Generalised diffusive delay logistic equations: Semi-analytical solutions

This paper considers semi-analytical solutions for a class of generalised logis- tic partial dierential equations with both point and distributed delays. Both one and two-dimensional geometries are considered. The Galerkin method is used to approximate the governing equations by a system of ordinary dierential delay equations. This method involves assuming a spatial structure for the solution and averaging to obtain the ordinary dierential delay equation models. Semi-analytical results for the stability of the system are derived with the critical parameter value, at which a Hopf bifurcation occurs, found. The results show that diusion acts to stabilise the system, compared to equivalent non- diusive systems and that large delays, which represent feedback from the distant past, act to destabilize the system. Comparisons between semi-analytical and numerical solutions show excellent agreement for steady state and transient solutions, and for the parameter values at which the Hopf bifurcations occur.