Metapopulation Biology
What is a metapopulation?
According to Levins’ original formulation, a metapopulation
is a “population of populations”. It is a set of populations
located in/on spatially distinct habitats (islands, patches) that
exchange individuals with (reasonable) likelihoods of
immigration between populations and of extinction of
individual populations.
Drawing from the text: “metapopulations are regional
assemblages of plant or animal species, with the long term
survival of the species depending on a shifting balance
between local extinctions and re-colonization in a
patchwork of fragmented landscapes.
The basic models make a number of unrealistic assumptions,
though the results are still valuable.
1. All patches are equally accessible to colonists from any
other patch, i.e. the patches are equidistant. If there are
more than 3 patches, how can this be?
2. Areas of patches are unimportant. We are interested only
in whether a patch is occupied or not. From what you
already know about the probability of extinction, isn’t
population size important?
3. The qualities of all patches are identical. Is this likely?
What is usually of interest in basic metapopulation models
is whether persistence of the metapopulation greatly
exceeds the expected persistence of individual local
populations. In other terms, whether probability of regional
extinction is much lower than the probability of local
extinction.
There are various ‘kinds’ of metapopulations. We’ll look at
them in ‘historical’ order.
Island-mainland (Island biogeography) models
You should have encountered the basics of this model in
Ecology. MacArthur and Wilson (1963, 1967) proposed a
model for isolated islands that determined species diversity
on them resulting from a balance between immigration onto
the islands and extinction on them.
The number of species occurring on an island was
determined by the area of the island. The equation for that
number also includes a constant set by the dispersal ability
of the taxonomic group in question. The equation:
S = CAz
In this model there is a source pool of species from which
colonists come – the mainland.
If this equation is transformed by taking logs, the resulting
relationship between (log) species number and (log) area is
linear…
Log S = log C + z log A
There is a classic example, cited in almost every text, of the
species-area curve for reptiles on Caribbean islands…
The slope z has been a point of great interest. Most speciesarea curves seem to have slopes of about 0.25.
There is an underlying theoretical reason for that. In
Ecology you may have also encountered the Preston log
normal abundance curve for species in a community. It
suggests that a plot of the log of individual species
abundance (usually plotted in base 2) versus the number of
species with those abundances should follow a normal
curve, as shown for moths captured at a light trap in
England:
Much different slopes have been observed. The question is
why?
Much lower slopes have been found when the patches or
islands are not really isolated, for example when larger and
larger areas are sampled from a mainland area as if they
were separate.
Much larger slopes have been observed when unexpectedly
high habitat diversity occurs within patches, or, in one
example, when pollution may have reduced species
interactions among populations in patches. There is, once
more, a classic plot, comparing bird diversity in patches on
New Guinea with that found on isolated oceanic islands
nearby…
What determines the number of species is the balance
between immigration and extinction. Over time the
immigration rate (beginning with a bare island) declines
from an initially high rate to potentially reach zero when all
species from the mainland pool have reached the island.
Meanwhile, as species accumulate, extinctions occur due to
chance events when population(s) are small or to
interactions among species as more are present.
The standard depiction of this relationship is:
When immigration and extinction rates are equal, an
equilibrium is reached. The number of species doesn’t
change, but there is still turnover. There are changes in the
names on the list.
There is one characteristic of the mainland-island
metapopulation model that may not be obvious: since there
is an unending source of potential colonists from the
mainland, the equilibrium occupancy of patches is greater
than zero – the island metapopulation should never go
regionally extinct.
That is not necessarily true of the Levins (or classical)
metapopulation.
In the Levins model there are only isolated patches, with no
mainland source. Whether the metapopulation persists or
not is determined by the balance between colonization and
extinction rates on patches. Of the total number of patches
available in the region, the number currently occupied by
this species is called P. By this definition of P, 0  P  1.
The value of P is determined by a balance between
immigration, I, which leads to the occupation of previously
unoccupied patches, and local extinction, E, which causes a
previously occupied patch to become unoccupied. There are
close analogies to both the basic equation for exponential
growth and the basic equation of island biogeography in this
formulation.
dP/dt = I - E
Immigration is determined by a characteristic rate, c, times
the number of occupied sites that can provide immigrants
(P), but can only occur onto unoccupied sites (1-P).
Extinction is set by a characteristic rate  and the number of
occupied sites at which extinction could occur. Thus:
dP/dt = cP(1-P) - P
What we are really interested in from this model is the
equilibrium occupancy of patches. We can find the
equilibrium if we set dP/dt to 0 and solve.
cP(1-P) = P
P = 1 – /c
Unless c >  an uninteresting equilibrium of regional
extinction results. If c >  then there is a positive
equilibrium and persistence of the metapopulation. It diesn’t
matter what the initial proportion of sites occupied is, the
metapopulation will ‘converge’ on a proportion determined
by whatever the fixed  and c are.
