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An activator in an activator-inhibitor reaction–diffusion system is a species whose local feedback enhances activation. Near a homogeneous equilibrium, positive self-feedback is reflected in the corresponding diagonal entry of the reaction Jacobian matrix. Cross-coupling to an inhibitor and differing diffusion rates can generate a Turing instability.

This nondimensional two-variable reaction–diffusion system combines a fast cubic local response with slow negative feedback. With $D>0$, $0<\alpha <1$, $\beta >0$ and $\epsilon \ll 1$, its spatially uniform rest state at $I=0$ is linearly stable: the Jacobian has trace $-\alpha -\epsilon \beta$ and determinant $\epsilon (1+\alpha \beta )$. For suitable parameters a finite perturbation can nevertheless cause a large excursion followed by recovery. Frozen recovery gives a bistable cubic reaction-diffusion equation; the full coupled travelling-wave boundary-value problem determines whether front pairs form a pulse. This is a generic excitable-medium formulation of the FitzHugh-Nagumo model, without assigning any biological variable or application.

An inhibitor in an activator-inhibitor reaction–diffusion system suppresses the activator. Its diffusion and reaction feedback can stabilize a spatially uniform state while allowing some nonzero wavenumbers to grow through a Turing instability.

Choose the nondimensionalizationThe chain rule gives ${\partial }_{\tau }={\beta }_u{\partial }_t$ and ${\partial }_x^2=({\beta }_u/D_u){\partial }_s^2$. Dividing the first equation by ${\beta }_u$ and the second by ${\beta }_v$ gives the required scaled reaction–diffusion system. Positive degradation and diffusion constants are assumed.

The reaction–diffusion system for amoebae contains local reproduction $\alpha n_0^{2}n$ and density-dependent loss $-\alpha n^3$, giving carrying density $n_0$. The term ${\nabla }^{2}n$ describes diffusion with coefficient one. The chemotaxis flux is $J_{\mathrm{chem}}=\chi (n)n\nabla a$, so its contribution to the conservation equation is $-\nabla \cdot J_{\mathrm{chem}}$. Attraction towards greater chemical concentration requires $\chi (n)>0$ at the relevant positive densities; negative values would mean repulsion, and zero removes the chemotactic response.

In the chemical equation, $\beta n$ is secretion proportional to amoeba density, $-\gamma a$ is first-order chemical degradation, and $D{\nabla }^{2}a$ is chemical diffusion. The cell reaction term permits births and deaths, so the total number of amoebae is not conserved.

For linear stability analysis of a general stable two-species reaction–diffusion system, take Fourier modes proportional to $e^{\lambda t+ikx}$. Their matrix is $J_k=J-k^2\mathrm{diag}(D_u,D_v)$. Put $s=k^2$, $\tau =\mathrm{tr}J$, $\Delta =\det J$ and $B=D_v{a}_{11}+D_u{a}_{22}$. ThenThe full system is stable precisely when ${\tau }_k<0$ and ${\Delta }_k>0$ for every allowed mode. If the homogeneous state is stable, the trace already decreases with $s$; only the determinant can change sign. If $B\le 0$, its minimum for $s\ge 0$ is at zero. If $B>0$, it is at $s_{\ast }=B/(2D_u{D}_v)$. Thus the two-species Turing criterion for instability on a continuum of wavenumbers isStrict stability has $B<2\sqrt{D_u{D}_v\Delta }$ together with the homogeneous conditions. Equality at positive $B$ marks a stationary neutral mode. At that first onset,Beyond threshold, the growing band lies between the positive roots $s_{\pm }=[B\pm \sqrt{B^2-4D_u{D}_v\Delta }]/(2D_u{D}_v)$, as long as the homogeneous trace stays negative.

