I'm a biologist and I'm working on a mathematical model that describe similarity between colonies (i.e. cells) occupying a circular habitat.
Migration occur only between adjacent colonies at rate m (m/2 for each of the two directions available) per generation.
In the paper, I'm studying the authors define a Migration Matrix:
\begin{align} M = \{\frac{m^2}{2} + (1 - m^2)\} I + m(1-m)(R + R^{-1}) + \frac{m^2}{4}(R^2 + R^{-2})\\\\ \end{align}
Where I is the identity matrix and R is a n-by-n circular matrix (As an example I'm assuming a 4x4 matrix):
$$ \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ \end{pmatrix} $$
I would like to understand how to 'verbally' interpret the Migration Matrix. For instance, the first term means that:
- either two individuals move from the same (adjacent) colony \begin{align}\frac{2m^2}{4} =\frac{m^2}{2} \end{align}
- or both individuals did not migrate \begin{align}(1-m)^2\end{align}
The paper I'm referring to is this one: https://www.sciencedirect.com/science/article/pii/004058097090047X