There is a useful parallel to growth equations that can be
drawn from this. c, the colonization rate, is like a birth rate;
 is like a death rate. In this notion of a parallel, r = c -  and
carrying capacity is 1 – /c. As a parallel to the logistic
equation:
dP/dt = (c- ) P (1 – [P/(1 - /c)]
The same parabolic graph
that was useful in evaluating
the effect of harvesting for
the logistic model can be used
to evaluate the effects of
changes in colonization and
extinction rates on equilibrium
occupancy…
How can we compare the probability of regional extinction
in a metapopulation versus one which is not subdivided?
As a first step, we can assess the probability of regional
extinction using the basic rules for combinations of
independent events. Local populations may have fairly large
probabilities of extinction, even when probabilities are
measured on a short term basis, i.e. probabilities for a time
period of one year. What if the probability of local extinction
in a single year is 0.7 (pe). Then (1-pe) is the probability of
population persistence. If we want to know what the
probability of persistence is over a two year period, it is:
P2 = (1-pe)(1-pe) = (1-pe)2
or, for an y year period
Py = (1-pe)y
The probability of persistence for a single local population
can become very small over very few years. However, what
happens when we consider a group of local populations
which comprise a metapopulation? Assume that patches are
identical, with identical probabilities of local extinction. If
there are n such populations functioning independently with
regard to extinction, then the probability of all going extinct
in any one year is:
P(0)n = (pe)n
And of persistence is:
Pn = 1- (pe)n
Now with only a few local populations as components of the
metapopulation, the probability of persistence can be quite
high, even with a high probability of local extinction in each
of the component populations.
This all works based on the assumption that processes in
separate populations are asynchronous.
Remember what happened when time lags were important,
and produced chaotic behaviour in population size? That
chaotic behaviour can be mitigated by exchange between
component populations of a metapopulation.
Thus far we’ve looked at two kinds of metapopulations:
mainland-island and Levins’ classic metapopulations. It’s
time to move on to the third type,
Source-sink metapopulations
There is a partial similarity to island-mainland models here.
Though there is no mainland, there are patches with superior
habitat and consistent growth rates > 0. They are sources of
colonists. Those colonists move to patches where conditions
are not so positive, and growth rates average < 0. They are
sinks. Populations would not persist there without colonists
from sources.
What happens in sink populations?
Without new immigrants from source populations, they
would go locally extinct. However, immigrants arrive moreor-less frequently. Those immigrants prevent local
extinction. This is called the rescue effect.
The text distinguishes between a ‘true sink’ population that
would decline to extinction without rescue from a source
population and a ‘pseudo-sink’ population that would
decline to a clearly lower equilibrium size in the absence of
a source. In real world, field situations there is no easy way
to distinguish the two types of sink populations.
If you remember the MacArthur-Wilson model of island
biogeography, you should see a great deal of similarity
between source-sink metapopulations and the islandmainland model of island biogeography. The mainland (e.g.
New Guinea for bird populations) is a source, and the
isolated islands are likely to be sinks, at least in the long run
(and particularly for distant, isolated islands).
A neat example of a real world sourcesink system is rare tropical tree species
in the forests of Barro Colorado island.
The distribution of abundance among
species is a very good fit to the Preston
log-normal.
There are usually more rare than common species, and here,
if there is a deviation from the log-normal, it is to have even
more rare species than predicted by the theoretical
distribution of abundances.
Quoting from the summary of the paper:
“Many (at least one third) of the rare species (fewer than 50
total individuals) do not appear to have self-maintaining
populations in the plot [50 hectares with approximately 238,000
individual trees and shrubs]. Their presence appears to be the
result of immigration from population centers outside the plot,
and their numbers are probably kept low by a combination of
unfavorable regeneration conditions, lack of appropriate
habitat, or both, in the plot” (Hubbell and Foster 1986)
The last type of metapopulation takes two forms, linked in
the text. The first is described as a non-equilibrium
metapopulation. It could probably be better described as a 0
equilbrium form. The rate of extinction exceeds the rate of
colonization. Unless something changes (e.g. rescue from
some source), the equilibrium state is regional extinction.
The second form is called a patchy population. Here the
opposite is occurring, colonization (or migration) rate is so
high that the population is not really subdivided effectively
into a metapopulation.
When metapopulation models are made more realistic,
particularly with regard to space, one is moved back in the
direction of studies associated with the MacArthur-Wilson
island biogeography approach.