For the Brusselator, $B=D_v(b-1)-D_u{a}^2$. Write $r=D_u/D_v$. The stationary determinant threshold gives the Turing threshold of the Brusselator in dimensional diffusivities:For this to be a genuine diffusion-driven loss of stability, it must occur before the homogeneous Hopf bifurcation threshold. The Brusselator Turing-before-Hopf diffusivity condition isIndeed $a^{2}r+2a\sqrt{r}<a^2$ is equivalent to $\sqrt{r}<(\sqrt{1+a^2}-1)/a$. With this strict inequality, $b_T<b<b_H$ contains a spatially unstable but homogeneously stable interval. If the ratio condition fails, the first linear instability as $b$ increases is homogeneous at $b_H$, rather than a Turing instability; equality gives simultaneous neutral homogeneous and finite-wavenumber modes. Equal diffusivities cannot produce a Turing instability.

These continuum formulas assume the onset wavenumber is available. On a finite domain with a specified discrete mode set, the discrete-mode Turing threshold for the Brusselator isprovided this minimum is below $b_H$. The question supplies no boundary geometry, so the continuous-mode onset above is the usual answer; this last expression states how a finite-size restriction changes it.

With the travelling wave coordinate $\xi =x-ct$, the reaction–diffusion system becomesAt the advancing leading edge the healthy-tissue density is small. Put $f(\xi )\sim e^{-\lambda \xi }$ with $\lambda >0$. The linearization at $f=0$ givesso a nonoscillatory decaying front requires real positive roots,The pulled travelling front selects the Fisher-KPP minimum wave speed, and therefore

For the initial wound $0<x<W$, two mirror-image fronts enter from its endpoints. The left front moves right from $x=0$ and the right front moves left from $x=W$, each with speed approximately $c_{\min }$. A sequence of sketches would show the initial rectangular depression becoming two smooth monotone transition layers, the zero-density plateau shrinking, and the two layers meeting near $x=W/2$ before the profile relaxes to $S=1$. Their meeting time isThere is an additional local relaxation time of order $1/r$ if “close to one” is assigned a fixed numerical tolerance, so the displayed estimate is the dominant healing time when $W$ is large compared with the front width $\sqrt{D/r}$.

Write the reaction terms asA nonzero homogeneous equilibrium satisfies $v_{\ast }=\beta u_{\ast }$ and $1+u_{\ast }-\gamma v_{\ast }=0$, henceFor physically positive populations it exists exactly when

The reaction Jacobian matrix at this equilibrium isIts trace and determinant areThe equilibrium is therefore stable to spatially uniform perturbations when

For a spatial Fourier mode of wavenumber $k$, put $q=k^2$. The linearized reaction–diffusion system has matrixIts trace is smaller than $\mathrm{tr}J$, whileThe two-species diffusion-driven instability criterion says that this upward-opening quadratic becomes negative for some $q>0$ precisely whenCombining all conditions, a Turing instability may occur in the region

At onset the discriminant vanishes, and the double root isThe threshold relation givesso the critical wavenumber isIf $d=1$, uniform stability requires $\beta >1$ whereas diffusion-driven instability requires $d>\beta$. These inequalities are incompatible, so the Turing region vanishes.

TODO: biological interpretations and organism-specific propagation are user-deferred. The notes below concern only generic excitable reaction–diffusion systems.

A front solution joins different asymptotic states in space. In a reaction–diffusion system, local nonlinear dynamics select possible states while diffusion couples neighboring points. A travelling wave with profile $u(x,t)=U(\xi )$, $\xi =x-ct$, converts a partial differential equation into a boundary value problem for the shape and speed. A pulse instead returns to the same resting state on both sides and can contain leading and trailing fronts.

A useful nondimensional cubic fast-slow reaction-diffusion model, in a convention of the FitzHugh-Nagumo model, iswhere $D>0$, $\epsilon \ll 1$, and $\beta >0$. Here $u$ is a fast activation variable, $w$ a slow negative-feedback variable and $I$ a generic drive; no physiological variable is assigned. Without spatial coupling, nullclines are $w=f(u)+I$ and $w=u/\beta$. At $I=0$, the resting state $(0,0)$ has Jacobian matrixIt has linear stability. With appropriate parameter choices a sufficiently large perturbation makes a fast excursion to the other outer branch of the cubic, followed by slow feedback and return. Thus a stable rest state can be excitable without being linearly unstable. Other parameters or applied drive can yield multiple equilibria or oscillations; these behaviours should not be asserted for every parameter set.

