Metapopulations

1
Metapopulations

• Objectives
• Determine how e and c parameters influence
  metapopulation dynamics
• Determine how the number of patches in a system
  affects the probability of local extinction and
  probability of regional extinction
• Compare propagule rain vs. internal
  colonization metatpopulation dynamics
• Evaluate how the rescue effect affects
  metapopulation dynamics

2
Metapopulations

• There are relatively few examples where the
  entire population resides within a single patch
• Most species are patchily distributed across
  space
• Hence it becomes a population of
  populationsmetapopulations

3
Metapopulations

• Metapopulation theory (Levin 1969, 1970)
  describes a network of patches, some occupied
  some not, where subpopulations are interacting
  (winking)
• The classic model is then based upon
  presence-absence, not a demographic model like
  source-sink dynamics

4
Fragmentation Heterogeneity
5
Fragmentation Heterogeneity
6
Fragmentation Heterogeneity
7
Metapopulations

• Lets define extinction and colonization
  mathematically
• Extinction pe thus persistence is 1-pe
• Colonization is pi with vacancy is 1-pi
• We can consider the fate of a single patch over
  time or the entire metapopulation over time

8
Metapopulations

• For a given patch, the likelihood of persistence
  for n years is simply
• pn (1 pe)n
• E.g. if a patch has a probability of persistence
  0.8 in a given year, the probability for 3
  years 0.83 0.512
• If we had 100 patches, approximately 52 would
  persist and 48 would go extinct

9
Metapopulations

• To consider the fate of the entire metapopulation
  (i.e. the probability of extinction of the entire
  population)
• If all patches have the same probability of
  extinction, it is simply pex
• For example, if pe0.5 across 6 patches then Px
  1-(pe)x or 0.56 0.0156 or 1.5

10
Metapopulations

• Now that we have defined e and c, let us consider
  the basic metapopulation model
• where f is the fraction of patches occupied in
  the system (e.g. 5/25 0.2)
• If f is the fraction of patches occupied, then
  1-f is the fraction empty, the we can compute I
  as I pi ( 1 - f )

df / dt I -E
11
Metapopulations

• Focusing on E, the rate at which occupied patches
  go extinct
• E should depend upon the number of patches
  occupied as well as the extinction probability
  (pe)
• Substituting our new values of I and E

E pef
df/dt pi (1 f) - pef
12
Metapopulations

• This is called the propagule rain model or an
  island-mainland model, because the colonization
  rate does NOT depend on patch occupancy
  patterns-it is assumed that colonists are
  available to populate and empty patch and they
  can come from within or outside the metapopulation

df/dt pi (1 f) - pef
13
Metapopulations

• At equilibrium the fraction of patches is
  constant, although the exact combination is
  dynamic
• The equilibrium fraction can be derived by
  setting the rate to 0

dt/dt 0 pi pif pef and then f pi / (pi
pe)
14
Metapopulations

• There are many important assumptions (as with any
  model), the most important being all patches are
  created equal pe and pi are constant over time
  and apply to patches irrespective of population
  size and finally, spatial arrange and proximity
  are not important to pe or pi
• You probably can imagine how when f is high, pi
  is probably large

15
Metapopulations

• This type of model is called the internal
  colonization model because colonization rates
  depend on current status (f) of the
  metapopulation system
• Extinction of a patch may depend on the fraction
  of patches occupied in the system
• When f is high, there are many potential
  colonists and pe decreases
• This is termed the rescue effect

16
Metapopulations

• Rescue effect and Internal Colonization Model

17
Metapopulations

• Objectives
• Determine how e and c parameters influence
  metapopulation dynamics
• Determine how the number of patches in a system
  affects the probability of local extinction and
  probability of regional extinction
• Compare propagule rain vs. internal
  colonization metatpopulation dynamics
• Evaluate how the rescue effect affects
  metapopulation dynamics