Fisher–KPP equation
The Fisher–KPP equation is one of the classical models of biological invasion. It combines two simple mechanisms: individuals spread through space, and the population grows locally.
The equation
\[\boxed{\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}+ru\left(1-\frac{u}{K}\right).}\]Here \(u(x,t)\) is population density at position \(x\) and time \(t\).
| Quantity | Meaning |
|---|---|
| \(D>0\) | diffusion coefficient: strength of spatial spreading |
| \(r>0\) | intrinsic local growth rate |
| \(K>0\) | local carrying capacity |
Where does the equation come from?
Start with logistic growth at one location:
\[\frac{du}{dt}=ru\left(1-\frac{u}{K}\right).\]This describes growth but contains no spatial movement.
Now add diffusion:
\[D\frac{\partial^2u}{\partial x^2}.\]The result is the Fisher–KPP equation. It is therefore a reaction–diffusion equation:
\[\text{change}=\text{spatial spreading}+\text{local growth}.\]What each term does
The diffusion term smooths spatial differences and moves population into neighbouring empty regions. The logistic term makes a small positive population grow when \(u<K\), while growth slows as \(u\) approaches \(K\).
Neither mechanism alone gives the same invasion behaviour. Logistic growth alone cannot colonise an empty location if no individuals reach it. Diffusion alone spreads the existing population but does not reproduce it.
How an invasion front forms
Imagine a population occupying the left side of a habitat while the right side is initially empty. Diffusion moves a small number of individuals just beyond the occupied region. Because density there is low, logistic growth is approximately exponential. Those new individuals reproduce, diffusion carries some farther ahead, and the process repeats.
The boundary between the populated and unpopulated regions therefore advances through space.
Travelling-wave idea
A travelling wave is a profile that moves without changing its basic shape. Write
\[u(x,t)=U(z),\qquad z=x-ct,\]where \(c>0\) is the wave speed.
As time increases, keeping the same value of \(z\) requires \(x\) to increase at speed \(c\). Thus \(U(x-ct)\) represents a wave moving to the right.
Turning the PDE into an ODE
For
\[z=x-ct,\]the derivatives become
\[u_t=-cU'(z),\qquad u_{xx}=U''(z).\]Substituting into the Fisher–KPP equation gives
\[-cU'=DU''+rU\left(1-\frac{U}{K}\right),\]or
\[\boxed{DU''+cU'+rU\left(1-\frac{U}{K}\right)=0.}\]The travelling-wave substitution has converted the spatial PDE into an ODE for the wave shape \(U(z)\).
What states does the wave connect?
For an invasion moving into empty habitat, the typical wave satisfies
\[U(-\infty)=K,\qquad U(+\infty)=0.\]Far behind the front, the population is close to carrying capacity. Far ahead of the front, the habitat is almost empty.
Why the leading edge is important
At the front's leading edge, population density is very small:
\[u\approx0.\]Then
\[1-\frac{u}{K}\approx1,\]so logistic growth becomes approximately
\[ru\left(1-\frac{u}{K}\right)\approx ru.\]Therefore the leading edge is approximately governed by the linear equation
\[u_t\approx Du_{xx}+ru.\]Deriving the minimum wave speed
At the leading edge, look for an exponentially decreasing travelling profile
\[U(z)=e^{-\lambda z},\qquad \lambda>0.\]Then
\[U'=-\lambda U,\qquad U''=\lambda^2U.\]Substituting into the linearised travelling-wave equation gives
\[D\lambda^2-c\lambda+r=0.\]Solving for \(c\),
\[c=D\lambda+\frac{r}{\lambda}.\]The smallest possible value occurs at
\[\lambda=\sqrt{\frac{r}{D}},\]which gives
\[\boxed{c_{\min}=2\sqrt{rD}.}\]What does the minimum speed mean?
For the classical Fisher–KPP equation, monotone travelling invasion fronts exist for
\[\boxed{c\ge 2\sqrt{rD}.}\]For sufficiently localised initial populations under the standard assumptions, the asymptotic invasion speed selected by the dynamics is typically the minimum speed
\[c^*=2\sqrt{rD}.\]Why both growth and movement affect speed
The formula
\[c^*=2\sqrt{rD}\]has a clear biological interpretation. Increasing \(D\) gets individuals into new territory faster. Increasing \(r\) allows low-density populations at the leading edge to establish faster.
