Reaction–diffusion equations
Reaction–diffusion equations describe systems in which quantities change locally and also move through space. They are fundamental in mathematical biology because biological processes rarely occur without spatial movement.
The basic equation
For one quantity \(u(x,t)\), a reaction–diffusion equation has the form
\[\boxed{\frac{\partial u}{\partial t}=D\nabla^2u+f(u).}\]The two terms have different roles:
| Term | Meaning |
|---|---|
| \(D\nabla^2u\) | diffusive movement through space |
| \(f(u)\) | local biological change |
Why is it called reaction–diffusion?
The word reaction comes historically from chemical systems, but in mathematical biology it can mean any local process: birth, death, infection, recovery, predation, competition, differentiation or biochemical conversion.
Diffusion represents undirected spatial movement.
Build the equation from two simpler models
First suppose there is no space. Local dynamics are described by an ODE:
\[\frac{du}{dt}=f(u).\]Now suppose there is movement but no local birth or death:
\[\frac{\partial u}{\partial t}=D\nabla^2u.\]Putting the mechanisms together gives
\[\frac{\partial u}{\partial t}=D\nabla^2u+f(u).\]What happens at one location?
The density at a particular point can change for two reasons. Individuals can be created or removed locally by \(f(u)\), and they can enter or leave through diffusion.
This distinction is important: an increase at one location does not necessarily mean local reproduction occurred. It may simply mean individuals moved there.
Diffusion smooths; reaction can oppose smoothing
Diffusion tends to flatten spatial differences. A reaction term can increase, decrease or otherwise transform local density.
A population example
Take logistic growth:
\[f(u)=ru\left(1-\frac{u}{K}\right).\]Adding diffusion gives
\[\boxed{u_t=D u_{xx}+ru\left(1-\frac{u}{K}\right).}\]This is the Fisher–KPP equation in one dimension. It can describe a population that grows locally and spreads into new territory.
Why partial derivatives appear
The state now depends on both position and time:
\[u=u(x,t).\]Therefore \(u_t\) measures change through time while holding position fixed, and \(u_{xx}\) measures spatial curvature at a fixed time.
Why the second spatial derivative?
Diffusive flux follows Fick's law:
\[J=-D u_x.\]Conservation gives
\[u_t=-J_x.\]Substitution gives
\[u_t=D u_{xx}.\]Reaction adds a local source or sink:
\[\boxed{u_t=-J_x+f(u)=D u_{xx}+f(u).}\]Source and sink interpretation
If \(f(u)>0\), the reaction creates or increases the quantity locally. If \(f(u)<0\), it removes or decreases it locally. If \(f(u)=0\), the local reaction is in equilibrium.
Diffusion can still change the local density even when \(f(u)=0\).
Spatially uniform solutions
If \(u\) is the same everywhere, then
\[\nabla^2u=0.\]The reaction–diffusion equation reduces to
\[\frac{du}{dt}=f(u).\]Thus the corresponding non-spatial ODE is contained inside the spatial model as the special case of a uniform population.
Equilibria of the reaction term
Constant spatial equilibria satisfy
\[f(u^*)=0.\]For logistic growth,
\[u^*=0,\qquad u^*=K.\]But spatial models can also have non-uniform steady states where diffusion and reaction balance each other.
Steady spatial patterns
A steady state satisfies
\[u_t=0.\]Therefore
\[D\nabla^2u+f(u)=0.\]This does not require \(u\) to be spatially constant. Local reaction and diffusion can sometimes balance to maintain spatial structure.
Several interacting variables
Many biological systems require more than one state variable. For two quantities,
\[\boxed{u_t=D_u\nabla^2u+f(u,v),}\]\[\boxed{v_t=D_v\nabla^2v+g(u,v).}\]The reaction functions describe local interaction, while \(D_u\) and \(D_v\) allow the components to move at different rates.
A predator–prey example
Suppose prey \(u\) and predators \(v\) interact locally:
\[u_t=D_u\nabla^2u+au-buv,\]\[v_t=D_v\nabla^2v+cbuv-dv.\]The interaction terms change local abundances, while the diffusion terms allow both populations to move through the habitat.
A spatial epidemic example
For susceptible and infectious densities, one possible reaction–diffusion structure is
\[S_t=D_S\nabla^2S-\beta SI,\]\[I_t=D_I\nabla^2I+\beta SI-\gamma I.\]The infection and recovery terms act locally, while diffusion represents movement of the host populations.
The exact infection term and movement assumptions should be chosen according to the biological system.
