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Innovation diffusion model in patch environment. (English) Zbl 1052.91076

A mathematical model is proposed to describe the dynamics of users of one product in two different patches. Advertisement force, contact rate between users and non-users, population dispersal rate and returning rate from user class to non-user class are chosen as key parameters. For the product with a long life-span, it is assumed that the returning rate is proportional to the number of users. It is shown that this model has a unique positive equilibrium which is a globally stable. For the product with a short life-span, time delays are introduced to represent the duration of the product in two different markets and the stability of a positive equilibrium is analyzed in one reasonable case. Periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated.

The model designs in assuming, that two patches are isolated, and the dynamics of both classes are governed by

$\begin{array}{cc}\hfill \frac{d{N}_{1}}{dt}& =-{\gamma }_{1}{N}_{1}-{\lambda }_{1}{A}_{1}{N}_{1}-{\delta }_{1}{N}_{1}+{\nu }_{1}{A}_{1}+{\beta }_{1},\hfill \\ \hfill \frac{d{A}_{1}}{dt}& ={\gamma }_{1}{N}_{1}+{\lambda }_{1}{A}_{1}{N}_{1}-{\delta }_{1}{A}_{1}-{\nu }_{1}{A}_{1},\hfill \end{array}$

where ${N}_{1}$ is the number of non-users in the first patch, ${A}_{1}$ is the number of users in the first patch, ${\beta }_{1}$ is the birth rate of the population in the first patch, ${\delta }_{1}$ is the death rate of the population in the first patch, ${\gamma }_{1}$ represents the intensity of advertisement for promoting product-users in the first patch, and ${\lambda }_{1}$ is the valid contact rate of users of the product with non-users in the first patch. Similarly, it is supposed that the dynamics of non-user-class and user-class in the second patch are governed by

$\begin{array}{cc}\hfill \frac{d{N}_{2}}{dt}& =-{\gamma }_{2}{N}_{2}-{\lambda }_{2}{A}_{2}{N}_{2}-{\delta }_{2}{N}_{2}+{\nu }_{2}{A}_{2}+{\beta }_{2},\hfill \\ \hfill \frac{d{A}_{2}}{dt}& ={\gamma }_{2}{N}_{2}+{\lambda }_{2}{A}_{2}{N}_{2}-{\delta }_{2}{A}_{2}-{\nu }_{2}{A}_{2},\hfill \end{array}$

where ${N}_{2}$ is the number of non-users in the second patch, ${A}_{2}$ is the number of users in the second patch, ${\beta }_{2}$ is the birth rate of the population in the second patch, ${\delta }_{2}$ is the death rate of the population in the second patch, ${\gamma }_{2}$ represents the intensity of advertisement for promoting product-users in the second patch, and ${\lambda }_{2}$ is the valid contact rate of users of the product with non-users in the second patch.

But in case, when the two patches are connected, ${\theta }_{1}$ is the probability that an individual migrates from the first patch to the second patch, ${\theta }_{2}$ the probability that an individual migrates from the second patch to the first patch. Further, it is assumed that ${k}_{2}$ is the fraction of new immigrants from the second patch who remain in user-class and that ${k}_{1}$ is the fraction of new immigrants from the first patch who remain in user-class. Under these assumptions, the model is:

$\begin{array}{cc}\hfill \frac{d{N}_{1}}{dt}& =-{\gamma }_{1}{N}_{1}-{\lambda }_{1}{A}_{1}{N}_{1}-{\delta }_{1}{N}_{1}+{\nu }_{1}{A}_{1}+{\beta }_{1}+{\theta }_{2}\left({N}_{2}+\left(1-{k}_{2}\right){A}_{2}\right)-{\theta }_{1}{N}_{1},\hfill \\ \hfill \frac{d{A}_{1}}{dt}& ={\gamma }_{1}{N}_{1}+{\lambda }_{1}{A}_{1}{N}_{1}-{\delta }_{1}{A}_{1}-{\nu }_{1}{A}_{1}-{\theta }_{1}{A}_{1}+{k}_{2}{\theta }_{2}{A}_{2},\hfill \\ \hfill \frac{d{N}_{2}}{dt}& =-{\gamma }_{2}{N}_{2}-{\lambda }_{2}{A}_{2}{N}_{2}-{\delta }_{2}{N}_{2}+{\nu }_{2}{A}_{2}+{\beta }_{2}+{\theta }_{1}\left({N}_{1}+\left(1-{k}_{1}\right){A}_{1}\right)-{\theta }_{2}{N}_{2},\hfill \\ \hfill \frac{d{A}_{2}}{dt}& ={\gamma }_{2}{N}_{2}+{\lambda }_{2}{A}_{2}{N}_{2}-{\delta }_{1}{A}_{2}-{\nu }_{2}{A}_{2}-{\theta }_{2}{N}_{2}+{k}_{1}{\theta }_{1}{A}_{1}·\hfill \end{array}$

