Travelling waves
A travelling wave is a spatial pattern that moves through space while keeping approximately the same shape. Instead of the profile changing everywhere independently, the whole pattern is translated at a constant speed.
What does a travelling wave look like?
Suppose a population front moves to the right. At later times the profile appears in a new position, but its shape is nearly unchanged.
The travelling-wave form
We look for solutions of the form
\[\boxed{u(x,t)=U(z),\qquad z=x-ct.}\]Here \(U\) is the fixed wave shape and \(c\) is the wave speed.
The new variable \(z\) measures position relative to the moving wave.
Why use z = x - ct?
If the wave moves right at speed \(c>0\), then a fixed feature that begins at \(x_0\) moves to
\[x=x_0+ct.\]Therefore
\[x-ct=x_0.\]So the same feature has a constant value of \(z\) as the wave moves.
Moving frame intuition
Imagine standing beside a road and watching a wave pass. In the laboratory frame, the profile moves. Now imagine travelling alongside it at the same speed. Relative to you, the wave shape no longer moves.
Travelling-wave analysis does exactly this mathematically.
From PDE to ODE
Consider a reaction–diffusion equation
\[u_t=D u_{xx}+f(u).\]Set
\[u(x,t)=U(z),\qquad z=x-ct.\]By the chain rule,
\[u_t=-cU'(z),\] \[u_x=U'(z),\qquad u_{xx}=U''(z).\]Substituting gives
\[-cU'=DU''+f(U).\]Rearranging,
\[\boxed{DU''+cU'+f(U)=0.}\]Why this reduction is useful
The original problem depends on both space and time. After the travelling-wave substitution, we solve for a function of one variable \(z\).
So a PDE becomes an ODE describing the wave shape.
Travelling fronts connect different states
Many biological travelling waves are fronts rather than pulses. A front connects one long-term state behind the wave to another state ahead of it.
For example, an invasion front may satisfy
\[U(-\infty)=K,\qquad U(+\infty)=0.\]Behind the wave, the population is established. Ahead of it, the habitat is nearly empty.
Front versus pulse
| Travelling front | Travelling pulse |
|---|---|
| connects two different states | returns to the same background state |
| often models invasion or replacement | often models a moving localised signal |
| example: occupied to empty habitat | example: excitation pulse in tissue |
Front and pulse visually
The Fisher–KPP travelling wave
For
\[u_t=D u_{xx}+ru\left(1-\frac{u}{K}\right),\]the travelling-wave equation becomes
\[DU''+cU'+rU\left(1-\frac{U}{K}\right)=0.\]For an invasion to the right, the classical front connects
\[U(-\infty)=K,\qquad U(+\infty)=0.\]Minimum speed in Fisher–KPP
For the classical Fisher–KPP equation, monotone travelling fronts exist for
\[\boxed{c\ge2\sqrt{rD}.}\]The wave speed depends on both dispersal \(D\) and low-density growth \(r\).
For sufficiently localised initial conditions under the standard assumptions, the dynamics typically select the minimum speed
\[c^*=2\sqrt{rD}.\]Why the leading edge controls speed
At the leading edge, \(U\) is very small. The nonlinear term can be approximated by
\[f(U)\approx f'(0)U.\]In Fisher–KPP,
\[f'(0)=r.\]This low-density growth, together with diffusion, controls the classical spreading speed.
Travelling-wave speed from front position
If a front moves at constant speed, a chosen level of the wave, such as the point where \(u=K/2\), moves approximately according to
\[x_f(t)=x_0+ct.\]So plotting front position against time gives an approximately straight line whose slope is \(c\).
Direction of travel
If \(c>0\), \(U(x-ct)\) moves to the right. If \(c<0\), it moves to the left.
The sign of \(c\) therefore gives direction, while \(|c|\) gives speed.
Travelling waves with advection
Suppose directed transport is also present:
\[u_t=D u_{xx}-v u_x+f(u).\]A travelling wave can still be sought in the form \(U(x-ct)\). Advection shifts the observed propagation speed because the wave is being carried by the background flow as well as generated by reaction–diffusion dynamics.
Travelling waves in epidemic models
Spatial epidemic systems can support travelling infection fronts. For example, susceptible and infectious densities may vary through space and local transmission can create infection behind an advancing front.
Unlike the scalar Fisher–KPP equation, epidemic models usually involve several coupled equations, so the wave structure can be more complicated.
Travelling waves in genetics
The original Fisher model described spread of an advantageous gene through space. Selection provides local growth of the advantageous type while dispersal spreads it geographically.
The wave then represents replacement of one genetic composition by another.
Travelling waves in physiology
Travelling pulses appear in excitable biological systems such as nerve or cardiac tissue. Here the travelling object is not a population invasion front but a propagating electrical or chemical signal.
This shows that travelling-wave mathematics is broader than ecology.
Wave speed is not always constant
Real fronts can accelerate, slow down or become irregular if the environment varies, dispersal is nonlocal, parameters change, or the geometry is complex.
The travelling-wave assumption is therefore an idealisation representing approximately constant-shape, constant-speed propagation.
Spatial heterogeneity
If
\[D=D(x),\qquad r=r(x),\]the environment itself changes with location. A single fixed profile \(U(x-ct)\) may then no longer describe the motion exactly.
Fronts may speed up, slow down, become distorted or become pinned.
Wave pinning
A front can sometimes stop moving when spatial heterogeneity or nonlinear interactions balance the forces driving propagation.
Then the system has a stationary spatial interface rather than a travelling one.
Stochastic travelling fronts
At the leading edge of a biological invasion, density may be very low. Individual births, deaths and movements can then be strongly random.
A deterministic travelling wave describes an average or continuum-scale pattern, while a stochastic model can show variable speed, irregular front shape or extinction of the leading population.
Travelling waves versus advection
These ideas should not be confused. In pure advection, a profile is transported because an external velocity carries it. In a reaction–diffusion travelling wave, the front can propagate even without a bulk flow because growth and diffusion regenerate the pattern ahead.
Travelling waves versus diffusion alone
Diffusion alone generally broadens a profile rather than preserving a fixed travelling shape. A reaction term can balance that spreading to produce a coherent front.
How numerical simulations reveal waves
Plot \(u(x,t)\) at several times. If later curves are nearly horizontal translations of one another, a travelling-wave description may be appropriate.
You can also track a fixed density level through time and estimate the speed from its position.
A practical workflow
Identify the PDE and the biological states the wave might connect. Introduce \(z=x-ct\). Calculate the transformed derivatives and reduce the PDE to an ODE. Specify boundary conditions at \(z\to\pm\infty\). Determine which speeds are mathematically and biologically admissible. Finally, check numerically whether the spatial profile actually approaches a constant-shape, constant-speed wave.