Metapopulation Biology Concept and Application

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Metapopulation Biology - Concept and Application
A metapopulation is a set of local populations
which interact via individuals moving among
populations. According to Levins' original
definition, a metapopulation is "a population of
populations". Each component is a local
population. A single local population occupies a
patch. If fragments (local populations) rarely,
if ever, exchange migrants, the islands are
isolated, and better considered in the context of
island biogeography. A patch is an area of some
type of habitat, considered in the simple models
as uniform and homogeneous, which is occupied by
a local population. The basic models do not
consider population size within a patch, only
presence or absence.
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Turnover is the term used to describe local
extinction and colonization to form new local
populations within the metapopulation by means of
dispersal from other local populations in
occupied patches. Most basic metapopulation
models are occupancy models. The fraction of
patches occupied by a species is what is counted
or modeled. What is usually of interest is
whether persistence of the metapopulation
greatly exceeds the expected persistence of
individual local populations. In other terms,
whether the probability of regional extinction is
much lower than the probability of local
extinction.
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The Basic Levins Model We begin with a
large' number of homogeneous patches suitable
for occupation by the species under study. Of the
total number of patches available in the region,
the number currently occupied by this species is
called f 0 lt f lt 1. The value of f 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. df/dt I - E
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Unlike the similar equation for exponential
growth, there are constraints on the parameter
measuring growth the value of f lies between 0
and 1. One of the basic uses for the model is
to assess the impact of metapopulation structure
on the probability of regional (i.e.
metapopulation) extinction. 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.
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A simple example given is one in which 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 0.32 .09
or, for an n year period
Pn (1-pe)n
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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
persistence (not all going extinct in any one
year) 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 high probability of local extinction in
each of the component populations. For that same
example, where pe 0.7, lets assume there
are 5 component populations.
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• Then
• Pn 1 - (0.7)5 1 - 0.168 0.832
• A useful, and at least somewhat realistic model
  needs to go deeper into factors which influence
  the immigration and extinction rates.
• Immigration rates are clearly affected by a
  number of physical and biological characteristics
  of the patches). Among the important physical and
  biological variables are
• patch size
• the distance separating patches
• current occupants of the patch
• immigration rates

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• patch size - a larger patch is a more likely
  target for dispersing organisms. Patch size will
  also affect the size of a saturated local
  population, but that is not considered in the
  basic occupancy model.
• The distances separating patches. The basic model
  assumes there is no spatial structure, that
  patches are spread uniformly over the landscape.
  Under that assumption the likelihood of
  dispersers reaching patches is also equal for all
  patches. In reality patches are unlikely to be
  uniformly distributed, and more isolated patches
  are less likely to receive immigrants. Though
  patches may be uniformly distributed, it is
  practically impossible for all patches to be
  equidistant.

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• The current occupants of the patch. Immigrants
  become part of communities which may contain
  competitors, predators, and pathogens of the new
  arrival. The presence and relative abundance of
  any or all of these interacting species may
  determine the success of the immigrant.
• Finally, immigration rates in a system of patches
  almost certainly depends on the number of patches
  already occupied. As well, even if spacing is
  uniform, real dispersal rates onto unoccupied
  patches probably don't depend on a total fraction
  occupied, but on the fraction of nearby patches
  occupied. Dispersal probabilities are almost
  certainly
• a function of distance.

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If only a few patches are occupied, the number of
dispersers is probably low, and therefore the
probability of immigrating onto the large number
of unoccupied patches is also low. If almost
all patches are occupied, then there is a
profusion of dispersers, and most (if not all)
unoccupied sites are likely to be reached. That
view can be quantified
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The probability of immigration onto a site, pi,
is a function of f, the fraction of sites already
occupied. The immigration rate depends on both pi
and the fraction of unoccupied sites (1-f),
i.e. I pi(1-f) The
extinction rate depends on a characteristic
probability, pe, for all local populations and
the fraction of sites currently occupied  
E pef 
Combining these simple improvements on the
basic equation for the rate of change in f over
time  df/dt
I E pi(1-f) - pef
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• Recognize an assumption youve seen before (and
  other assumptions)
• This model assumes no time lags. Effects of
  changes in f, pi, or pe on the rate of change
  in occupancy fraction occur immediately.
• The basic model is deterministic, and assumes
  that there are a large number of patches, a
  necessity of a continuous model.
• Patches are homogeneous (size, isolation, habitat
  quality, resource availability)
• There is no spatial structure in the distribution
  of patches. (This is not possible in reality)
• Probabilities of extinction, pe, and immigration,
  pi, are constant across time.

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• Finally, although it was already presented in the
• improved equation, recognize that were
  assuming regional occurrence, f, affects the
  probabilities of extinction and immigration. The
  actual values of pe and pi are functions of f.

