A mathematical model to describe the cross-feeding dynamics in an algal-bacterial co-culture

The non-dimensional model ODEs:
\(\dot{a}=\eta\ a\ \left(\frac{v}{1+v}\right)\ \left(1-a\right)\)
\(\dot{b}=b\ \left(\frac{c_o}{1+c_o}\right)\ \left(1-b\right)\)
\(\dot{c_o}=r_e\ -\ \left(1-X\right)r_u\)
\(\dot{c_i}= -r_p - X r_u + r_r\)
\(\dot{v} = s_v b - r_v\)
The conversion between cell density and carbon biomass in dimensionless units and assuming a constant linear relationship.
\(c_b = k_{b,c}\ b\)
\(c_a = k_{a,c}\ a\)
\(c_{a,s} = \phi_s\ c_a\)
The non-dimensional model rates
\(r_e=\left(1-\phi_s\right)\ s_c\ a\)  
\(r_s=\phi_s\ k_{a,c}\ \dot{a}\)
\(r_p=k_{a,c}\ \dot{a} + r_e\)
\(r_u=\frac{k_{b,c}\ b}{\eta}\left(\frac{c_o}{1+c_o}\right)\)
\(r_r = \left[1 - \eta\left(1-b\right)\right]r_u\)
\(r_v=k_{a,v}\ \epsilon\ a \left(\frac{v}{1+v}\right)\)