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...
We insert the between-species competion terms as:\\
\begin{equation}
\beta = \epsilon T \nonumber
\end{equation}
where T = INSERT THE MATRIX HERE!!\\ \end{equation}\\
where T = INSERT THE MATRIX HERE!! \noindent \\
The modified equations for the equilibrium solutions
become:\\ become: \noindent \\
\begin{equation}
A_i^* + \sum_{j} \beta_{ij}A_j^* - \frac {\alpha^{AP}}{\lambda_i^q} \sum_{j<=i} P_j^* = K_a
\label{denklem3}
...
\label{denklem4}
\end{equation}\\
When we plug in the \ref{denklem4} into \ref{denklem3}, we
get\\ get \noindent \\
\begin{equation}
A_i^* + \sum_{j} \beta_{ij} A_j* - \frac{\alpha^{AP}}{\lambda_i^q} \sum_{j<=i} (\alpha^{AP} A_j^* + K_p ) = K_a \nonumber
\end{equation}
...
\label{denklem5}
\end{equation}\\
Using \ref{denklem5}, we can write the differential equation for pollunator species explicitely as
follows:\\ follows: \noindent\\
\begin{equation}
\frac{dA_i}{dt} = A_i ( K_a - A_i - \epsilon A_{i+1} - \epsilon A_{i-1} + \frac{\alpha^{AP}}{\lambda_i^q} \sum_{j<=i} P_j ) \nonumber
\end{equation}\\
At $\epsilon = 0 $ (i.e no competition between different pollunator species), the equilibrium populations
satisfy:\\ satisfy: \noindent\\
\begin{equation}
A_i^* +\frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} A_j^* - i K_p \frac{\alpha^{PA}}{\lambda_i^q} = K_a
\label{denklem6}
\end{equation}\\
When we write $\beta$ explicetly, we have the
equation:\\ equation: \noindent\\
\begin{equation}
A_i^ + \epsilon \sum_{j} T_{ij} A_j - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} A_j - i K_p \frac{\alpha^{PA}}{\lambda_i^q} = K_a
\label{denklem7}
\end{equation}\\
We make a perturbation to the equilibrium population by $\epsilon \delta
A_i$:\\ A_i$: \noindent \\
\begin{equation}
A_i = A_i^* + \epsilon \delta A_i \nonumber
\end{equation}\\
...
A_i^* + \epsilon \delta A_i + \epsilon \sum_{j} T_{ij} ( A_j^* + \epsilon \delta A_j) - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} (A_j^* + \epsilon \delta A_j) - i K_p \frac{\alpha^{PA}}{\lambda_i^q} = K_a \nonumber
\end{equation}\\
From \ref{denklem6}, the equilibrium terms vanish and we end up
with:\\ with: \noindent\\
\begin{equation}
\epsilon \delta A_i + \epsilon \sum_{j} T_{ij} A_j^* + \epsilon^2 \sum_{j} T_{ij} \delta A_j - \epsilon( \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} \delta A_j) = 0
\end{equation}\\
We neglect the second order $\epsilon^2$ term ($\epsilon^2 ~ 0$). Finaly we
get:\\ get: \noindent\\
\begin{eqnarray}
\epsilon ( \delta A_i + \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} \delta A_j) = - \epsilon \sum_{j} T_{ij} A_j^* \nonumber \\
...
\label{denklem8}
\end{eqnarray}\\
We can rewrite the above equation as a matrix
equation:\\ equation: \noindent \\
\begin{equation}
\sum_{j} M_{ij} \delta A_k = - \sum_{j} T_{ij} A_i^* = C_i^* \nonumber
\end{equation}\\
...
-\frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} & \mbox{if $j
0 & \mbox{if $j>i$} \end{array} \right. \nonumber \] \\
If we can invert the matrix M, we can
write:\\ write: \noindent \\
\begin{equation}
\delta A_i = (M^{-1} C^*)_{i} \nonumber
\end{equation}\\
M is a lower-triangular matrix where the diagonal elements are the
eigenvalues\\ eigenvalues \noindent \\
$$ \mu_i = 1 - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} > 0 \nonumber $$ \\
since in our model $ \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} = \frac{1-0.00032}{(2i-1)^{1.64}} <1$ for all i.
\noindent
...