The fast-subsystem front can be obtained explicitly by freezing $w=I=0$. For the bistable cubic reaction-diffusion equation, $u=0,1$ are stable and $u=\alpha$ is unstable. The decreasing logistic front of a bistable cubic equation isTo check it, $U^{\prime }=-U(1-U)/\sqrt{2D}$ and substitution into $D{U}^{\prime \prime }+c{U}^{\prime }+f(U)=0$ cancels every term. For $\alpha <1/2$, the high state advances to the right; for $\alpha >1/2$, the direction reverses. At $\alpha =1/2$ the front is stationary. More generally, multiplying the travelling-wave equation by $U^{\prime }$ and integrating yieldsThis links propagation direction to the bias between the two stable states and recovers Maxwell balance for a scalar reaction-diffusion front at zero speed. It is a bistable selected-speed problem, rather than a universal Fisher-type minimum-speed formula.

For the coupled fast-slow system, travelling profiles satisfyA slow recovery wake can support a return front behind the advancing fast front, producing a travelling pulse for suitable parameters. Its width reflects the slow recovery time and propagation speed, while each sharp edge has the faster diffusion–reaction scale. The frozen-recovery speed above is a controlled subsystem result, not an exact speed for every coupled pulse. A definite pulse speed requires parameters and connecting boundary condition; none are specified by this brief-note question.

A generic chemical reaction–diffusion system couples local kinetics to spatial transport:A homogeneous stationary state satisfies $f=g=0$. Let its reaction Jacobian be $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Local well-mixed stability requires $a+d<0$ and $ad-bc>0$. A Fourier perturbation of wavevector magnitude $q$ evolves under $J-q^2\mathrm{diag}(D_u,D_v)$, whose determinant isSince its trace remains negative, a diffusion-driven stationary instability appears if the determinant becomes negative for some allowed wavevector. In a spatially large domain the conditions areTogether with reaction-only stability, these give a finite unstable wavelength range. Equal diffusion coefficients cannot generate this instability: they simply shift each reaction eigenvalue by $-D{q}^2$. On a finite domain, an admissible spatial eigenmode must actually lie in the unstable band. Nonlinearities then determine saturation and whether a predicted pattern is stable, so a linear unstable band alone does not calculate a final pattern.^[ref]

Other kinetics support travelling fronts or oscillatory waves rather than a stationary pattern. For instance, the scalar abstract equation ${\partial }_{t}u=D{\partial }_x^{2}u+ru(1-u/u_s)$ has a pulled-front speed $2\sqrt{Dr}$ for a sufficiently steep invading initial profile. Here it is used as an abstract kinetics-and-diffusion example, without a biological interpretation. The ratio of diffusion time $L^2/D$ to reaction time characterizes whether spatial mixing is rapid enough to justify a well-mixed model.

For positive concentrations and positive parameters, this reaction–diffusion system has homogeneous equilibrium $(u_{\ast },v_{\ast })=(1/b,1/b^2)$ and reaction Jacobian matrix $\left(\begin{array}{cc}b& -b^2\\ 2/b& -1\end{array}\right)$. Its trace is $b-1$ and determinant is $b$, making the equilibrium reaction-stable exactly when $0<b<1$. Stable complex eigenvalues permit damped oscillations, which must be distinguished from sustained periodic oscillations.

A spatially homogeneous equilibrium of a reaction–diffusion system is a constant state $u_{\ast }$ satisfying $f(u_{\ast })=0$. Since its spatial derivatives vanish, its stability to homogeneous perturbations is determined by the Jacobian matrix $Df(u_{\ast })$; diffusion can additionally destabilize nonconstant modes.