The wave speed therefore depends on both dispersal and reproduction.
Effect of changing r and D
| Change | Effect on classical wave speed |
|---|---|
| increase \(r\) | faster invasion |
| increase \(D\) | faster invasion |
| multiply \(r\) by 4 | speed doubles |
| multiply \(D\) by 4 | speed doubles |
This follows from the square-root dependence rather than a linear dependence.
What role does K play in wave speed?
For the classical Fisher–KPP equation, the minimum speed
\[2\sqrt{rD}\]does not contain \(K\). The reason is that speed selection is controlled by growth at the low-density leading edge, where the nonlinear carrying-capacity effect is negligible.
The parameter \(K\) still determines the populated density behind the invasion front.
Wave position through time
If the front travels at approximately constant speed \(c\), its position behaves approximately as
\[x_{\mathrm{front}}(t)\approx x_0+ct.\]Historical note
In 1937, Ronald Fisher introduced this reaction–diffusion equation in a model for the spatial spread of an advantageous gene. In the same year, Kolmogorov, Petrovskii and Piskunov studied a broad class of related reaction–diffusion equations. This is the origin of the name Fisher–KPP.
Dimensionless form
The equation can be simplified by scaling density, space and time. Let
\[w=\frac{u}{K},\qquad \tau=rt,\qquad \xi=x\sqrt{\frac{r}{D}}.\]Then the equation becomes
\[\boxed{w_\tau=w_{\xi\xi}+w(1-w).}\]This shows that the same basic dynamics can describe many systems after appropriate rescaling.
Why this model is important
The Fisher–KPP equation is one of the simplest models that links individual dispersal, local population growth and a measurable macroscopic invasion speed.
It provides a foundation for more complicated models of ecological range expansion, gene spread, microbial colonies, epidemics and biological invasions.
Biological assumptions
The classical model assumes local logistic growth, ordinary diffusion, homogeneous space and constant parameters. It also treats population density continuously.
These assumptions make the mathematics clear, but they may be unrealistic for strongly heterogeneous habitats, long-distance dispersal, discrete small populations or species with low-density growth difficulties.
What happens with an Allee effect?
The Fisher–KPP model assumes that a very small positive population can grow because
\[f'(0)=r>0.\]Species with a strong Allee effect may instead decline below a critical density. In that case, invasion dynamics and wave speed can differ substantially from Fisher–KPP behaviour.
What if dispersal is not ordinary diffusion?
Some organisms make rare long-distance movements. Standard diffusion may then underestimate how rapidly the range expands.
Alternative dispersal kernels, integrodifference equations or other nonlocal models can be more appropriate.
Spatial heterogeneity
If habitat quality changes with location, parameters can depend on space:
\[r=r(x),\qquad D=D(x),\qquad K=K(x).\]The front may then accelerate, slow down, become distorted or fail to invade some regions.
Fisher–KPP and epidemic spread
The same reaction–diffusion idea helps build spatial epidemic models: movement carries infection geographically while local transmission creates new infections.
However, a realistic epidemic generally requires several compartments rather than one logistic population variable, so the Fisher–KPP equation is better viewed as a foundational invasion model than as a complete epidemic model.
Deterministic interpretation
The Fisher–KPP equation describes a continuous deterministic density. At the very low-density leading edge of a real invasion, there may be only a few individuals and demographic randomness can matter strongly.
Stochastic spatial models can therefore give different establishment probabilities and front behaviour when populations are small.
A practical workflow
Identify the population density \(u(x,t)\). Interpret \(D\), \(r\) and \(K\) biologically. Separate the movement and growth terms. Examine the low-density leading edge. Use the travelling coordinate \(z=x-ct\) to study wave shape. Interpret \(2\sqrt{rD}\) as the classical minimum wave speed, and finally check whether logistic growth and ordinary diffusion are reasonable for the biological system.