arXiv subjects

Hai-Yang Jin

Publications and source records attributed to Hai-Yang Jin.

12 recordsLinked to original sources

arXiv2026

Global boundedness and stabilization for a three-component reaction-diffusion model with dual-dependent motility

In this paper, we consider the initial-boundary value problem of a three-component reaction-diffusion system with dual-dependent motility, which depends on both the chemical concentration and the nutrient level. We systematically establish the global existence, boundedness, and asymptotic behavior of the classical solutions to the system with no-flux boundary conditions through classified discussion and by exploiting specific structural properties of the motility function. In turn, by comparing the results obtained in the distinct cases, we demonstrate that the solution converges to different constant steady states, thereby indicating the crucial role of the dual-dependent motility function in the long-term behavior of the system.

math.AP↗

arXiv2026

Parabolic-elliptic reduction suppresses Hopf bifurcation in a forager-exploiter system with cascade taxis

We study a parabolic-elliptic forager-exploiter model describing the cascade interaction between two species through a shared environmental resource with a constant renewal rate. We first establish the global existence and uniform boundedness of classical solutions in arbitrary dimensions {\color{black}for large initial population data and large taxis sensitivity coefficients. We then investigate the long-time behavior of solutions.} When the resource renewal rate is small, all solutions will converge exponentially to the unique constant steady state. Furthermore, for large resource renewal rates, the parabolic-elliptic system does not undergo Hopf bifurcation, which is in sharp contrast to the corresponding fully parabolic case, where temporal oscillations arise via Hopf bifurcation.

math.AP↗

arXiv2026

Periodic dynamics in a forager-exploiter system under homogeneous and heterogeneous resource environments

We investigate time-periodic dynamics in a forager-exploiter system with a taxis cascade under both homogeneous and heterogeneous resource environments. The model describes the interactions among foragers, exploiters, and environmental resources, where foragers move toward higher resource densities while exploiters aggregate toward regions with higher forager densities. Our results show that different resource renewal mechanisms shape periodic dynamics in fundamentally different ways. Precisely, for time-periodic resource renewal rates, we establish the existence of positive time-periodic solutions for any positive renewal rate and further prove their global stability under suitable conditions on the parameters. In contrast, for homogeneous environments, only large resource renewal rates can destabilize the constant steady state through the Hopf bifurcation, thereby generating non-constant time-periodic solutions. Interestingly, for spatially heterogeneous and temporally homogeneous environments, our numerical simulations indicate that spatial heterogeneity exhibits opposing effects on periodic dynamics depending on total resource availability. When resources are sufficiently abundant, spatial heterogeneity tends to suppress the emergence of temporal oscillations, whereas when resources are relatively scarce, it may instead promote oscillatory behaviors, and sufficiently concentrated local resource supplies can trigger local or even global temporal oscillations. These findings reveal a delicate interplay among resource renewal mechanisms, resource availability, and spatial heterogeneity in shaping dynamics behavior.

math.AP↗

arXiv2023

Global solvability and stability of an alarm-taxis system

This paper is concerned with the global boundedness and stability of classical solutions to an alarm-taxis system describing the burglar alarm hypothesis as an important mechanism of anti-predation behavior when species are threaten by predators. Compared to the existing prey-taxis systems, the alarm-taxis system has more complicated coupling structure and additionally requires the gradient estimate of the primary predator density to attain the global boundedness of solutions. By the sophisticated coupling energy estimates based on the Neumann semigroup smoothing properties, we establish the existence of globally bounded solutions in two dimensions with Neumann boundary conditions and furthermore prove the global stability of co-existence homogeneous steady states under certain conditions on the system parameters.

math.AP↗

arXiv2020

Negligibility of haptotaxis effect in a chemotaxis-haptotaxis model

In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotaxis on asymptotical behavior, we show that any global-in-time haptotaxis solution component vanishes exponentially as time approaches infinity, and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, the aforementioned critical mass blow-up phenomenon for the chemotaxis-only model is almost undestroyed even with arbitrary introduction of haptotaixs, showing negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and longtime behavior in the chemotaxis-haptotaxis model.

math.AP↗

arXiv2020

Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility

We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility \begin{equation*}\label{e1}\tag{$\ast$} \begin{cases} u_t=Δ(γ(v)u)+αu F(w) -θu, &x\in Ω, ~~t>0,\\ v_t=DΔv+u-v,& x\in Ω, ~~t>0,\\ w_t=Δw-uF(w),& x\in Ω, ~~t>0, \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}= \frac{\partial w}{\partial ν}=0,&x\in \partialΩ, ~~t>0,\\ (u,v,w)(x,0)=(u_0,v_0,w_0)(x), & x\inΩ, \end{cases} \end{equation*} in a bounded domain $Ω\subset\R^2$ with smooth boundary, $α$ and $θ$ are non-negative constants and $ν$ denotes the outward normal vector of $\partial Ω$. The random motility function $γ(v)$ and functional response function $F(w)$ satisfy the following assumptions: \begin{itemize} \item $γ(v)\in C^{3}([0,\infty)),~0<γ_{1}\leqγ(v)\leq γ_2, \ |γ'(v)|\leq η$ for all $v\geq0$; \item $F(w)\in C^1([0,\infty)), F(0)=0,F(w)>0 \ \mathrm{in}~(0,\infty)~\mathrm{and}~F'(w)>0 \ \mathrm{on}\ \ [0,\infty)$ \end{itemize} for some positive constants $γ_1, γ_2$ and $η$. Based on the method of weighted energy estimates and Moser iteration, we prove that the problem \eqref{e1} has a unique classical global solution uniformly bounded in time. Furthermore we show that if $θ>0$, the solution $(u,v,w)$ will converge to $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time tends to infinity, while if $θ=0$, the solution $(u,v,w)$ will asymptotically converge to $(u_*,u_*,0)$ in $L^\infty$ with $u_*=\frac{1}{|Ω|}(\|u_0\|_{L^1}+α\|w_0\|_{L^1})$ if $D>0$ is suitably large.