From local disturbance to spatial spread
Suppose a population is initially present only near the centre of a habitat. Diffusion moves some individuals outward. If local reaction supports growth, the newly occupied regions can increase in density.
The disturbance may therefore spread rather than simply flatten.
Travelling waves
Some reaction–diffusion equations support solutions
\[u(x,t)=U(x-ct).\]The profile \(U\) moves at speed \(c\) while keeping its shape. These travelling waves can represent biological invasion fronts, gene spread or epidemic propagation.
Reaction–diffusion is not advection
Diffusion represents undirected spreading. Advection represents systematic transport by a velocity field.
If both occur, a model can take the form
\[u_t=D u_{xx}-v u_x+f(u).\]Now diffusion spreads, advection transports and reaction changes density locally.
Boundary conditions matter
A PDE needs information about the spatial boundary. A no-flux condition
\[\frac{\partial u}{\partial n}=0\]represents no diffusive movement across the boundary. A fixed-value condition specifies \(u\) at the edge.
The choice is biological, not merely mathematical.
Initial conditions matter
We also specify the initial spatial distribution:
\[u(x,0)=u_0(x).\]A localised introduction, a nearly uniform state and a strongly patterned initial condition can produce very different transient behaviour.
Two spatial dimensions
For \(u(x,y,t)\),
\[u_t=D\nabla^2u+f(u),\]where
\[\nabla^2u=u_{xx}+u_{yy}.\]This allows spatial patterns to develop over a surface rather than along a line.
Reaction–diffusion and pattern formation
With several interacting components, diffusion can sometimes destabilise a spatially uniform equilibrium and create a persistent spatial pattern.
This may seem surprising because diffusion normally smooths differences. The key is that interacting components can diffuse at different rates and react with each other.
Turing's idea
Alan Turing showed in 1952 that a state stable without spatial diffusion can, under suitable conditions, become unstable when diffusion is added. This mechanism is called a diffusion-driven instability or Turing instability.
It provides a mathematical mechanism for spontaneous pattern formation from small spatial disturbances.
Intuition for two-component pattern formation
Imagine one component locally activates production while another suppresses it. If their spatial ranges differ, a small local increase can reinforce itself nearby while inhibition acts farther away.
Under suitable parameter conditions, this competition between reaction and diffusion can amplify a spatial disturbance rather than erase it.
Diffusion does not always create patterns
Turing patterns require particular interactions and parameter relationships. Simply adding diffusion to an ODE system does not automatically produce pattern formation.
Stability and spatial modes
To analyse pattern formation mathematically, a small spatial perturbation can be decomposed into spatial modes. Each mode has a wavelength.
Linear stability analysis asks whether each mode decays or grows. If some non-zero spatial modes grow while the uniform mode remains stable, diffusion-driven pattern formation is possible.
Wavelength and biological scale
A spatial pattern has a characteristic wavelength: the distance between repeating peaks. Model parameters and domain size can determine which wavelengths grow most strongly.
This connects mathematical stability analysis to the physical spacing of biological structures.
Dimensionless form
Rescaling space, time and state variables can reduce the number of independent parameters. This makes it easier to identify which combinations of growth, interaction and diffusion actually control the dynamics.
Dimensionless analysis is particularly useful when comparing systems with different physical units or spatial scales.
Numerical solutions
Most nonlinear reaction–diffusion systems do not have simple explicit solutions. Numerical methods divide space into a grid or mesh and advance the solution through time.
A simulation should use sufficiently small spatial and temporal steps, appropriate boundary conditions and checks that the numerical pattern is not an artefact of the discretisation.
What reaction–diffusion models can represent
| Biological setting | Reaction | Diffusion |
|---|---|---|
| population invasion | birth and density dependence | individual dispersal |
| epidemic spread | infection and recovery | host movement |
| predator–prey system | predation and reproduction | movement of both species |
| developmental pattern | biochemical interactions | molecular transport |
| cell population | division and death | cell motility |
Model assumptions
Ordinary reaction–diffusion models usually treat density as continuous and movement as local, undirected diffusion. Parameters are often assumed constant and the environment homogeneous.
These assumptions should be checked against the biological problem. Long-distance dispersal, directional movement, discrete individuals or strongly heterogeneous environments may require different models.
A practical modelling workflow
Define the state variables and spatial domain. Write the local biological dynamics first. Add diffusion only to quantities that genuinely move. Choose diffusion coefficients with appropriate units. Specify initial and boundary conditions. Analyse equilibria and their stability. Then investigate waves, spatial patterns or other structures relevant to the biological question.