Dynamics of constant and variable stepsize methods for a nonlinear population model with delay

Hutchinson's equation is a reaction-diffusion model where the quadratic reaction term involves a delay. It is a natural extension of the logistic equation (no diffusion, no delay) and Fisher's equation (no delay), both of which have been used to illustrate the potential for spurious long-term dynamics in numerical methods. For the case where initial conditions and periodic boundary conditions are supplied, we look at the use of central differences in space and either Euler's method or the Improved Euler method in time. Our aim is to investigate the impact of the delay on the long-term behaviour of the scheme. After studying the fixed points of the methods in constant stepsize mode, we consider an adaptive time-stepping approach, using a standard local error control strategy. Applying ideas of Hall (1985) we are able to explain the fine detail of the time-step selection process.