1. Metapopulation Biology (Chap. 15 pp.302-308)
Metapopulation Biology has developed from two fronts:
1) Island biogeography - a view that, like Darwin, we
can learn a lot by studying island populations. You
will see more of this later. Here, just some basics.
2) A recognition that much of human impact has been
to fragment what was formerly continuous habitat
into ‘islands’.

2. The theory of island biogeography was developed by
Robert MacArthur and E.O. Wilson (1967).
It establishes the concept of an equilibrium diversity
(but not a set list of species) on islands.
The theory considers only the early history of species -
immigration and establishment, but not what happens when
biological interactions among species become important.
It turns out that two physical factors dominate the results -
island area and isolation (how far it is from sources of
colonists).

3. The basic graphical model that describes an equilibrium
is comprised of two curves:
1) An immigration curve that is concave downwards. Its
shape results from the best colonists (best dispersers) from
the source pool immigrating to any island more rapidly
than others. Once they’ve immigrated, others do so also,
but at a lower and lower rate, until (eventually, theoretically)
all species from the source pool have immigrated.
2) An extinction curve that increases in slope as time and
diversity increase on the island. As the island becomes
‘packed’, the extinction rate increases because the population
size of each species is smaller, and extinctions become more
likely.
Now…the graph:

4. Where the immigration rate and extinction rate are equal, the
number of species (S hat) is at equilibrium.
Note that neither the immigration nor extinction rate is 0. Thus
the list of species is ever changing, though the number of
species doesn’t change.

5. There should be more species closer to a source (due to a
higher immigration rate), and more species on larger islands
(decreasing extinction rate). MacArthur and Wilson verified
that for islands in the Sunda group ‘near’ New Guinea...

6. It should be apparent that the quality of habitat on islands is
also important, though not part of the basic theory.
It should also be apparent that the concept of ‘island’ can
(and should) be broadened to include all kinds of isolated
patches of habitat...
Lakes are ‘islands’ in a matrix of land.
Forests are islands in a matrix of agricultural fields…
Finally, when we consider patches, rather than a ‘mainland’
source, and recognize that there may be repeated movement
among patches, we end up with various sorts of
metapopulation-like models…
Among them (with resemblances) are: 1) metapopulation
models, 2) source-sink models, and 3) landscape models.

7. Note that each model is considering the same set of patches.
Source-sink models consider the quality of the patches.
Landscape models consider the environmental context.

8. Basic metapopulation models are simpler, in part because
they make simplifying assumptions about conditions evident
in source-sink and landscape models…
Patches are assumed to uniform and homogeneous.
Patches are assumed to be all equally distant from each other.
The assumption can be called one of no spatial stucture.
(This is impossible in reality, but it enormously simplifies
the math)
There are no time lags in the dynamics.
There is a large number of patches.
There is no concern with population size in a patch, only with
presence there.
There is no temporal variation in probabilities of
immigration (pi) or extinction (pe)

9. In considering population dynamics earlier, we were
describing a closed population. We eliminated immigration
and emigration (or assumed them equal), so that all dynamics
resulted from births and deaths within the population.
In moving to metapopulation models, we recognize that
populations may be subdivided among patches, but that
there will be movement among patches involving a fraction
of individuals in each sub-population.
Once this is accepted, it has important consequences for the
persistence of the population as a whole…
We separate local extinction (of a subpopulation) from
regional extinction (of the whole population).

10. If pe is the probability of local extinction in any year,
what is the probability of persistence? It is (1 - pe) for a
single year.
What is the probability of persistence in a patch for two
subsequent years? Years are independent of each other. The
probability of multiple independent events is the product of
the probabilities of the individual events. For two years…
P2 = (1 - pe) x (1 - pe) = (1 - pe)2
or, for n years…
Pn = (1 - pe)n
but, what if we want to look at persistence in a set of patches?
Now we have to look at a number of simultaneous events in
independent patches...

11. The probability of regional persistence for two patches is:
P2 = (1 - [pe x pe]) = 1 - pe2
and similarly for a larger number of patches.
Now consider what this means for real numbers…
Assume that pe = 0.7.
The probability of a single patch persisting for two years is
0.3 x 0.3 = 0.09
The probability of a set of two patches persisting for a year
is And for two years is = 0.26.
For a larger number of patches, the probability of persistence
increases, and note the increase is a power function.

