  The current paper is to investigate the numerical approximation of logistic
type chemotaxis models in one space dimension with a free boundary. Such a
model with a free boundary describes the spreading of a new or invasive species
subject to the influence of some chemical substances in an environment with a
free boundary representing the spreading front (see Bao and Shen [1], [2]). The
main challenges in the numerical studies lie in tracking the moving free
boundary and the nonlinear terms from chemical. To overcome them, a front
fixing framework coupled with finite difference method is introduced. The
accuracy of the proposed method, the positivity of the solution, and the
stability of the scheme are discussed.The numerical simulations agree well with
theoretical results such as the vanishing spreading dichotomy, local
persistence, and stability. These simulations also validate some conjectures in
our future theoretical studies such as the dependence of the
vanishing-spreading dichotomy on the initial solution u0, initial habitat h0,
the moving speed {\nu} and the chemotactic sensitivity coefficients
\c{hi}1,\c{hi}2.
