An Efficient and Accurate numerical Scheme for Turing Instability on a predator-prey Model

We present an efficient and accurate numerical method for solving a ratio-dependent predatorprey model with a Turing instability. The system is discretized by a finite difference method with a semi-implicit scheme which allows much larger time step sizes than those required by a standard explicit scheme. A proof is given for the positivity and boundedness of the numerical solutions depending only on the temporal, but not on the spatial step sizes. Finally, we perform numerical experiments demonstrating the robustness and accuracy of the numerical solution for the Turing instability. In particular, we show that the numerical non-constant stationary solutions exist.