\[ \left\{ \begin{aligned} \frac{\mathrm{d}c_m}{\mathrm{d}t}&=-r_ac_{m,eq}\\ \frac{\mathrm{d}c_p}{\mathrm{d}t}&=-r_dc_{p,eq}+a_1\frac{\mathrm{d}m}{\mathrm{d}t}\\ \frac{\mathrm{d}c_h}{\mathrm{d}t}&=mr_h\\ \frac{\mathrm{d}m}{\mathrm{d}t}&=m(1-\frac{m}{K(c_{h,eq})}) \end{aligned} \qquad\qquad \begin{aligned} c_m(0)&=c_{m0}\\ c_p(0)&=c_{p0}\\ c_h(0)&=c_{h0}\\ m(0)&=m_0 \end{aligned} \right. \]