\begin{equation}\label{rate mass chsange}
\begin{aligned}
&c_{ij}\int_0^L \frac{\partial}{\partial t}(n_{ij}+m_{ij})(x,t)\,dx
=c_{ij}\left(\int_0^L q_{ij}(x,t)\,dx+\int_{(0,x_3)\cup(x_4,L)}\frac {\gamma_1 n_{ij}(2\beta-\gamma_2  n_{ij})}{(\beta-\gamma_2  n_{ij})^2}q_{ij}(x,t)\,dx\right).
\end{aligned}
\end{equation}
