Diffusion
Diffusion is a mathematical model of random local movement that produces large-scale spreading. Individual organisms or molecules may move in many directions, but when density is uneven the net effect is usually movement from more crowded regions toward less crowded regions.
Start with an uneven population distribution
Suppose many individuals are concentrated in one region and few are present nearby. Individuals move randomly left and right. Because there are more individuals in the crowded region, more individuals are available to leave that region than to enter it from the sparse region.
The net movement is therefore away from high density even though each individual movement may be random.
What is the spatial variable?
In one dimension, let
\[u(x,t)\]be population density at position \(x\) and time \(t\).
Because \(u\) depends on both space and time, we use partial derivatives.
The spatial gradient
The quantity
\[\frac{\partial u}{\partial x}\]measures how density changes as we move through space at one fixed time.
If the gradient is positive, density rises as \(x\) increases. If it is negative, density falls as \(x\) increases.
Flux: how much crosses a location?
Diffusive movement is described using flux. Flux measures the amount of population crossing a unit boundary per unit time.
Fick's law is
\[\boxed{J=-D\frac{\partial u}{\partial x}}.\]Here \(J\) is flux and \(D>0\) is the diffusion coefficient.
Why is there a minus sign?
Suppose density increases as we move to the right, so
\[\frac{\partial u}{\partial x}>0.\]Then
\[J<0.\]A negative flux means net movement to the left, toward the lower-density region.
If the density gradient is negative, the flux is positive, meaning net movement to the right.
Gradient and flux visually
From flux to the diffusion equation
Now consider a small interval. The population inside changes when more individuals enter than leave, or vice versa.
Conservation gives the local balance law
\[\frac{\partial u}{\partial t}=-\frac{\partial J}{\partial x}.\]Substitute Fick's law:
\[\frac{\partial u}{\partial t}=-\frac{\partial}{\partial x}\left(-D\frac{\partial u}{\partial x}\right).\]If \(D\) is constant,
\[\boxed{\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}}.\]What does the second derivative mean?
The second derivative
\[\frac{\partial^2u}{\partial x^2}\]measures curvature of the density profile.
At the top of a sharp peak, the curve bends downward, so the second derivative is negative. Then
\[\frac{\partial u}{\partial t}<0,\]and the peak decreases.
In a trough, the curve bends upward, so the second derivative is positive. Then local density increases.
Curvature intuition
What does the diffusion coefficient D mean?
The coefficient \(D\) controls how rapidly the density spreads.
A larger \(D\) means faster smoothing and faster spatial spread. A smaller \(D\) means slower spread.
In one spatial dimension, the units of \(D\) are typically
\[\frac{\text{distance}^2}{\text{time}}.\]Why distance squared?
For ordinary diffusion, the characteristic squared displacement grows approximately linearly with time:
\[\langle X(t)^2\rangle\propto Dt.\]So the typical distance travelled grows like
\[\sqrt{Dt}.\]This square-root scaling is characteristic of diffusive spread.
Microscopic randomness and macroscopic diffusion
Diffusion can emerge from many small random movements. At the individual level, motion may resemble a random walk. At the population level, the density can approach the smooth diffusion equation.
This connects stochastic movement with deterministic partial differential equations.
Diffusion does not mean individuals move smoothly
The density \(u(x,t)\) may evolve smoothly even though individual organisms make irregular movements.
Diffusion conserves total population in a closed domain
If there are no births, deaths or losses through the boundaries, diffusion only redistributes individuals.
For a domain \([a,b]\), the total population is
\[N(t)=\int_a^b u(x,t)\,dx.\]With no-flux boundaries, diffusion conserves this total.
No-flux boundary condition
A closed boundary has
\[J=0.\]Using Fick's law, this becomes
\[\frac{\partial u}{\partial x}=0\quad\text{at the boundary}.\]Biologically, individuals cannot cross that edge.
Absorbing boundaries
An absorbing boundary allows individuals to be lost on reaching the edge. One common mathematical condition is
\[u=0\quad\text{at the boundary}.\]This can represent a lethal habitat edge or removal from the modelled region.
Two-dimensional diffusion
If density depends on two spatial coordinates,
\[u=u(x,y,t),\]the diffusion equation becomes
\[\boxed{\frac{\partial u}{\partial t}=D\nabla^2u},\]where
\[\nabla^2u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}.\]Diffusion plus biological reactions
Real populations also reproduce, die, compete or become infected. A reaction–diffusion model combines these local processes with movement:
\[\boxed{\frac{\partial u}{\partial t}=D\nabla^2u+f(u)}.\]For logistic growth,
\[\frac{\partial u}{\partial t}=D\nabla^2u+ru\left(1-\frac{u}{K}\right).\]Diffusion can create invasion fronts
When local growth is positive and individuals diffuse into nearby empty space, a population can spread as a moving front.
The Fisher–KPP equation is a classic example:
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}+ru\left(1-\frac{u}{K}\right).\]It can support travelling-wave solutions representing an advancing population.
Diffusion is not always the correct movement model
Ordinary diffusion assumes undirected local movement with no preferred destination. Animals may instead move toward food, away from predators, along rivers, with wind, or in response to chemical signals.
These mechanisms may require advection, taxis or more structured movement models.
Density-dependent diffusion
The diffusion coefficient can itself depend on density:
\[D=D(u).\]This can represent organisms moving differently in crowded and sparse conditions.
The resulting diffusion is nonlinear and can produce behaviour different from ordinary Fickian diffusion.
Spatially varying diffusion
Movement conditions can also vary by location:
\[D=D(x).\]For example, movement through open habitat may be faster than through dense vegetation or fragmented terrain.
Diffusion in epidemic models
Different compartments can have different diffusion coefficients:
\[S=S(x,t),\qquad I=I(x,t).\]A spatial epidemic model may therefore contain terms such as
\[D_S\nabla^2S\qquad\text{and}\qquad D_I\nabla^2I.\]These describe movement of susceptible and infectious individuals through space while infection and recovery occur locally.
Diffusion versus advection
| Diffusion | Advection |
|---|---|
| random or undirected spreading | directed transport |
| smooths density differences | moves density with a preferred velocity |
| often uses \(D\nabla^2u\) | often uses a first spatial derivative term |
Diffusion versus migration between patches
Diffusion uses continuous space. Patch models instead move individuals between discrete locations.
Both represent spatial movement, but the mathematical structure and interpretation of movement parameters differ.
A practical interpretation workflow
Identify what \(u(x,t)\) measures. Check the units of density and \(D\). Decide whether movement is genuinely undirected. Interpret Fick's law as movement down the density gradient. Use the conservation law to understand how flux changes local density. Then add births, deaths or interactions only if the biological question requires them.