Metapopulation models
A metapopulation is a collection of local populations living in separate habitat patches but connected by movement. Instead of treating space as continuous, we represent it as a set of distinct locations.
What is a patch?
A patch is a spatial unit within which the population is modelled locally. Depending on the problem, a patch could be an island, woodland fragment, lake, farm, village, city or administrative region.
The important assumption is that movement between patches can be distinguished from dynamics within patches.
local population
local population
local population
Population-size formulation
Let \(N_i(t)\) be the population in patch \(i\). A general deterministic model is
\[\boxed{\frac{dN_i}{dt}=f_i(N_i)+\sum_{j\ne i}m_{ji}N_j-\sum_{j\ne i}m_{ij}N_i.}\]| Term | Meaning |
|---|---|
| \(f_i(N_i)\) | births, deaths and other local dynamics in patch \(i\) |
| \(m_{ji}N_j\) | movement from patch \(j\) into patch \(i\) |
| \(m_{ij}N_i\) | movement from patch \(i\) into patch \(j\) |
Why there are two movement sums
The population in patch \(i\) increases when individuals arrive from other patches and decreases when individuals leave. Therefore movement contributes
\[\text{movement change}=\text{immigration}-\text{emigration}.\]This is the discrete-patch analogue of spatial redistribution in a continuous-space model.
A two-patch example
Suppose two patches have logistic growth and symmetric movement at rate \(m\):
\[\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{K_1}\right)+mN_2-mN_1,\]\[\frac{dN_2}{dt}=r_2N_2\left(1-\frac{N_2}{K_2}\right)+mN_1-mN_2.\]The local logistic terms describe growth inside each patch. The movement terms couple the equations.
What coupling means
Without movement, each patch follows its own equation independently. With movement, \(N_1\) appears in the equation for \(N_2\), and \(N_2\) appears in the equation for \(N_1\).
The patches are therefore no longer independent dynamical systems.
A model-generated two-patch example
The graph below is calculated numerically from the two-patch logistic equations above using \(r_1=0.35\), \(K_1=100\), \(r_2=0.22\), \(K_2=60\), \(m=0.04\), \(N_1(0)=10\) and \(N_2(0)=50\).
Movement conserves the regional total
If movement only transfers individuals between patches, movement by itself does not change the total population. For two symmetric patches, adding the equations cancels the movement terms:
\[mN_2-mN_1+mN_1-mN_2=0.\]Birth and death can change the regional total, but movement merely redistributes individuals.
Movement matrix
For many patches, connectivity can be represented by a matrix. If \(m_{ij}\) is the per-capita movement rate from patch \(i\) to patch \(j\), then the collection of values \(m_{ij}\) describes the movement network.
Not every pair of patches needs to be connected. Geographic distance, barriers and transport routes can all affect the matrix.
Source and sink patches
A source patch can produce a local surplus of individuals. A sink patch may not sustain itself without immigration.
Movement from source patches can therefore maintain populations in locations where local conditions alone would lead to decline.
The rescue effect
A local population may become very small or disappear. Immigration from another occupied patch can recolonise it or prevent extinction.
This is called the rescue effect. Regional persistence can therefore be greater than persistence of any isolated local population.
Occupancy models
Some metapopulation models ignore exact population sizes and record only whether patches are occupied.
If \(p(t)\) is the fraction of habitat patches occupied, a classical Levins-type model is
\[\boxed{\frac{dp}{dt}=cp(1-p)-ep.}\]Here \(c\) is a colonisation parameter and \(e\) is a local extinction rate.
Understanding the colonisation term
The factor \(p\) represents occupied patches that can supply colonists, while \(1-p\) represents empty patches available for colonisation. Thus
\[cp(1-p)\]increases occupancy.
The term
\[ep\]removes occupied patches through local extinction.
Equilibria of the Levins model
Set
\[cp(1-p)-ep=0.\]Then
\[p[c(1-p)-e]=0.\]The equilibria are
\[p^*=0\]and, when \(c>e\),
\[\boxed{p^*=1-\frac{e}{c}.}\]Thus positive long-term occupancy requires colonisation to be sufficiently strong relative to extinction.
Threshold interpretation
If \(c\le e\), the positive equilibrium is absent and the simple model predicts eventual regional extinction. If \(c>e\), colonisation can balance local extinction and maintain a positive fraction of occupied patches.
Metapopulation epidemics
The same patch idea can be used for infectious disease. Each patch may contain its own susceptible, infectious and recovered populations:
\[S_i(t),\qquad I_i(t),\qquad R_i(t).\]Transmission occurs locally, while movement connects the epidemic dynamics between patches.
Why spatial coupling changes epidemics
One region may experience an outbreak first and seed infection into other regions. Movement can synchronise regional epidemics, produce repeated introductions, or allow infection to persist regionally even if it disappears temporarily from individual patches.
Metapopulation versus reaction–diffusion
| Metapopulation model | Reaction–diffusion model |
|---|---|
| space divided into discrete patches | space treated as continuous |
| movement between named locations | local movement represented by spatial derivatives |
| natural for islands, cities or regions | natural for continuous habitats |
| connectivity can be irregular | classical diffusion assumes local spatial spreading |
Metapopulation versus network model
The ideas overlap strongly. A metapopulation model emphasises local populations within patches, while a network representation emphasises the pattern of connections between those patches.
Many modern metapopulation models are naturally written as networks.
Deterministic and stochastic versions
Deterministic models describe continuous population sizes or occupancy fractions. For small local populations, extinction and colonisation are discrete random events, so stochastic models can be more realistic.
In a stochastic occupancy model, each individual patch is either occupied or empty and changes state probabilistically.
Connectivity is not always beneficial
Movement can rescue declining populations, but it can also spread disease, invasive species or disturbances. Strong connectivity may synchronise local fluctuations, reducing the chance that some patches remain unaffected when others decline.
Habitat fragmentation
Fragmentation can reduce patch size, increase isolation or remove connections. A metapopulation model allows these effects to be studied separately.
Two landscapes with the same total habitat area can have different persistence if their connectivity differs.
Model assumptions
Patch boundaries must be meaningful at the chosen scale. Local populations are usually assumed to mix sufficiently within patches, and movement rates are often treated as averages.
If within-patch spatial structure is important, or movement varies strongly through time, a more detailed model may be needed.
A practical workflow
Choose biologically meaningful patches. Specify the local dynamics in each patch. Determine which patches are connected and estimate movement rates. Decide whether population sizes or occupancy states are required. Analyse local and regional persistence separately. Finally, test how conclusions change when connectivity, extinction or colonisation rates vary.