What determines the likelihood of a patch being colonized,
or, alternatively, of a propagule reaching the patch? How
far away are sources? What are the dispersal characteristics
of potential colonists?
These questions were initially studied by Diamond in
modeling the probabilities of birds colonizing New Guinea
satellite islands. The important characteristics were the
number of dispersers and the distance to the island,
embodied in the equation:
C = e-d
C is the number of propagules reaching the isolated patch
 is the number of propagules produced by the source
 is the characteristic describing the dispersal function, and
d is the distance between source and isolated patch.
Diamond also recognized that patch area makes a
difference. Larger patches are more likely to be found (and,
in island biogeography, can support a larger number of
species).
Diamond also recognized that different species have
differing dispersal characteristics and tendencies to
disperse.
From all this, he constructed incidence functions.
Some species, called super tramps, are very likely to
disperse, can move long dsitances, and are thus early
colonists, as well as the most likely to reach very isolated
patches. They tend to be weak competitors, and disappear
from patches when they are colonized by species more
adapted to diverse communities.
There are a range of different groups of tramp species, not
such extreme dispersers, but even they eventually are
displaced by species with limited dispersal capability, but
strong ability to compete for resources.
Hanski and his collaborators, critical in the development of
metapopulation theory, turned this basic idea on its head,
and developed incidence functions that assess the longterm probability of a patch being occupied.
One of the key questions (particularly to conservation
biologists) is what conditions lead to long-term survival of a
metapopulation. To have a metapopulation survive at least
100x the expected survival time for a local component
population, it has been found that:
Pˆ H  3
Where P is the equilibrium proportion of patches occupied
and H is the total number of patches available.
The last thing to consider is data that supports the
metapopulation approach…
First is a basically mainland-island example of the
checkerspot butterfly, Euphydras edita, which lives in the
area around San Francisco, and is monophagous, at least as
a caterpillar, using an annual plantain, Plantago erecta, as
both a site for egg laying and a food source. This plantain
grows on serpentine rock outcrops. There are many small
outcrops, and one large one, called Morgan Hill. There are
estimates of hundreds of thousands of butterflies on Morgan
Hill, but only a shifting mosaic of presence on the other
sites. The causes of local extinctions vary. Many local
extinctions occurred during an extended severe drought that
lasted from 1975-7. The figure on the next slide maps the
outcrops around Morgan Hill, and indicates a few recent
events for the metapopulation.
Maps from other sources make it apparent that isolation
(distance from Morgan Hill) is an important factor. Most
colonization events recorded occurred on outcrops close to
the source 'mainland'. Extinction events are more widely
distributed.
A second example is carabid ground beetles. They have been
studied for more than 3 decades in the Netherlands.
Radioactive tags indicate they remain fairly localized,
moving less than 100 m per day. As a result, the relative
isolation of populations required for effective metapopulation
dynamics is likely, even over short distances. The figure that
follows shows dynamics in two different beetle species.
The upper panel shows numbers of individuals in different
component populations of Pterostichus versicolor. The
different populations fluctuated asynchronously, so that when
one population declined in size, another was likely to be
increasing. Sources were increasing populations likely to be
throwing off emigrants. The sinks were declining
populations; extinction was prevented by immigrants from
source populations.
The lower panel shows the dynamics of component
populations in a different species, Calathus
melanocephalus. There is a much greater synchrony among
components. When one population is declining, so do most,
if not all others. Whatever caused the declining periods,
they drove populations locally extinct; since there were no
source populations providing emigrants to colonize sink
populations. Local extinction (and even regional extinction)
is much more likely when populations within a
metapopulation function synchronously.
References
Den Boer, P. 1981. On the survival of populations in a heterogeneous and variable
environment. Oecologia, 50:39-53.
Diamond, J.M. 1975a. Assembly of species communities. In M.L. Cody and J.M.
Diamond, eds. Ecology and Evolution of Communities. pp.342-444. Belknap Press,
Cambridge, MA
Diamond, J.M. 1975b. The island dilemma: Lessons of modern biogeographic
studies for the design of natural reserves. Biol. Conserv. 7:129-146.
Hanski, I. 1999. Metapopulation Ecology. Oxford Univ. Press, Oxford.
Harrison, D., D.D. Murphy and P.R. Ehrlich. 1988. Distribution of the bay
checkerspot butterfly, Euphydras editha bayensis: evidence for a metapopulation
model. Amer. Natur. 132:360-382.
Hubbell, S.P. and R. Foster. 1986. Commoness and rarity in a Neotropical forest:
implications for tropical tree conservation. Pp. 205-231 in M.E. Soule, ed.
Conservation Biology: The Science of Scarcity and Diversity. Sinauer Assoc.,
Sunderland, MA.