Depleting the signal: Analysis of chemotaxis-consumption models -- A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

Depleting the signal: Analysis of chemotaxis-consumption models—A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

Equilibration in a two-species-two-chemicals chemotaxis-competition system

This paper deals with the two-species--two-chemicals chemotaxis-competition system with signal-dependent sensitivity, where Ω is a bounded domain in Rn (n ≥ 2) with smooth boundary, χ1,χ2 and μ1,μ2 are constants satisfying some conditions. About this system Tu–Mu–Zheng–Lin (Discrete Contin. Dyn. Syst.;2018;38;3617– 3636) showed global existence and stabilization of solutions under some smallness conditions for χ1 and χ2. Here energy arguments for seeing stabilization in the previous work were based on ideas in Bai–Winkler (Indiana Univ. Math. J.;2016;65;553–583); however, these ideas were recently improved by the first author (Discrete Contin. Dyn. Syst. Ser. S;2020;13;269–278), which implies that the result about stabilization in the previous work seems not to be the best. This paper gives an improve- ment of conditions for stabilization in the previous work. The feature of the proof is to use the Sylvester criterion in deriving energy estimates

Traveling wave solutions for two species competitive chemotaxis systems

In this paper, we consider two species chemotaxis systems with Lotka–Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number c∗. We also show the non-existence of such traveling waves with speed less than some critical number c∗0 , which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range for the chemotaxis sensitivity coefficients ensuring c∗ = c∗0 , which implies that the minimum wave speed exists and is not affected by the chemoattractant

A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features