Methods
The model (1) is comprised of four variables: number of plants in a site
(Nx), fecundity (Si) determined by how
many seeds are produced by a single plant, dispersal rate
(di), and death rate (\(\mu\)). The number of plants in
a site in the following year (\(N_{x}^{{}^{\prime}}\)) is determined by the number
of seeds that do not disperse (\(1-d_{x}S_{x}\ N_{x}\)) plus the
number of seeds that survive dispersal from the other site
(1\(-\mu d_{x}S_{x}N_{x}\)). The individuals are presumed to be
annuals, and therefore do not survive into the following year. A
proportion of seeds that disperse fail to do so and remain in the native
sub-habitat. This percentage is denoted by c.
\begin{equation}
N_{1}^{{}^{\prime}}=1-d_{1\ }\left(1-c\right)S_{1}\ N_{1}+1-\mu d_{2}\left(1-c\right)S_{2}N_{2}\nonumber \\
\end{equation}(1)
\begin{equation}
N_{2}^{{}^{\prime}}=1-\mu d_{1}\left(1-c\right)S_{1}N_{1}+1-d_{2}\left(1-c\right)S_{2}N_{2}\nonumber \\
\end{equation}The model was then arranged to form a vector-matrix multiplication (2
and 3). Sub-habitat 2, is a constant environment so only experiences
good years. Sub-habitat 1 fluctuates in environmental condition and the
frequency of which it experiences good year is \(\frac{1}{f}\ \). The
proportion of S1 that survive, or the severity of the
year, is \(v\). In good years (2), the fecundity is increased by\(\frac{v}{f}\) and in bad years (3), the fecundity is decreased by\(\left(1-\left(\frac{1}{f}\right)\right)v\).