Quantitative Evaluation of an Active Chemotaxis Model in Discrete Time

摘要

A system of N particles in a chemical medium in R~d is studied in a discrete time setting. Underlying continuous time interacting particle system can be expressed asdX_i(t) = [-{I - A)X_i(t) + ▽h(t,X_i(t))]dt + dW_i(t), (0.1)Xi(0) = x_i∈R~d?i = 1,...,N?/(?t)h(t, x) = -αh(t, x) + D△h(t, x) + β/n∑_(i=1)~NG(X_i(t), x), h(0, ·) = h(·),where Xi(t), h(t, x) are respectively the location of the i-th particle and the concentration of the chemical medium at location x at time t with h(0, x) = h(x). In this article we describe a general discrete time non-linear formulation of the model (0.1) and a strongly coupled particle system approximating it. Similar models have been studied before (Budhiraja et al.(2010)) 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 of the measure valued coupled dynamical system (driven by some contractivity assumptions on non-random matrix A like "‖A‖ < ω_0 for some 0 < ω_0 < 1" along with other parameters) obtained through the large N asymptotics of the particle empirical measure. We also provide uniform in time quantitative convergence rates for the particle empirical measure to the corresponding limit measure under suitable conditions on relevant model parameters.