Travelling wave

For $c>2\eta$, the triangle $0\le a\le \eta$, $-\eta a\le b\le 0$ traps the unstable branch leaving the saddle $(\eta ,0)$ toward lower population. On its top edge $b^{\prime }<0$. On the sloping edge, $(b+\eta a{)}^{\prime }=\eta (c-\eta )a-a({\eta }^2-a^2)/(1+a^2)>0$, so that edge is also inward. The right edge has $a^{\prime }=b\le 0$. Monotonicity of $a$ inside the compact region leaves only $(0,0)$ as the forward limit. This proves a nonnegative monotone travelling front. For $0<c<2\eta$, the target equilibrium is a focus, so any approaching nonconstant trajectory crosses into negative population and cannot be a physical front.

A travelling front is a travelling wave joining distinct limiting states ahead of and behind a moving interface. Its spatial profile depends on a coordinate such as $x-ct$.

The dimensionless bacterial growth and substrate depletion system $b_t=b_{xx}+gb$, $g_t=-gb$ has travelling-wave equations $b\prime \prime +cb\prime +gb=0$, $cg\prime =gb$ in coordinate $y=x-ct$. For a front with $(b,g)\to (b_0,0)$ behind and $(b,g)\to (0,1)$ ahead, its first integral is $b\prime +c(b+g)=c$, forcing $b_0=1$. Positivity of the leading edge requires $c\ge 2$.

With inhibited bacterial growth $b_t=b_{xx}+gb(1-\eta b)$ and unchanged depletion $g_t=-gb$, a positive travelling wave leaving density $b_1$ obeys $c{b}_1=c-\eta {\int }_{\mathbb{R}}g{b}^{2}dy$. This follows by integrating both travelling-wave equations between their constant endpoint states. Thus $b_1<1$ for $\eta >0$: the yield behind the front is smaller than for uninhibited growth.

A propagating chemical front can persist when only one species diffuses. For $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv$, set $z=x-ct$, $k^2=1-\delta /2$, $c=-\delta /(2k)$ and $\beta v_0=2-\delta$, with $0<\delta <2$. The displayed profiles satisfy both equations because $U^{\prime }=kU(1-U)$ and $V^{\prime }/V=-2kU$. They connect $(u,v)=(0,v_0)$ to $(1,0)$: the second concentration changes by local reaction, even without its own diffusion.

A solitary wave is a spatially localized travelling wave that retains its profile as it propagates. The displayed form uses a zero background; other constant backgrounds are possible. A soliton additionally has the characteristic elastic interaction property with other solitons. The existence of a single localized travelling profile does not alone establish that stronger property.

A front is a spatial profile connecting different asymptotic states on its two sides. A travelling wave front moves with a constant speed; a stationary front has zero speed. In a one-dimensional stationary reduction it is a heteroclinic orbit between the corresponding homogeneous equilibria. Equal first-integral values are necessary for a stationary front in a scalar conservative spatial equation.

A stationary front solution of $u_t=u_{xx}+f(u)$ connecting equilibria $u_{-}$ and $u_{+}$ requires ${\int }_{u_{-}}^{u_{+}}f(s)ds=0$. Multiply the stationary equation by $u_x$ and integrate: the derivative term has zero endpoint contribution. This equal-potential condition is the Maxwell balance. For $f(u)=\mu u+\alpha u^3-u^5$, a front from zero to a nonzero constant requires $u_{+}^2=3\alpha /4$ and $\mu =-3{\alpha }^2/16$.

A mesa is a localized broad plateau joined to its surrounding state by two separated fronts. Its leading spatial average equals the plateau fraction times the plateau value; the transition layers give corrections proportional to their width divided by the domain length. A conserved mean can determine the fraction occupied by the plateau, as in the coexistence fraction of a conserved-field amplitude mesa.

Fisher's equation is a reaction-diffusion equation combining diffusion with logistic growth. For positive $D,r,K$, one-dimensional fronts joining $K$ to zero have minimum speed $2\sqrt{Dr}$. The phase-plane construction of a Fisher travelling front and the linearized leading edge explain this threshold.

In $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv+v_{xx}$, the line $v=1-u$ is invariant when $\beta =1-\delta$. On it $u$ satisfies Fisher's equation with growth coefficient $\delta$. The reduction also requires compatible initial and boundary data: parameter tuning alone does not reduce arbitrary two-field solutions. Positive concentrations and positive $\beta$ in this normalization require $0<\delta <1$.

For $u_t=u_{xx}+u(1-u^q)$, $q>0$, the displayed decreasing travelling wave has speed $c=(q+4)/\sqrt{2(q+2)}$, with $a>0$ setting its translation. Writing $Y=U^{q/2}$ gives $U^{\prime }=-kU(1-Y)$, $k=\sqrt{2/(q+2)}$, which verifies the wave equation by equating constant and $Y$ coefficients. The explicit speed increases with $q$, the leading-edge decay rate decreases, and the filled-state decay rate $q/\sqrt{2(q+2)}$ increases. Aligning fronts at half height distinguishes these changes from a translation.

For $u_t=u_{xx}+u(k-u)$ with $k>0$, a decreasing travelling wave connecting $k$ to $0$ satisfies $w^{\prime }=p$, $p^{\prime }=-cp-w(k-w)$. If $c\ge 2\sqrt{k}$, choose $a>0$ with $a(c-a)\ge k$. The triangle $0\le w\le k$, $-aw\le p\le 0$ is forward invariant: on its sloping side, $(p+aw{)}^{\prime }=w[a(c-a)-k+w]\ge 0$. The descending unstable manifold of the saddle equilibrium $(k,0)$ stays in this triangle and tends to $(0,0)$, producing a positive heteroclinic orbit. Below the threshold the origin is a stable spiral for positive $c$, forcing sign changes; at $c=0$ conserved energy also excludes the connection.

The substitution $U(x,t)=U(x-ct)$ turns time derivatives into $-c{U}^{\prime }$ and spatial diffusion into $U^{\prime \prime }$, reducing a reaction-diffusion PDE to an ODE system.

The linear spreading speed is obtained from exponentially decaying modes at an unstable state's leading edge and the condition that their spatial exponents become repeated.

This criterion concerns travelling-front solutions of Fisher's equation.

For diffusion coefficient $D$ and leading-edge growth rate $r>0$, a Fisher-KPP-type pulled front has minimum speed $2\sqrt{Dr}$.

A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.