Qualitative behaviour of numerical approximations to Volterra integro-differential equations

In this paper, we investigate the qualitative behaviour of numerical approximations to a nonlinear Volterra integro-differential equation with unbounded delay. We consider the simple single-species growth model (d)/(dt) N(t) = lambdaN(t) (1 - c(-1) integral(-infinity)(t) k(t - s)N(s) ds) We apply the (composite) theta-rule as a quadrature to discretize the equation. We are particularly concerned with the way in which the long-term qualitative properties of the analytical solution can be preserved in the numerical approximation. Using results in (S.N. Elaydi and S. Murakami, J. Differ. Equations Appl. 2 (1996) 401; Y. Song and C.T.H. Baker, J. Differ. Equations Appl. 10 (2004) 379) for Volterra difference equations, we show that, for a small bounded initial function and a small step size, the corresponding numerical solutions display the same qualitative properties as found in the original problem. In the final section of this paper, we discuss how the analysis can be extended to a wider class of Volterra integral equations and Volterra integro-differential equations with fading memory. (C) 2004 Elsevier B.V. All rights reserved.