D_{B}

. Their additional chemotaxis flux is

J_{B, chem} = \frac{χ _{B} B}{( N _{0} + N ) ^{2}} N_{X} .

It points up the nutrient concentration gradient, and its negative spatial derivative enters the balance equation. The positive

N_{0}

prevents singular sensitivity at zero nutrient concentration; increasing the background nutrient concentration reduces the drift response to a fixed gradient. The logistic growth equation term

r B (1 - B / K)

represents net reproduction with low-population growth rate

r

K

D_{N}

. Its production term

a B^{2} / (B_{0}^{2} + B^{2})

is an increasing, bounded response to the bacteria, reaching half its maximum

a

B = B_{0}

. It has the algebraic form of a Hill equation with exponent two; the equation alone does not specify a molecular mechanism. The term

d BN

removes nutrient at a rate proportional to both bacteria abundance and nutrient concentration.

The full fluxes are

J_{B} = - D_{B} B_{X} + \frac{χ _{B} B}{( N _{0} + N ) ^{2}} N_{X}, J_{N} = - D_{N} N_{X} .

The homogeneous Neumann boundary conditions make both fluxes vanish at the endpoints, so neither bacteria nor nutrient crosses the boundaries. The initial conditions specify their starting spatial distributions. This model includes nutrient production by bacteria, rather than a separate external supply term.