For spatially homogeneous perturbations write $q=1+u$ and $c=\eta +v$. The linearized reaction equations areThis triangular matrix has eigenvalues $-1$ and $-\kappa$, both negative. Hence the homogeneous steady state is linearly asymptotically stable against homogeneous perturbations. Even when $\kappa =1$, where the two eigenvalues coincide and a polynomial factor may occur in the matrix exponential, all such perturbations still decay.

The term $\alpha {\nabla }^{2}q$ is cell diffusion arising from random movement. Chemotaxis up the concentration gradient gives the cell flux $\beta q\nabla c$, whose negative divergence is the transport contribution $-\beta \nabla \cdot (q\nabla c)$. The reaction term $q(1-q)$ gives growth governed by the logistic differential equation with dimensionless carrying capacity one.

The chemoattractant itself diffuses, is produced by cells at rate $\kappa \eta q$, and decays at rate $\kappa c$. The homogeneous Neumann boundary conditions suppress the normal diffusive flux of each quantity. They also suppress the normal chemotactic cell flux because the normal derivative of $c$ vanishes. Thus the rectangular boundary is impermeable, and the spatially integrated cell count changes through cell proliferation rather than transport across that boundary.

Substitute $gb=c{g}^{\prime }$ into the bacterial profile equation. This giveswhich integrates once toAt the asymptotic constant states the profile derivative tends to zero. The state ahead of the travelling wave, where $b\to 0$ and $g\to 1$, therefore gives $C=c$. The state behind it, where $g\to 0$ and $b\to b_0$, gives $C=c{b}_0$. Since $c>0$,This first integral expresses the matching between biomass produced and biological growth factor consumed after the chosen nondimensionalization.

Take profiles $b=b(y)$ and $g=g(y)$ with $y=x-ct$, and use a prime for differentiation with respect to $y$. The chain rule gives $b_t=-c{b}^{\prime }$, $b_{xx}=b^{\prime \prime }$ and $g_t=-c{g}^{\prime }$. ThereforeThese are coupled ordinary differential equations for a right-moving travelling wave with speed $c>0$. A first-order formulation, using $v=b^{\prime }$, is $b^{\prime }=v$, $v^{\prime }=-cv-gb$, $g^{\prime }=gb/c$. In particular a positive profile has $g^{\prime }>0$, consistent with depleted substrate behind the travelling front and fresh biological growth factor ahead.

Choose a positive reference biological growth factor concentration $G_{ref}$, for example its concentration ahead of the travelling wave. SetwhereThe chain rule then transforms the equations intoEach displayed coefficient is one with the chosen scales. Hence the dimensionless system isThe free choice of $G_{ref}$ expresses the freedom to normalize the as-yet unspecified concentration scale.

An unbiased random walk of bacteria leads, on the continuum scale, to diffusion with coefficient $D$, represented by $D{B}_{XX}$. The cell proliferation rate $\nu GB$ is proportional both to the available biological growth factor concentration and to the bacterial density. Consumption of biological growth factor at rate $\mu GB$ gives its negative reaction term.

There is no diffusion term for $G$ because the biological growth factor is attached to the substrate rather than freely transported. There is also no chemotactic drift in this model: the specified motion is random, while the biological growth factor changes cell proliferation rather than directional movement. Both reaction terms vanish if either bacteria or biological growth factor are absent.

Write $P(N)=f(N,0)$. Its three positive zeros of a function are $1/2,1,2$, and expansion and differentiation giveThus its local minimum and maximum occur respectively atwith valuesThe function graph approaches the value $1$ as $N\downarrow 0$, decreases through its first zero to the minimum, increases through its second zero to the maximum, and then decreases through $2$ to $-\infty$. This fixes the shape and all axis crossings of the cubic polynomial.

Changing $a$ translates the function graph vertically without moving either stationary point. For $a=1/2$ both extrema are above the horizontal axis: their heights are approximately $0.421109$ and $0.764076$. For $a=-1/2$ both are below it, at approximately $-0.578891$ and $-0.235924$. The function graph for $a=1/2$ therefore has only its final positive-axis crossing, whereas the function graph for $a=-1/2$ has only its first positive-axis crossing. The original plots mark the extrema and also show the equilibrium point branches used below.