses on the density and tubule orientation field, while the second model considers the Active model C. Unique global-in-time solutions are proven for both models when the initial data is sufficiently small.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Yuan Chen, Keith Promislow
Summary: Freezing of salt water leads to separation into ice and brine. The salt ejection from the ice phase resembles chemotaxis. We analyze the gradient flow of a regularized free energy for brine inclusions on a periodic domain in R-d(d = 2,3) and prove uniqueness and global existence of classical solutions from initial data in an energy space for positive time.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2023)
Article
Mathematics, Applied
Yina Lin, Qian Zhang, Meng Zhou
Summary: In this paper, we investigate the incompressible chemotaxis-Navier-Stokes equations with a logistic source in two-dimensional space. We firstly establish a blow-up criterion and then prove the global existence of classical solutions to the system for the Cauchy problem under certain rough conditions on the initial data.
Article
Mathematics, Applied
Yafeng Li, Chunlai Mu, Qiao Xin
Summary: In this paper, we discuss a hyperbolic-parabolic system on a network. The global existence of solution to this problem with suitable the transmission conditions at interior is obtained by energy estimates. Moreover, for the case of acyclic network, we prove the existence and uniqueness of stationary solution to the system and show that the stationary solution provides asymptotic profiles for a class of global solutions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Hantaek Bae, Kyungkeun Kang
Summary: This paper considers the coupling of the Keller-Segel model and the incompressible fluid equations to describe the dynamics of swimming bacteria. The existence of local-in-time solutions is proven for large data in scaling invariant Besov spaces. The paper also shows the existence of global-in-time solutions when smallness conditions are imposed to initial data. The authors further derive temporal decay rates of the bacteria density and the fluid velocity by changing the fluid part to the Stokes equations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Yimamu Maimaiti, Wang Zhang, Ahmadjan Muhammadhaji
Summary: This paper explores a predator-prey model that incorporates prey-taxis and a general functional response in a bounded domain. The stability, pattern formation, and global bifurcation of the model are examined. It is found that the inclusion of nonlocal terms enhances linear stability and can generate patterns due to prey-taxis. When the prey-tactic sensitivity is repulsive, a branch of nonconstant solutions emerges from the positive constant solution.
Article
Mathematics, Applied
Haixia Li, Jie Jiang
Summary: This paper investigates the global existence of weak solutions to a degenerate kinetic model of chemotaxis. By modifying the comparison method and introducing a suitable approximation scheme, the authors establish the global existence of solutions with higher regularity compared to previous literature.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Kentaro Fujie, Takasi Senba
Summary: This paper investigates the fully parabolic chemotaxis system of local sensing in higher dimensions. It proves the absence of finite-time blow-up phenomenon in this system even in the supercritical case. For any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. In the case of exponential decaying motility, it is established that solutions may blow up at infinite time for any magnitude of mass.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
Renkun Shi, Guoqiao You
Summary: This paper focuses on the attraction-repulsion chemotaxis system and establishes the global existence of classical solutions under different parameter conditions using a modified free energy functional.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Mengyao Ding, Wenbin Lyu
Summary: This paper studies the global solvability of a two-dimensional chemotaxis-fluid model, and shows that persistent Dirac-type singularities can be ruled out without imposing any critical superlinear exponent restriction on the logistic source function.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics, Applied
Xiaoyu Chen, Jijie Zhao, Qian Zhang
Summary: This paper considers the Cauchy problem for the three-dimensional axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion Delta n(m). By utilizing the structure of axisymmetric flow without swirl, the author shows the global existence of weak solutions for the chemotaxis-Navier-Stokes equations with m = 5/3.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Xin Xu
Summary: Chemotaxis is an important biological mechanism that can be studied by analyzing spiky steady states, which can be used to model phenomena like cell aggregation.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Engineering, Mechanical
Jianping Gao, Shangjiang Guo, Li Ma
