$d_1l_1\displaystyle\frac{d}{dt}c_{11}=\displaystyle\frac{l_1}{l_1+l_2}(j(j_{s0},c_{s0},c_0,c_{11}) - j(j_{s1},c_{s1},c_{11},c_{21})) -\displaystyle\frac{d_1}{d}D(c_{11}-c_{12})$

d_1l_1dc_11/dt=l_1(j(j_s0,c_s0,c_0,c_11)-j(j_s1,c_s1,c_11,c_21))/(l_1+l_2)-(d_1/d)D(c_11-c_12)

$d_1l_2\displaystyle\frac{d}{dt}c_{12}=\displaystyle\frac{l_2}{l_1+l_2}j(j_{s0},c_{s0},c_0,c_{12})+\displaystyle\frac{d_1}{d}D(c_{11}-c_{12})-\displaystyle\frac{l_2}{h}D(c_{12}-c_{22})$

d_1l_2dc_12/dt=l_2j(j_s0,c_s0,c_0,c_12)/(l_1+l_2)+(d_1/d)D(c_11-c_12)-(l_2/h)D(c_12-c_22)

$d_2l_1\displaystyle\frac{d}{dt}c_{21}=\displaystyle\frac{l_1}{l_1+l_2}(j(j_{s1},c_{s1},c_{11},c_{21})-j(j_{s2},c_{s2},c_{21},c_3))-\displaystyle\frac{d_2}{d}D(c_{21}-c_{22})$

d_2l_1dc_21/dt=l_1(j(j_s1,c_s1,c_11,c_21)-j(j_s2,c_s2,c_21,c_3))/(l_1+l_2)-(d_2/d)D(c_21-c_22)

$d_2l_2\displaystyle\frac{d}{dt}c_{22}=\displaystyle\frac{l_2}{h}D(c_{12}-c_{22})+\displaystyle\frac{d_2}{d}D(c_{21}-c_{22})-\displaystyle\frac{l_2}{l_1+l_2}j(j_{s2},c_{2s},c_{22},c_3)$

d_2l_2dc_22/dt=(l_2/h)D(c_12-c_22)+(d_2/d)D(c_21-c_22)-l_2j(j_s2,c_2s,c_22,c_3)/(l_1+l_2)

$j(j_s,c_s,c_1,c_2)=\displaystyle\frac{j_s}{c_s}(c_1-c_2)\left(\displaystyle\frac{c_s}{c_1+c_2}\theta(c_1+c_2 - c_s)+\theta(c_s-c_1-c_2)\right)$

j(j_s,c_s,c_1,c_2)=(j_s/c_s)(c_1-c_2)(c_s theta(c_1+c_2 - c_s)/(c_1+c_2)+theta(c_s-c_1-c_2))

$\displaystyle\frac{dN}{dt}=\displaystyle\frac{1}{l_1+l_2}((l_1j(j_{s0},c_{s0},c_0,c_{11})+l_2j(j_{s0},c_{s0},c_0,c_{12}))-(l_1j(j_{s2},c_{s2},c_{21},c_3))+l_2j(j_{s2},c_{s2},c_{22},c_3)))$

dN/dt=(l_1j(j_s0,c_s0,c_0,c_11)+l_2j(j_s0,c_s0,c_0,c_12))-(l_1j(j_s2,c_s2,c_21,c_3))+l_2j(j_s2,c_s2,c_22,c_3)))/(l_1+l_2)