Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation



Publication Details

Alfifi, H., Marchant, T. R. & Nelson, M. I. (2014). Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 79 (1), 175-199.


Semi-analytical solutions are developed for the diffusive Nicholson's blowflies equation. Both one and two-dimensional geometries are considered. The Galerkin method, which assumes a spatial structure for the solution, is used to approximate the governing delay partial differential equation by a system of ordinary differential delay equations. Both steady-state and transient solutions are presented. Semi-analytical results for the stability of the system are derived and the critical parameter value, at which a Hopf bifurcation occurs, is found. Semi-analytical bifurcation diagrams and phase-plane maps are drawn, which show the initial Hopf bifurcation together with a classical period doubling route to chaos. A comparison of the semi-analytical and numerical solutions shows the accuracy and usefulness of the semi-analytical solutions. Also, an asymptotic analysis for the periodic solution near the Hopf bifurcation point is developed, for the one-dimensional geometry. 2012 The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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