Summary: This paper investigates a nutrient-microorganism system with nutrient-taxis in the sediment. The study establishes the global existence and uniform-in-time bound of classical solutions in a domain with any dimension. It also analyzes the linearized stability of the constant interior steady state, showing that different types of nutrient-taxis can (de-)stabilize the positive constant equilibrium. Additionally, the local existence and stability of nonconstant steady states near the positive constant equilibrium are obtained over a 1D domain using bifurcation theory. Several numerical simulations on one- and two-dimensional spatial domains are presented to demonstrate the system's ability to exhibit various interesting spatio-temporal patterns.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics, Applied
Yongli Cai, Qian Cao, Zhi-An Wang
Summary: This paper investigates the global dynamics of a ratio-dependent predator-prey system with prey-taxis. By establishing a mathematical model and conducting numerical simulations, we find that pattern formation may occur and prey-taxis drives the evolution of spatial inhomogeneity.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Hongyun Peng, Zhi-An Wang, Changjiang Zhu
Summary: This paper investigates the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. The system is transformed into a parabolic-hyperbolic system using a Cole-Hopf type transformation, and it is shown that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. The Cole-Hopf transformation is then reversed to obtain relevant results for the pre-transformed PDE-ODE hybrid system. The use of the effective viscous flux is a key ingredient in the proof, enabling the desired energy estimates and regularity for chemotaxis systems with initial data of low regularity.
SCIENCE CHINA-MATHEMATICS
(2022)
Article
Mathematics, Applied
Zhi-An Wang, Leyun Wu
Summary: This paper investigates a class of reaction-diffusion systems with cross diffusion, establishing the global existence and asymptotic behavior of solutions in a two-dimensional bounded domain with Neumann boundary conditions by imposing suitable structure assumptions. These systems have various applications such as autocatalytic chemical reaction, predator-prey interactions, combustion, etc.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Zhi-An Wang
Summary: This passage discusses the use of Fourier transform to derive key a priori estimates for the temporal gradient of chemical signal, and establishes the existence of global solutions and hydrodynamic limit of a chemotactic kinetic model in one dimension with internal states and temporal gradient.
KINETIC AND RELATED MODELS
(2022)
Article
Mathematics, Applied
Zhi-An Wang, Xin Xu
Summary: By treating the chemotactic coefficient as a bifurcation parameter, this paper establishes the existence of nonconstant monotone radial stationary solutions using global bifurcation theory and the Helly compactness theorem. It further shows that the radial stationary solution tends to a Dirac delta mass as the chemotactic coefficient tends to infinity. The linearized stability of bifurcating stationary solutions near the bifurcation points is also proven using the stability criterion of Crandall and Rabinnowitz.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Mathematics
Qingqing Liu, Hongyun Peng, Zhi-An Wang
Summary: This paper investigates a quasi-linear hyperbolic-parabolic system modeling vasculogenesis, showing the existence of a nonlinear diffusion wave under suitable structural assumptions on the pressure function. The study demonstrates that the solution of the system will locally and asymptotically converge to this wave if the wave strength is small. Additionally, using time-weighted energy estimates, it is further proven that the convergence rate of the nonlinear diffusion wave is algebraic.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Qingqing Liu, Hongyun Peng, Zhi-An Wang
Summary: This paper derives the large-time profile of solutions to the Cauchy problem of a hyperbolic-parabolic system modeling the vasculogenesis in R-3. By constructing a time-frequency Lyapunov functional and employing the Fourier energy method and delicate spectral analysis, the paper shows that solutions of the Cauchy problem tend time-asymptotically to linear diffusion waves around the constant ground state with algebraic decaying rates under suitable conditions on the density-dependent pressure function.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics
Hai-Yang Jin, Zhi-An Wang, Leyun Wu
Summary: In this paper, we study the initial-boundary value problem of a three-species spatial food chain model in a bounded domain. We establish the global existence of classical solutions and prove the global stability of prey-only steady state, semi-coexistence, and coexistence steady states.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Biology
De Tang, Zhi-An Wang
Summary: In this paper, the authors investigate population models with resource-dependent dispersal, considering both single-species and two-species competition. They find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This result is supported by yeast experiment observations and suggests that resource-dependent dispersal may produce different outcomes compared to random dispersal. For the two-species competition model, the authors classify the global dynamics based on the dispersal strategy and show that resource-dependent dispersal allows for coexistence even when one species has slower diffusion, contrary to the result with random dispersal where the slower diffuser always eliminates its fast competitor. The authors conclude that resource-dependent dispersal has a profound impact on population dynamics and evolutionary processes.
