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Generalised diffusive delay logistic equations: Semi-analytical solutions

Journal Article


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Abstract


  • 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.

Publication Date


  • 2012

Citation


  • Alfifi, H. Y., Marchant, T. R. & Nelson, M. I. (2012). Generalised diffusive delay logistic equations: Semi-analytical solutions. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (4-5), 579-596.

Scopus Eid


  • 2-s2.0-84865192266

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9468&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/2133

Number Of Pages


  • 17

Start Page


  • 579

End Page


  • 596

Volume


  • 19

Issue


  • 4-5

Abstract


  • 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.

Publication Date


  • 2012

Citation


  • Alfifi, H. Y., Marchant, T. R. & Nelson, M. I. (2012). Generalised diffusive delay logistic equations: Semi-analytical solutions. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (4-5), 579-596.

Scopus Eid


  • 2-s2.0-84865192266

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9468&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/2133

Number Of Pages


  • 17

Start Page


  • 579

End Page


  • 596

Volume


  • 19

Issue


  • 4-5