12. Now we know why there is great importance attached to the
occurrence of metapopulations, and why conservation biology
uses metapopulation concepts to increase the probability of
persistence of endangered populations.
Given immigration onto unoccupied patches and extinction of
subpopulations on some occupied patches, the next step is to
see if there is an equilibrium occupancy (the fraction of
patches occupied)…
The variables we use are:
f - the fraction of sites occupied
pi - the probability of colonizing an unoccupied patch
pe - the probability of local extinction on an occupied
patch

13. The total immigration rate is a function of the probability of
immigration - and the number of unoccupied sites available
to be colonized. Thus:
I = pi (1 - f)
Similarly, the total extinction rate is a function of the
probability of extinction - and the number of occupied sites
where an extinction could occur.
E = pe f
An equilibrium will occur when the fraction of sites occupied
does not change over time, or…
df/dt = 0 = pi (1 - f) - pe f
Solving for the equilibrium f -
f eq = pi / (pi + pe)

14. This result is called the island - mainland model. It results
from the assumption that pi is a constant, i.e. that there is
a continuous source of propagules, a propagule rain. This is
only possible if there is a large, mainland source.
If, instead, the only source of propagules is already colonized
patches, then we need to revise the value of pi
If each occupied site contributes propagules, then the
probability of immigration should increase in direct proportion
to the number of occupied sites...
pi = if
This produces what is called the internal colonization model.
The equation is then:
df/dt = 0 = if (1 - f) - pe f
f eq = 1 - pe / i

15. Note that the ‘terminology’ (the variables) has different names
in this derivation than in the text (p …). The text only
deals with the internal colonization model.
There are important differences in the results predicted by the
two versions of the model…
In the island-mainland model persistence of the metapopulation
is guaranteed as long as the mainland source continues to
disperse propagules (pi > 0).
In the internal colonization model, the metapopulation will
persist only if the strength of the colonization effect (i) is
greater than the probability of local extinction (pe).

16. Here’s an example of the island-mainland situation. The
organism involved is the checkerspot butterfly that lives on
serpentine grasslands near Santa Clara, California.
Morgan Hill is the
‘mainland’.
Surrounding patches
marked by arrows are
occupied. The others are
suitable, but unoccupied.

17. This still considers all patches as being homogeneous and
equal. In the real world quality, isolation, and area vary.
Larger patches can hold larger subpopulations, and are,
therefore, less likely to go extinct. There is good evidence of
the effect of patch size in studies of a shrew on islands in 2
Finnish lakes. Size varied from 0.1 to 1000 ha. Larger
islands were more likely to support populations at any time...

18. The same size effect, and also the effect of isolation, is
evident in plants from dry grassland areas along the Rhine...

19. Isolation makes a difference, not only for initial colonization,
but also in what is called the rescue effect.
If a population is dwindling toward extinction, but is near
sources of new colonists, then there is a chance that new
immigrants of species already present will rescue the
population from extinction. James Brown (not the soul singer)
demonstrated it in populations of arthropods on thistles in his
backyard in Tucson…
Site category # plants # species mean turnover
large-near
large-far
small-near
small-far

20. We can insert the concept of rescue into the metapopulation
models by including a formula for pe .
Let the pe be a function of some characteristic extinction
rate e multiplied by a number that decreases as a greater
fraction of sites is occupied…
pe = e (1 - f)
You can insert this function in either model and solve for a
new equation for equilibrium. What is different here is that
there are conditions that lead to “saturation”, i.e. f = 1, all
sites are predicted to be occupied. In the internal colonization
model, this is predicted if i > e. In the island-mainland model,
that is the result if pi > e.

21. Why are we interested in metapopulation concepts (or
extensions into source-sink and lanscape models)?
One important application is in conservation biology. There
is great danger of extinction for a large number of species.
If they are all in one ‘preserve’, what are their chances?
Far smaller than if we conserve them in the form of a number
of subpopulations, particularly if we find ways of permitting
individuals in the separate subpopulations to disperse among
‘patches’ or parks in which they survive.
How do we achieve this end?
1) in nature
2) under controlled conditions (i.e. zoos and parks)