In order to incorporate duration of the product into the model, described above was modified. The assumption that the returning rate is proportional to the number of users seems reasonable when the length of the life-span of the product is large. Otherwise, the effect of the length will be significant. Because of this, time delays due to the life-span of the product was introduced into the model and it was obtained:

$\begin{array}{c}\frac{d{N}_{1}}{dt}=-{\gamma }_{1}{N}_{1}\left(t\right)-{\lambda }_{1}{A}_{1}\left(t\right){N}_{1}\left(t\right)-{\delta }_{1}{N}_{1}\left(t\right)+{e}^{-{\delta }_{1}{\tau }_{1}}\left({\gamma }_{1}{N}_{1}\left(t-{\tau }_{1}\right)\hfill \\ \hfill +{\lambda }_{1}{A}_{1}\left(t-{\tau }_{1}\right){N}_{1}\left(t-{\tau }_{1}\right)\right)+{\beta }_{1}+{\theta }_{2}\left({N}_{2}+\left(1-{k}_{2}\right){A}_{2}\right)-{\theta }_{1}{N}_{1},\end{array}$ $\begin{array}{c}\frac{d{A}_{1}}{dt}={\gamma }_{1}{N}_{1}\left(t\right)+{\lambda }_{1}{A}_{1}\left(t\right){N}_{1}\left(t\right)-{\delta }_{1}{A}_{1}\left(t\right)-{e}^{-{\delta }_{1}{\tau }_{1}}\left({\gamma }_{1}{N}_{1}\left(t-{\tau }_{1}\right)\hfill \\ \hfill +{\lambda }_{1}{A}_{1}\left(t-{\tau }_{1}\right){N}_{1}\left(t-{\tau }_{1}\right)\right)-{\theta }_{1}{A}_{1}\left(t\right)+{k}_{2}{\theta }_{2}{A}_{2}\left(t\right),\end{array}$ $\begin{array}{c}\frac{d{N}_{2}}{dt}=-{\gamma }_{2}{N}_{2}\left(t\right)-{\lambda }_{2}{A}_{2}\left(t\right){N}_{2}\left(t\right)-{\delta }_{2}{N}_{2}\left(t\right)+{e}^{-{\delta }_{2}{\tau }_{2}}\left({\gamma }_{2}{N}_{2}\left(t-{\tau }_{2}\right)\hfill \\ \hfill +{\lambda }_{2}{A}_{2}\left(t-{\tau }_{2}\right){N}_{2}\left(t-{\tau }_{2}\right)\right)+{\beta }_{2}+{\theta }_{1}\left({N}_{1}+\left(1-{k}_{1}\right){A}_{1}\right)-{\theta }_{2}{N}_{2},\end{array}$ $\begin{array}{c}\frac{d{A}_{2}}{dt}={\gamma }_{2}{N}_{2}\left(t\right)+{\lambda }_{2}{A}_{2}\left(t\right){N}_{2}\left(t\right)-{\delta }_{2}{A}_{2}\left(t\right)-{e}^{-{\delta }_{2}{\tau }_{2}}\left({\gamma }_{2}{N}_{2}\left(t-{\tau }_{2}\right)\hfill \\ \hfill +{\lambda }_{2}{A}_{2}\left(t-{\tau }_{2}\right){N}_{2}\left(t-{\tau }_{2}\right)\right)-{\theta }_{2}{A}_{2}\left(t\right)+{k}_{1}{\theta }_{1}{A}_{1}\left(t\right)·\end{array}$

Here ${\tau }_{1}$ represents the life-span of the product in the first patch and ${\tau }_{2}$ represents the life-span of the product in the second patch.

##### MSC:
 91D25 Spatial models (mathematical sociology) 34K60 Qualitative investigation and simulation of models
##### Keywords:
stability; periodics; delay