Title: 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.