@Article{EAJAM-7-3,
author = {},
title = {Dynamics of an Innovation Diffusion Model with Time Delay},
journal = {East Asian Journal on Applied Mathematics},
year = {2017},
volume = {7},
number = {3},
pages = {455--481},
abstract = {<p style="text-align: justify;">A nonlinear mathematical model for innovation diffusion is proposed. The
system of ordinary differential equations incorporates variable external influences (the
cumulative density of marketing efforts), variable internal influences (the cumulative
density of word of mouth) and a logistically growing human population (the variable
potential consumers). The change in population density is due to various demographic
processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem
involves two dynamic variables viz. a non-adopter population density and an adopter
population density. The model is analysed qualitatively using the stability theory of
differential equations, with the help of the corresponding characteristic equation of the
system. The interior equilibrium point can be stable for all time delays to a critical value,
beyond which the system becomes unstable and a Hopf bifurcation occurs at a second
critical value. Employing normal form theory and a centre manifold theorem applicable
to functional differential equations, we derive some explicit formulas determining the
stability, the direction and other properties of the bifurcating periodic solutions. Our numerical
simulations show that the system behaviour can become extremely complicated
as the time delay increases, with a stable interior equilibrium point leading to a limit
cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit
cycles with more local maxima and minima per cycle, and finally chaotic solutions.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.201216.230317a},
url = {https://global-sci.com/article/82727/dynamics-of-an-innovation-diffusion-model-with-time-delay}
}