ROUGH DRAFT authorea.com/48305
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  • Competition Model

    The original model

    The equations in the original model are as follows:
    \[\frac{dA_i}{dt} = A_i (K_a - A_i + \frac{\alpha^{PA}}{\lambda^q} \sum_{j<=i} P_j ) \label{denklem1}\] \[\frac{dP_i}{dt} = P_i (K_p - P_i + \alpha^{AP} A_i) \label{denklem2}\]

    Model with competition

    We insert the between-species competition terms as:
    \[\beta = \epsilon T \nonumber\]
    where T = INSERT THE MATRIX HERE!!
    The modified equations for the equilibrium solutions become:
    \[A_i^* + \sum_{j} \beta_{ij}A_j^* - \frac {\alpha^{AP}}{\lambda_i^q} \sum_{j<=i} P_j^* = K_a \label{denklem3}\] \[P_i^* = \alpha^{AP} A_i^* + K_p \label{denklem4}\]
    When we plug in the \ref{denklem4} into \ref{denklem3}, we get
    \[A_i^* + \sum_{j} \beta_{ij} A_j* - \frac{\alpha^{AP}}{\lambda_i^q} \sum_{j<=i} (\alpha^{AP} A_j^* + K_p ) = K_a \nonumber\]

    \[A_i^* + \sum_{j} \beta_{ij} A_j* - (\frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} \sum_{j<=i} \alpha^{AP} A_j^*) - \frac{\alpha^{PA}}{\lambda_i^q} \sum_{j<=i} K_p = K_a \label{denklem5}\]


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

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