Numerical Analysis For A Cooperative Model With Diffusion

In this paper, we give some numerical analysis for a cooperative system with diffusion. The system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. The paper is divided into two parts. In the first part, a monotone iterative scheme is developed for solving the nonlinear finite difference solution of the steady-state problem. The existence and uniqueness of a nonnegative finite differnce solution of the steady-state problem is investigated. It is shown that the system may have four types of nonnegative solutions for different parameters, and each of these solutions can be computed by a suitable choice of initial iteration in the monotone iterative scheme. The second part is for time-dependent problem. The investigation includes a monotone iterative algorithm, and the existence and uniqueness of the solutions. Also discussed is the asymptotic behavior of the time-dependent solutions, including the convergence of time-dependent solution to a steady-state solution. Some numerical results are given in every part.