## Metapopulations

'Metapopulation' is defined as two or more populations of the same species (subpopulations) occupying different defined areas that exchange individuals between them through emigration and immigration. When the rate of migration into and out of the subpopulation is great, changes within the subpopulation mirror those of the metapopulation. When no individuals move among the subpopulations, each subpopulation behaves independently and is affected to a greater extent by environmental catastrophes. Relatively small, isolated subpopulations may suffer local extinction, especially if the habitat patch is small. Moderate movement of individuals into an empty patch may result in colonization of that patch. Immigration and emigration from occupied patches into unoccupied or under-occupied patches result in 'metapopulation dynamics' where each patch is subject to different rates of extinction (e) and colonization (c).

### The Levins model

In 1969, Levins described the dynamics of a metapopulation as a mathematical model based on the following assumptions: (1) all patches are identical; (2) each patch is either occupied or empty; (3) all subpopulations are equally likely to go extinct; and (4) rate of colonization of each patch is proportional to the number of occupied patches (p) and empty patches (1 — p):

where p represents the fraction of habitat patches occupied by a species, e is the rate of local extinction, and m is the rate of colonization of empty patches. A stable metapopulation is reached when the fraction of occupied patches, p, is equal to 1 — e/m.

In 1982, Hanski developed a more complex model of colonization/extinction by correlating the rate of extinction with the number of occupied patches:

He further correlated m with degree of isolation of one patch from another (a function of distance):

where a and m0 are parameters and D is the average distance between patches (isolation). Finally, he correlated e with patch size:

where e0 and b are parameters and A is the average patch size. Combining the two equations relates the rate of extinction with patch size, its isolation, and the degree to which other patches are occupied:

Further manipulation of Hanski's model of metapopulation introducing stochasticity concludes that metapopulation dynamics results in two types of species: a 'core species' which is adequately spaced out in its niche, is common and regionally abundant; and a 'satellite species' which is rare and present only in patches.

The source-sink (BIDE) model

In 1988, Pulliam proposed the BIDE metapopulation model to explain the movement of individuals from patches that experience population growth (sources) into patches where the population is decreasing (sinks). In habitats where the species is actively dispersing, a stable equilibrium can be reached between source and sink habitats, where dispersal from one to another maintains the species within that metapopulation habitat. The BIDE model can be mathematically illustrated as dN/dt — N(B + I - D - E)

where N is the size of the population, B stands for birth, I for immigration, D for death, and E for emigration.

The metapopulation accounts for all subpopulations where j counts the number of m subpopulations.

The total birth (B) is equal to the sum of births in all habitats:

The total death (D) is equal to the sum of all deaths in all habitats:

Immigration (I) between habitats is equal to the sum of immigrations between all habitats (j — immigration from habitat i to habitat k):

where

Emigration (E) between habitats is equal to the sum of emigrations between all habitats (j — emigration from habitat j to habitat k):

When bj> dj (net births are greater than net deaths) and ej> j (net emigration is greater than net immigration), the subpopulation is considered a source and is a net exporter of individuals.

When bj< dj (net births are smaller than net deaths) and ej < j (net emigration is smaller than net immigration), the subpopulations are considered a sink and is a net importer of individuals.

If, for a given subpopulation, bj + j — dj — ej = 0, the subpopulation is considered in dynamic equilibrium.

The assumptions of the BIDE model are given as follows:

1. BIDE rates are deterministic;

2. BIDE rates vary among habitats; and

3. populations behave in an 'ideal free distribution' (individuals have perfect knowledge regarding the habitat which drives their habitat choice decision).

The status of the sink habitat depends on the status of the source habitat. This situation is called 'donor control'. Immigration from large or increasing subpopulation (source) to a small or declining subpopulation (sink) keeps the sink population from eventual extinction and is known as the 'rescue effect'.

### Dispersion - Spatial Scale

Individuals in a population can be found to be spatially distributed according to three general patterns: random, uniform, or clumped. In a very homogeneous environment where there are no advantages for individuals to aggregate around a resource or avoid a less favorable area within the habitat, individuals will be randomly distributed, a rare situation in natural settings. Uniform distribution is characterized with more regular spacing than random spacing and is found in environments with great competition between individuals or where close proximity between individuals has a negative effect on one or both of the individuals' fitness. In this situation even spacing between individuals will be found. Clumped patterns of spacing are the most common pattern found in nature, although within large groups occupying a habitat the overall group spacing may be closer to random (Figure 2).

One method to determine the type of dispersion pattern based on a series of samples compares the frequency of occurrence of different group sizes to a 'normal' distribution curve. If the occurrence (I = index of dispersion) of groups of different sizes is the same, the

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