Numerical solutions of a Michaelis--Menten-type ratio-dependent predator--prey system with diffusion

This paper is concerned with finite difference solutions of a Michaelis-Menten-type ratio-dependent predator-prey system with diffusion. The system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the various steady-state solutions. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. A simple and easily verifiable condition on the rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to a semitrivial steady-state solution. The above results lead to computational algorithms for the solution as well as the global asymptotic stability of the system. Some numerical results are given. All the conclusions are directly applicable to the finite difference solution of the corresponding ordinary differential system.