Spatial populations
Population models often describe how abundance changes through time. Spatial population models add another question: where are the individuals?
A population can be abundant overall but concentrated in only part of a habitat. Movement, local crowding, environmental differences and geographical barriers can therefore change the dynamics.
From population size to population density
A non-spatial model might use
\[N(t),\]the total number of individuals at time \(t\).
In one-dimensional continuous space we can instead use
\[u(x,t),\]where \(u(x,t)\) is population density at position \(x\) and time \(t\).
Thus \(u(2,5)\) means the density at location \(x=2\) at time \(t=5\).
Density is not the same as total population
If \(u(x,t)\) is density, the total population in a one-dimensional habitat from \(a\) to \(b\) is
\[N(t)=\int_a^b u(x,t)\,dx.\]The density tells us how population is distributed through space; integration adds the local densities to obtain total abundance.
Why space matters
Individuals do not usually experience the average conditions of the entire habitat. They interact locally. Food, predators, infection risk, shelter and competitors may all vary from place to place.
Spatial structure can therefore change persistence, invasion speed, coexistence, epidemic spread and extinction risk.
Three ingredients of a spatial population model
A useful way to organise spatial dynamics is
\[\text{local population change}+\text{movement through space}.\]Local change may include births, deaths, infection or competition. Movement may represent random dispersal, directed movement, migration or transport.
Start with a non-spatial growth model
Logistic growth is
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).\]This assumes the population can be represented by one abundance \(N(t)\). It contains no information about location.
Allow growth to occur locally
If density varies through space, local logistic growth can be written as
\[\frac{\partial u}{\partial t}=ru\left(1-\frac{u}{K}\right).\]Here the partial derivative is used because \(u\) depends on both \(x\) and \(t\).
This equation still does not move individuals between locations. Each point changes independently according to its local density.
Add movement
A common model of undirected dispersal is diffusion. In one dimension,
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}.\]The parameter \(D>0\) is the diffusion coefficient. Larger \(D\) means faster spatial spreading.
The second spatial derivative measures local curvature of the density profile. Diffusion tends to move density away from concentrated regions and smooth spatial differences.
Intuition for diffusion
Reaction–diffusion population model
Combining local logistic growth with diffusion gives
\[\boxed{\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}+ru\left(1-\frac{u}{K}\right).}\]This is a reaction–diffusion equation.
The two parts have distinct meanings:
| Term | Meaning |
|---|---|
| \(D u_{xx}\) | movement or dispersal through space |
| \(ru(1-u/K)\) | local population growth |
The word reaction here means local biological change; it does not require a chemical reaction.
What does a spatial derivative mean?
The derivative
\[\frac{\partial u}{\partial x}\]describes how density changes as we move through space at a fixed time.
The second derivative
\[\frac{\partial^2u}{\partial x^2}\]describes the curvature of that spatial profile and is the quantity appearing in ordinary diffusion.
Space can be one-, two- or three-dimensional
In two-dimensional space, density may be written
\[u(x,y,t).\]Random dispersal is then represented using the Laplacian:
\[\frac{\partial u}{\partial t}=D\nabla^2u+f(u).\]where
\[\nabla^2u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}.\]Continuous space versus patches
Not every spatial model uses a PDE. A landscape can instead be divided into discrete patches.
For two patches, let \(N_1(t)\) and \(N_2(t)\) denote population sizes. Individuals may move between them at rates \(m_{12}\) and \(m_{21}\).
A simple two-patch model
If both patches have local growth and individuals migrate between them, a general form is
\[\frac{dN_1}{dt}=f_1(N_1)-m_{12}N_1+m_{21}N_2,\]\[\frac{dN_2}{dt}=f_2(N_2)+m_{12}N_1-m_{21}N_2.\]Migration out of one patch appears as a loss there and as a gain in the destination patch.
Metapopulations
A metapopulation is a collection of local populations occupying habitat patches that are connected by dispersal.
Some local populations may become extinct while empty patches are recolonised. Consequently, regional persistence can differ from persistence within any single patch.
Source and sink habitats
A source is a habitat where local reproduction can support the population and may produce emigrants. A sink is a habitat where local demographic conditions alone would not maintain the population, but immigration can keep it occupied.
This means observing a population in a location does not necessarily imply that the location can sustain it without immigration.
Heterogeneous environments
Real landscapes are not uniform. Growth rate, carrying capacity or mortality can depend on location:
\[r=r(x),\qquad K=K(x).\]A model might therefore use
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}+r(x)u\left(1-\frac{u}{K(x)}\right).\]Spatial heterogeneity can create regions of high and low abundance even after transient dynamics have disappeared.
Boundaries matter
A spatial domain must usually be supplied with boundary conditions.
A no-flux boundary means individuals cannot cross the edge. In one dimension this can be written
\[\frac{\partial u}{\partial x}=0\quad\text{at the boundary}.\]An absorbing boundary can represent individuals being lost when they reach the edge. Periodic boundaries mathematically connect opposite edges of the domain.
Directed movement
Diffusion represents undirected movement. Organisms may instead move preferentially because of wind, water flow, resources, chemical signals or habitat quality.
Such movement requires additional terms. The spatial mechanism should therefore be chosen from the biology rather than adding diffusion automatically.
Spatial invasion
If a population is introduced locally and can grow and disperse, it may spread through the habitat as an invasion front.
In reaction–diffusion models, local growth determines whether the population can establish while dispersal determines how it reaches new locations. Together they can determine the speed and shape of spread.
Travelling waves
A travelling wave is a spatial profile that moves while approximately retaining its shape. It can be written as
\[u(x,t)=U(x-ct),\]where \(c\) is wave speed.
Travelling waves can represent advancing biological populations, epidemic fronts or invading species.
Spatial epidemic models
In an epidemic model, susceptible and infectious populations can each depend on location:
\[S=S(x,t),\qquad I=I(x,t).\]Local infection and recovery can be combined with movement. Spatial structure then allows an epidemic to spread geographically rather than appearing everywhere simultaneously.
Spatial scale matters
A model appropriate for movement between countries may be unsuitable for movement between neighbouring cells. The meaning of space, movement and interaction depends on the spatial scale of the biological question.
Choosing a spatial resolution that is too coarse can hide local structure, while an unnecessarily fine resolution can make a model difficult to parameterise and analyse.
Spatial versus well-mixed models
| Well-mixed model | Spatial model |
|---|---|
| state depends mainly on time | state depends on location and time |
| individuals effectively experience average conditions | local conditions and interactions matter |
| often uses ODEs | may use PDEs, patches, lattices or networks |
| no explicit invasion front | can represent geographical spread |
When is a spatial model needed?
Spatial structure is worth including when location changes the biological conclusions. Examples include localised outbreaks, habitat fragmentation, range expansion, movement corridors, heterogeneous resources or interactions that occur mainly between nearby individuals.
If movement is extremely fast relative to local dynamics and the habitat is effectively homogeneous, a simpler well-mixed model may sometimes be adequate.
What spatial models can tell us
Depending on the model, spatial analysis can address where populations persist, how quickly an invasion spreads, whether habitat fragmentation prevents persistence, how migration connects local populations, where infection concentrates, and whether spatial patterns form.
A practical modelling workflow
Define the biological state and spatial domain. Decide whether space should be continuous or divided into patches. Specify local biological processes. Choose a biologically appropriate movement mechanism. Set initial and boundary conditions. Analyse or simulate the resulting model. Finally, interpret the spatial pattern in terms of the biological question.