JOURNAL OF MATHEMATICAL BIOLOGY
(2023)
Article
Mathematics, Applied
Wenbin Lyu, Zhi-An Wang
Summary: This paper studies a class of singular prey-taxis models in a smooth bounded domain with homogeneous Neumann boundary conditions. The main challenge lies in the singularity that may occur as the prey density vanishes. By using the technique of a priori assumption, the comparison principle of differential equations, and semigroup estimates, it is shown that the singularity can be avoided if the intrinsic growth rate of the prey is sufficiently large, leading to the existence of global classical bounded solutions. Moreover, the global stability and convergence rates of co-existence and prey-only steady states are established using the method of Lyapunov functionals.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Hai-Yang Jin, Zhi-An Wang, Leyun Wu
Summary: This paper investigates the global boundedness and stability of classical solutions to an alarm-taxis system. It focuses on the burglar alarm hypothesis as an important mechanism of antipredation behavior when prey species are threatened by predators. By utilizing sophisticated coupling energy estimates based on Neumann semigroup smoothing properties, the paper establishes the existence of globally bounded solutions in two dimensions with Neumann boundary conditions and proves the global stability of coexistence homogeneous steady states under certain conditions on the system parameters.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Ru Hou, Zhian Wang, Wen-Bing Xu, Zhitao Zhang
Summary: This paper investigates the spreading speed of cooperative nonlocal dispersal systems with irreducible reaction functions and non-uniform initial data. The non-uniformity refers to the exponential decay of all components of the initial data, but with different decay rates. It has been established that different decay rates of initial data in a monostable reaction-diffusion or nonlocal dispersal equation result in different spreading speeds. The paper demonstrates that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will have a uniform spreading speed that depends solely on the smallest decay rate of the initial data. The decreasing property of the uniform spreading speed in the smallest decay rate implies that the component with the smallest decay rate can accelerate the spatial propagation of other components.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
Article
Mathematics, Applied
Wenbin Lyu, Zhi-An Wang
Summary: This paper investigates a parabolic-elliptic chemotaxis model with density-suppressed motility and general logistic source in an n-dimensional smooth bounded domain with Neumann boundary conditions. The main focus is on exploring the strength of logistic damping necessary to ensure the global boundedness of solutions, and establishing the asymptotic behavior of solutions under certain conditions.
ADVANCES IN NONLINEAR ANALYSIS
(2023)
Article
Mathematics, Applied
Xiumei Deng, Qihua Huang, Zhi-An Wang
Summary: Due to the importance of remediating contaminated ecosystems, numerous mathematical models have been developed to describe the interactions between populations and toxicants in polluted aquatic environments. However, these models often neglect the effects of toxicant-induced behavioral changes on population dynamics. In this study, we develop a diffusive population-toxicant model with toxicant-taxis, taking into account the possibility of individuals fleeing from high toxicant concentrations to low toxicant concentrations. We analyze the global well-posedness and stability of steady states under different conditions, and find that spatial pattern formations and bistable steady states may occur when the toxicant-taxis is strong. Our numerical simulations demonstrate the emergence of spatial aggregation and segregation patterns between populations and toxicants, illustrating the significant effects of toxicant-induced movement responses on the spatial distributions of populations in polluted aquatic environments.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Mathematics
Wenbin Lyu, Zhi-An Wang
Summary: This paper investigates a class of reaction-diffusion system with density-suppressed motility. The paper proves the existence of a unique global classical solution for the system and shows that the solution is uniformly bounded in time under certain conditions.
ELECTRONIC RESEARCH ARCHIVE
(2022)
