# Changing Dynamical System of a Technological Substitution Model

Topics: Differential equation, Mathematics, Parameter Pages: 4 (562 words) Published: January 22, 2013
CHANGING DYNAMICAL SYSTEM OF A TECHNOLOGICAL SUBSTITUTION MODEL A.L. Panya*, Enu N. Ekaka-a*
akpabokigh4real@yahoo.com
*Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt. Abstract
It is a well known fact that a dynamical system will change its stability behavior due to changes in the model parameters. In this study, we propose to analyse the fundamental changes in stability if the inter-specific coefficients are varied. Our results which we have not been seen elsewhere will be presented and discussed qualitatively. INTRODUCTION

The interaction between two technologies as proposed by the original author [1] (and several other cited references by [1]) follows a system of non-linear first order ordinary differential equations. In this scenario, the system we are analysing is a modified form of the popular Lotka-Volterra competition model [2].

Since the behaviour of the solution trajectory informs us about the dynamical system, we would think that a variation of a model parameter while other model parameters are fixed will cause the dynamical system to change in different patterns as a response to changing parameters.

The aim of this important subject in mathematics is to study the fundamental changes in the qualitative behavior of stability otherwise known as bifurcation analysis. In this talk, we will illustrate its application in a Technological Substitution Model (TSM)[1].

MODEL FORMULATION
Following [1], the model is
(1)

Where:
x1, x2 are the populations of technology 1 and technology 2. These can be expressed several ways, e.g., rate of units sold, volume of market share, etc. a1, a2 are the growth rates of technology 1 and technology 2. K1, k2 are the carrying capacity of technology 1 and technology 2(i.e the maximum population size of the technologies). , are inter-specific coefficients of competition of technology 1 and technology 2.

MODIFIED MODEL 1
We modify...