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\section{The original model}
The equations in the original model are as follows:
\begin{equation}
\frac{dA_i}{dt} = A_i (K_a - A_i + \frac{\alpha^{PA}}{\lambda^q} \sum_{j<=i} P_j )
\label{denklem1}
...
\label{denklem2}
\end{equation}
We insert the between-species competion terms
as: as:\\
\begin{equation}
\beta = \epsilon T \nonumber
\end{equation}
where T = INSERT THE MATRIX
HERE!! HERE!!\\
The modified equations for the equilibrium solutions
become: become:\\
\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}
...
\begin{equation}
P_i^* = \alpha^{AP} A_i^* + K_p
\label{denklem4}
\end{equation} \end{equation}\\
When we plug in the \ref{denklem4} into \ref{denklem3}, we
get get\\
\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}
\begin{equation}
A_i^* + \sum_{j} \beta_{ij} A_j* - (\frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=i} \alpha^{AP} A_j^*) - \frac{\alpha^{PA}}{\lambda_i^q} \sum_{j<=i} K_p = K_a
\label{denklem5}
\end{equation} \end{equation}\\
Using \ref{denklem5}, we can write the differential equation for pollunator species explicitely as
follows: follows:\\
\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} \end{equation}\\
At $\epsilon = 0 $ (i.e no competition between different pollunator species), the equilibrium populations
satisfy: satisfy:\\
\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} \end{equation}\\
When we write $\beta$ explicetly, we have the
equation: equation:\\
\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} \end{equation}\\
We make a perturbation to the equilibrium population by $\epsilon \delta
A_i$: A_i$:\\
\begin{equation}
A_i = A_i^* + \epsilon \delta A_i \nonumber
\end{equation} \end{equation}\\
Then we put it into \ref{denklem7} we
get: get:\\
\begin{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} \end{equation}\\
From \ref{denklem6}, the equilibrium terms vanish and we end up
with: with:\\
\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} \end{equation}\\
We neglect the second order $\epsilon^2$ term ($\epsilon^2 ~ 0$). Finaly we
get: get:\\
\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 \\
\delta A_i - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=1} \delta A_j) = -\sum_{j} T_{ij} A_j^*
\label{denklem8}
\end{eqnarray} \end{eqnarray}\\
We can rewrite \ref{denklem8} as a matrix
equation: equation:\\
\begin{equation}
\sum_{j} M_{ij} \delta A_k = - \sum_{j} T_{ij} A_i^* = C_i^*
\end{equation} \end{equation}\\
where
\[ $ M_{ij} = \left\{ \begin{array}{ll}
1 - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} & \mbox{if $j=i$}\\
-\frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} & \mbox{if $j
0 & \mbox{if $j>i$} \end{array} \right.
\] $\\
If we can invert the matrix M, we can
write: write:\\
\begin{equation}
\delta A_i = (M^{-1} C^*)_{i}
\end{equation} \end{equation}\\
M is a lower-triangular matrix where the diagonal elements are the
eigenvalues eigenvalues\\
$$ \mu_i = 1 - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} > 0 $$
\\
since in our model $ \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} = \frac{1-0.00032}{(2i-1)^{1.64}} <1$ for all i.
...
