Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

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.