Travelling bands of microorganisms, chemotactically directed, move into a food source, consuming it as they go. A model for this is given by

$$b_{t}=\frac{\partial}{\partial x}\left[D b_{x}-\frac{b \chi}{a} a_{x}\right], \quad a_{t}=-k b$$

where $b(x, t)$ and $a(x, t)$ are the bacteria and nutrient respectively and $D, \chi$, and $k$ are positive constants. Look for travelling wave solutions, as functions of $z=x-c t$ where $c$ is the wave speed, with the boundary conditions $b \rightarrow 0$ as $|z| \rightarrow \infty, a \rightarrow 0$ as $z \rightarrow-\infty$, $a \rightarrow 1$ as $z \rightarrow \infty$. Hence show that $b(z)$ and $a(z)$ satisfy

$$b^{\prime}=\frac{b}{c D}\left[\frac{k b \chi}{a}-c^{2}\right], \quad a^{\prime}=\frac{k b}{c}$$

where the prime denotes differentiation with respect to $z$. Integrating $d b / d a$, find an algebraic relationship between $b(z)$ and $a(z)$.

In the special case where $\chi=2 D$ show that

$$a(z)=\left[1+K e^{-c z / D}\right]^{-1}, \quad b(z)=\frac{c^{2}}{k D} e^{-c z / D}\left[1+K e^{-c z / D}\right]^{-2}$$

where $K$ is an arbitrary positive constant which is equivalent to a linear translation; it may be set to 1 . Sketch the wave solutions and explain the biological interpretation.