Skip to main content

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

Journal Article


Abstract


  • 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.

Publication Date


  • 2014

Citation


  • 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.

Scopus Eid


  • 2-s2.0-84893027941

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3450

Number Of Pages


  • 24

Start Page


  • 175

End Page


  • 199

Volume


  • 79

Issue


  • 1

Abstract


  • 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.

Publication Date


  • 2014

Citation


  • 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.

Scopus Eid


  • 2-s2.0-84893027941

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/3450

Number Of Pages


  • 24

Start Page


  • 175

End Page


  • 199

Volume


  • 79

Issue


  • 1