Accelerated monotone scheme for finite difference equations concerning steady-state prey–predator interactions

Abstract

In previous work, by adapting a suitable finite difference method to a particular monotone scheme, the authors and A. Lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the Volterra–Lotka equations describing prey–predator interactions with diffusion. In this paper, in order to improve the efficiency of the method, we show how Newton's method can be successfully combined with the previous scheme to greatly accelerate the convergence. In some particularly ‘difficult’ problems, the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3.