We are concerned with the following density-suppressed motility model: $u_t=\Delta (\gamma(v) u)+\mu u(1-u); v_t=\Delta v+ u-v,$ in a bounded smooth domain $\Omega\subset \mathbb{R}^2$ with homogen...

Boundedness, stabilization, and pattern formation driven by density-suppressed motility

The global existence and the instability of constant steady states are obtained together for a Keller-Segel type chemotactic aggregation model. Organisms are assumed to change their motility depending only on the chemical density but not on its gradient. However, the resulting model is closely related to the logarithmic model, ut=Δ(ź(v)u)=źź(ź(v)(źuźkvuźv)),vt=źΔvźv+u, $$\begin{aligned} u_{t}=\Delta \bigl(\gamma (v)u\bigr)=\nabla \cdot \biggl(\gamma (v) \biggl(\nabla u- \frac{k}{v}u\nabla v \biggr) \biggr),\quad v_{t}={\varepsilon }\Delta v-v+u, \end{aligned}$$ where ź(v):=c0vźk$\gamma (v):=c_{0}v^{-k}$ is the motility function. The global existence is shown for all chemosensitivity constant k>0$k>0$ with a smallness assumption on c0>0$c_{0}>0$ . On the other hand constant steady states are shown to be unstable only if k>1$k>1$ and ź>0${\varepsilon }>0$ is small. Furthermore, the threshold diffusivity ź1>0${\varepsilon }_{1}>0$ is found that, if ź<ź1${\varepsilon }<{\varepsilon }_{1}$, any constant steady state is unstable and an aggregation pattern appears. Numerical simulations are given for radial cases.

Global Existence and Aggregation in a Keller---Segel Model with Fokker---Planck Diffusion

This paper deals with a Keller–Segel type parabolic–elliptic system involving nonlinear diffusion and chemotaxis in a smoothly bounded domain , , under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function , . The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when , or , . In addition, we find the conditions on parameters, and , that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.

https://iopscience.iop.org/article/10.1088/1361-6544/aaf513/pdf

Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing

This paper concerns a higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth. It is shown that the strong logistic damping can prevent blow-up in the higher dimensions.

Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth

We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain.

Keller-Segel Chemotaxis Models: A Review

The global existence of a chemotaxis model for cell aggregation phenomenon is obtained. The model system belongs to the class of logarithmic models and takes a Fokker–Planck type diffusion for the equation of cell density. We show that weak solutions exist globally in time in dimensions n ∈ { 1 , 2 , 3 } and for large initial data. The proof covers the parameter regimes that constant steady states are linearly stable. It also partially covers the other parameter regimes that constant steady states are unstable. We also find the sharp instability condition of constant steady states and provide numerical simulations which illustrate the formation of aggregation patterns.

A logarithmic chemotaxis model featuring global existence and aggregation

Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis

This paper analyzes prey–predator models with indirect predator-taxis in such a way that chemical secreted by the predator triggers the repellent behavior of prey against the predator. Under the assumption of quadratic decay of predator, we prove the global existence and uniform boundedness of classical solutions up to two spatial dimensions. Moreover, via the linear stability analysis, we show that large chemosensitivity gives rise to the occurrence of pattern formations. We also obtain the global stability results for the nontrivial constant steady states by establishing proper Lyapunov functionals.

Changwook Yoon

Papers

Global Existence and Aggregation in a Keller---Segel Model with Fokker---Planck Diffusion

Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing

A logarithmic chemotaxis model featuring global existence and aggregation

Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis

Global solvability of prey–predator models with indirect predator-taxis