Now we can see how the details of the model
change when we examine the two kinds of models
that have been developed the island-mainland
model and the internal colonization model
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These two approaches add further detail to the
immigration and extinction functions, the pi and
pe. One considers the set of occupied patches
as the source of colonists (internal
colonization). In the other, colonists are
dispersed from a mainland' source external to
the metapopulation.
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Internal Colonization In this model the
probability of immigration is itself a function
of the number of occupied patches (f ) and a
constant which measures how much each patch
contributes to the probability (i). The
probability of immigration is
pi if and the rate of change in occupancy
df/dt if(1-f) - pef This is the
model (equation) proposed by Levins in the
original formulation of the metapopulation
concept.
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It can be solved for an equilibrium value of f.
Set df/dt 0 and solve for f.
pef if (1 - f)
pe i (1 - f)
if i - pe
and finally the solution f
1 - (pe/i)
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• According to this equation, the metapopulation
  will persist only if the effect of an occupied
  island on immigration, i, is greater than the
  characteristic probability of a local population
  to go extinct, pe.
• If we assume these characteristics correspond to
  rates of immigration and extinction, you can look
  at the system as having thresholds for
  persistence.
• For any given extinction rate, there is a
  minimum threshold rate for colonization if
  the system is to persist.
• For any given immigration rate, there is a
  maximum threshold rate of extinction which is
  tolerable if the system is to persist.

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Extinction rates in basic island biogeography are
functions of island area immigration rates are
functions of island isolation. So, if islands
are separated by some average distance
(isolation), persistence in the system requires
that the islands have a threshold minimum area,
limiting the extinction rate. For a given
average area, there is a threshold maximum
isolation which is tolerable if a species is to
persist on a group of islands. We can add this
into the metapopulation model.
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A logical model for the effect of distance on
immigration is a negative exponential
relationship, i.e.
i i0e-aD and, though it is not so
widely used or accepted, suggest a parallel
relationship between area and the extinction
parameter, i.e.
pe k0e-bA

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Now the formula for determining equilibrium
becomes f 1 - (k0/i0)eaD-bA Tests of
this model have run into problems, most traceable
to the assumptions of uniform area and absence of
any spatial structure in the distribution of
patches. Hanski (1991) studied how the
relationship between occupied and unoccupied
islands is affected by patch area and isolation.
The data are for occupation of islands by Sorex
araneus, a small shrew, in large lakes in
Finland. The line in the figure on the next page
is a best fit separating occupied from unoccupied
islands. Presence is indicated by filled points
and absence by open dots.
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Now we can draw on island biogeography to get an
idea about the biological patterns likely with
different patch sizes and isolation Diamond
introduced the idea of incidence functions. They
measure what fraction of islands is occupied by a
species as a function of island area. Since
Diamond used the approach to view differences
among species, he found some species present only
on small islands, and others present only on
large ones, plus a variety of intermediates. He
argued that this reflected the ecology of the
species. Those present only on small islands he
called super-tramps' or tramps'. These species
were superb immigrators, but very poor
competitors.
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On larger islands, as increasing diversity
accumulated, the tramp-type species were driven
extinct. On very small islands, with only small
population sizes possible, the probability of
local extinction is high, and for species with
slow growth and slow colonization, characteristic
of large islands, this is the likely result.
How can we incorporate this into metapopulation
biology?
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One way of relating extinction probability to
patch area is to make the effective e a function
of area. In the development of incidence
functions, area was incorporated not as a
negative exponential as indicated earlier, but
according to Diamond in the form of
pe k0A-x The parameter x
describes how fast extinction probability
decreases with area. Note that, since extinction
cannot exceed 1, there is a minimum value of A
which must occur to ensure this. Now we develop
an incidence function, and, with Diamond,
designate it J.
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The incidence function, J, bears a close
resemblance to the equilibrium occupancy, except
that we are describing only how the probability
of occupancy varies with area. J(A) 1/1
(k0/i)A-x Hanski fit values for x and the ratio
k0/i, as well as observed and predicted values of
rates of immigration and extinction for a group
of shrew species on lake islands. To give you a
sense of the values which fit a small mammal,
probably a relatively poor colonizer
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Parameter estimates for incidence functions of 3
shrew species on islands in large lakes in
Finland Species x k0/i
Observed pi Observed pe Sorex araneus
2.30 0.79 0.20
0.04 S. caecutiens 0.91 17.67
0.05 0.33 S. minutus
0.46 4.09 0.13
0.46 The values of x tell us something about
these species. For S. araneus x gt 1. Persistence
for this species increases rapidly (gt
exponentially) with increasing island area.
Persistence increases for the other species, as
well, but much more slowly. For the smallest
species, S. minutus, the exponent is only about
0.5, increasing like the square root.
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One of the most widely studied types of organisms
in metapopulation biology is butterflies. The
same sort of pattern observed for shrews occurs
in various species of butterflies, among them a
skipper, Hesperia comma. In this figure you can
see that both patch area and isolation are
important in predicting occupancy
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The same point can be made using a fritillary
butterfly on those Finnish islands Area
Effect Isolation Effect Average patch
Occupancy patches/ Occupancy
area prop. 4
sq.km. prop. lt0.01 23
0.24 1 61 0.21 0.01-0.1
138 0.24 2-3 70 0.32 0.10-1.0
88 0.40 4-7 58 0.25
gt1.0 6 0.56 gt7 66 0.41
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One more Fahrig and Merriam (1985) found local
population size in Peromyscus leucopus in Ontario
woodlots varied as a function of the isolation of
the woodlots (i.e. reflecting the importance of
immigrants and emigrants on dynamics) and
occupancy.
Thus, both patch size and isolation are
important. The distribution of these
distributions is continuous. However, the basic
model structures are from opposite ends of the
spectrum. The other model assumes
colonization from outside the set of patches,
from a continuously occupied mainland.