ScienceAsia 33 (2007): 363-366 |doi: 10.2306/scienceasia1513-1874.2007.33.363

The Existence and Stability of the Travelling Wave Solution of a Gompertz Growth with the Simplest Nonlinear Advection Model

Khadizah binti Ghazali (Khadizah bt. G.)^a and Shaharir bin Mohamad Zain (Shaharir b. M.Z.)^b*

 
ABSTRACT:     We show the existence of a travelling wave solution for a simplest nonlinear advection and the Gompertz reaction model. We prove that the solution is perturbatively stable without any restriction on the parameters of the Gompertz model but the stability of one of its trivial solutions is subject to a restriction on one of the values of the Gompertz reaction parameters. We also show that the solution is Poincare stable around one of it critical points provided the wave velocity of the traveling wave solution is greater than the product of the two reaction parameters.

KEYWORDS: Gompertz reaction model, Non-linear advection model, partial differential equation (PDE), traveling wave solution of a PDE, perturbative stability, Poincare stability.

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^a Faculty of Science and Technology, University Malaysia Sabah (UMS), Malaysia.
^b Faculty of Science and Technology, University Malaysia Terengganu (UMT, formerly KUSTEM), Malaysia.

* Corresponding author, E-mail: Shaharirzain711@hotmail.com

Received 9 Jun 2006, Accepted 25 Apr 2007