Quantitative evaluation of an active Chemotaxis model in Discrete time

Abstract

A system of $N$ particles in a chemical medium in $\mathbb{R}^{d}$ is studied in a discrete time setting. Underlying interacting particle system in continuous time can be expressed as \begin{eqnarray} dX_{i}(t) &=&[-(I-A)X_{i}(t) + \bigtriangledown h(t,X_{i}(t))]dt + dW_{i}(t), \,\, X_{i}(0)=x_{i}\in \mathbb{R}^{d}\,\,\forall i=1,\ldots,N\nonumber\\ \frac{\partial}{\partial t} h(t,x)&=&-\alpha h(t,x) + D\bigtriangleup h(t,x) +\frac{\beta}{n} \sum_{i=1}^{N} g(X_{i}(t),x),\quad h(0,\cdot) = h(\cdot).\label{main} \end{eqnarray} where $X_{i}(t)$ is the location of the $i$th particle at time $t$ and $h(t,x)$ is the function measuring the concentration of the medium at location $x$ with $h(0,x) = h(x)$. In this article we describe a general discrete time non-linear formulation of the aforementioned model and a strongly coupled particle system approximating it. Similar models have been studied before (Budhiraja et al.(2011)) under a restrictive compactness assumption on the domain of particles. In current work the particles take values in $\R^{d}$ and consequently the stability analysis is particularly challenging. We provide sufficient conditions for the existence of a unique fixed point for the dynamical system governing the large $N$ asymptotics of the particle empirical measure. We also provide uniform in time convergence rates for the particle empirical measure to the corresponding limit measure under suitable conditions on the model.