= Solution
Use $K$ as the population scale, $N_0$ as the <nutrient> <concentration> scale, $r^{-1}$ as the time scale, and $L$ as the length scale. The <nondimensionalization> is
$$
\boxed{B=Kb,\qquad N=N_0n,\qquad T=t/r,\qquad X=Lx.}
$$
For example, the <chemotaxis> term becomes
$$
-\chi_B\partial_X\left(\frac{B}{(N_0+N)^2}N_X\right)
=-\frac{\chi_BK}{N_0L^2}\partial_x\left(\frac b{(1+n)^2}n_x\right).
$$
Dividing the population equation by $rK$ and the <nutrient> equation by $rN_0$ gives the requested system with the following six <dimensionless quantities>:
$$
\boxed{d_b=\frac{D_B}{rL^2},\qquad d_n=\frac{D_N}{rL^2},\qquad
\chi_b=\frac{\chi_B}{rL^2N_0},\qquad
\lambda=\frac a{rN_0},\qquad b_0=\frac{B_0}{K},\qquad \mu=\frac{dK}{r}.}
$$
The rescaled domain is $0<x<1$, $t>0$. Its <boundary conditions> and <initial conditions> are
$$
\boxed{b_x(0,t)=b_x(1,t)=n_x(0,t)=n_x(1,t)=0,}
$$
$$
\boxed{b(x,0)=\frac{B_{\mathrm{initial}}(Lx)}K,\qquad
n(x,0)=\frac{N_{\mathrm{initial}}(Lx)}{N_0},\qquad 0\le x\le1.}
$$
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