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The modified equations for the equilibrium solutions 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}  \end{equation}  \begin{equation}  P_i^* = \alpha^{AP} A_i^* + K_p   \label{denklem4}  \end{equation}  When we plug in the \ref{denklem4} into \ref{denklem3}, we 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  \label{denklem4}  \end{equation}