math.AP↗

arXiv2020

The Keller-Segel system with logistic growth and signal-dependent motility

The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (γ(v)\nabla u- uχ(v)\nabla v)+μu(1-u), &x\in Ω, ~~t>0, 0=Δv+ u-v,& x\in Ω, ~~t>0,\\ u(x,0)=u_0(x), & x\in Ω, \end{cases} \end{equation} with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \R^2$ with smooth boundary, where the motility functions $γ(v)$ and $χ(v)$ satisfy the following conditions \begin{itemize} \item {\color{black}$(γ,χ)\in [C^2[0,\infty)]^2$} with $γ(v)>0$ and {\color{black} $\frac{|χ(v)|^2}{γ(v)}$ is bounded for all $v\geq 0$.} %for all $v\geq 0$ and $\lim\limits_{v\to\infty}\frac{|χ(v)|^2}{γ(v)}$ exists. \end{itemize} By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref{0-1} with $μ>0$ for any $u_0 \in W^{1, \infty}(Ω)$. Then based on a Lyapunov function, we show that all solutions $(u,v)$ of \eqref{0-1} will exponentially converge to the unique constant steady state $(1,1)$ provided $μ>\frac{K_0}{16}$ with $K_0=\max\limits_{0\leq v \leq \infty}\frac{|χ(v)|^2}{γ(v)}$.

math.AP↗

arXiv2020

Critical mass on the Keller-Segel system with signal-dependent motility

This paper is concerned with the global boundedness and blowup of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is there is a number $m_*>0$, such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e. $L^1$-norm of the initial value of cell density) is less than $m_*$, while the solution may blow up if the initial cell mass is greater than $m_*$.

math.AP↗

arXiv2019

Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator-prey systems with density-dependent preytaxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusion) and mobility (preytaxis) of the predator are correlated through a prey density dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearized system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

math.AP↗

arXiv2019

Global stabilization of the full attraction-repulsion Keller-Segel system

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system \begin{equation}\label{ARKS}\tag{$\ast$} \begin{cases} u_t=Δu-\nabla\cdot(χu\nabla v)+\nabla\cdot(ξu\nabla w), &x\in Ω, ~~t>0, v_t=D_1Δv+αu-βv,& x\in Ω, ~~t>0, w_t=D_2Δw+γu-δw, &x\in Ω, ~~t>0,\\ u(x,0)=u_0(x),~v(x,0)= v_0(x), w(x,0)= w_0(x) & x\in Ω, \end{cases} \end{equation} in a bounded domain $Ω\subset \R^2$ with smooth boundary subject to homogeneous Neumann boundary conditions. %The parameters $D_1,D_2,χ,ξ,α,β,γ$ and $δ$ are positive. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system \eqref{ARKS} with large initial data. Precisely, we show that if the parameters satisfy $\frac{ξγ}{χα}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\fracβδ,\fracδβ\Big\}$ for all positive parameters $D_1,D_2,χ,ξ,α,β,γ$ and $δ$, the system \eqref{ARKS} has a unique global classical solution $(u,v,w)$, which converges to the constant steady state $(\bar{u}_0,\fracαβ\bar{u}_0,\fracγδ\bar{u}_0)$ as $t\to+\infty$, where $\bar{u}_0=\frac{1}{|Ω|}\int_Ωu_0dx$. Furthermore, the decay rate is exponential if $\frac{ξγ}{χα}> \max\Big\{\fracβδ,\fracδβ\Big\}$. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. $D_1\ne D_2$) in multi-dimensions.

math.AP↗

arXiv2018

Chemotaxis effect vs logistic damping on boundedness in the 2-D minimal Keller-Segel model

In this paper, we study chemotaxis effect vs logistic dampening on boundedness for the two-dimensional minimal Keller-Segel model with logistic source in a 2-D smooth and bounded domain. It is well-known that this model allows only for global and uniform-in-time bounded solutions for any chemotactic strength and logistic dampening. Here, we carefully employ a simple and new method to regain its boundedness and, with particular attention to how boundedness depends qualitatively on the coefficient of chemotactic strength and logistic dampening rate. Up to a scaling constant depending only on initial data and the domain, we provide explicit upper bounds for the the solution components of the corresponding initial-boundary value problem. This qualitative boundedness results seems to be the first result in the regard.

math.AP↗

arXiv2017

Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics

In this paper, we study the rates of convergence of supposedly given global bounded classical solutions to a two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics. Except in one case where the rate of convergence for the fluid component is expressed in terms of the Poincare constant and the model parameters, all other rates of convergence are shown to be expressible only in terms of the model parameters and the underlying space dimension.

math.AP↗