Chemotaxis and growth system with singular sensitivity function

Abstract

This paper continues the study of the initial value problem of a chemotaxisgrowth system. In the previous paper [13], we have handled the case when the sensitivity function χ(ρ) is regular. In this paper we are concerned with the case when the function has singularity at ρ = 0 like χ(ρ) = log ρ or − 1 ρ. We verify global existence of solutions and discuss some asymptotic behaviour of solutions.

Global existence for chemotaxis with finite sampling radius

“…As such, models based on the above equations have been applied to a wide range of biological pattern formation processes, including mound formation in the slime mold Dictyostelium, bacterial pattern formation, animal pigmentation patterns and limb bud patterning ( [16,33,26,18]). In this paper we study a simple modification of the classical chemotaxis model (1), where the gradient sensing term ∇s is replaced by the non-local gradient…”

“…Details behind this model and about the analysis will be given later. First we summarise some results of the classical chemotaxis model (1) relevant to the analysis here.…”

“…An extensive review article by D. Horstmann, [12], concerning (1) and related models provides greater detail, here we summarise the essentials. In one space dimension, solutions exist globally, a fact only recently proven (Osaki and Yagi [21], see also Hillen and Potapov [10] and Horstmann and Winkler [13]).…”

“…Several of these regularizations have been studied in the literature, including saturation effects (e.g. Othmer and Stevens [24], Rivero et al [31], Aida et al [1]), volume filling (e.g. Painter and Hillen [25]), attraction-repulsion mechanisms (e.g.…”

Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

“…There is also a large body of related work by the Japanese school of researchers in reaction-diffusion equations. For example, Mimura et al have studied: cross-diffusion competition systems [29,40], dynamics of models for chemotaxis growth [2,7,17], pattern formation in resource-consumer systems [15,39], asymptotic analysis of interface dynamics for competition-diffusion models [10][11][12], interaction of travelling pulse solutions [13,31,41], interface dynamics for two-phase Stefan like problems [23][24][25], and travelling waves in bistable reactiondiffusion systems [30,31,43].…”