1. Galapagos Islands

2. Metapopulation Models
Species area curves (islands as case study)
Review of Island Biogeography
Extension to subpopulations  Metapopulation
Classic (Levins)
Island-mainland (Gotelli)
Core-satellite (Hanski)
Habitat loss (Lande/Karieva)
More spatially realistic
Incidence Function models
Matrix metapopulation models
(Ozgul et al Am. Nat.)

3. Galapagos Islands

4. Island species richness
Bigger islands have more species than small islands
“Species-Area curves”
Documented for diverse taxa
Other types of habitat also follow this pattern.....island-like (mtn. tops, forest remnants, lakes, etc.)
Island area
# species

5. Species richness increases with island AREA
Log scale
Rich Glor
Log scale
MacArthur and Wilson (1967)

6. Galapagos land plants S = c · Az c is constant of spp./ area
z is the slope
z = (~0.3 for most islands)

7. An extension of this idea: Island biogeography
Dynamic equilibrium theory that explains species richness of islands
Island richness determined by colonization and extinction rates (number of species thru time)
Richness increases with size (more habiats to support more species, less extinction)....
decreases with isolation (less likely to be colonized)

8. Species richness decreases with isolation
More isolated islands less likely to receive colonists (immigration low)

9. small MAINLAND large near far
Extinction more likely for small populations (small islands)
MAINLAND
large
near
Colonization more likely for closer islands
far

10. Island biogeography
MacArthur and Wilson 1967

11. Experimental test of island biogeography
Defaunation experiment by Simberloff and Wilson (1969 Ecology)
Methods:
Survey small mangrove islands for arthropods.
Cover islands with plastic and spray with insecticide (gets rid of all arthropods)
Observe colonization/ succession over one year.
How many and what species return?

13. Simberloff and Wilson’s experiment Florida Keys

14. Results
Species richness on islands returned to levels similar to before defaunation
Closer, larger islands had more species
The precise species identity was not consistent, only the total number of species
Order of colonization and species interactions important for “who” composes the community

15. And now for a big mental leap
And now for a big mental leap... from diversity to individual population dynamics
Metapopulations
Collection of subpopulations
Proportion of sites “occupied” determined by colonization and extinction rates at each site
Connected by individual movement (dispersal between sites provides colonists)
Individual sites may be colonized in one year and extinct the next
Individual site dynamics are variable, but overall “metapopulation” can be stable
Rana cascadae

16. Metapopulation structure
Levins (1969) introduced the concept of metapopulations to describe the dynamics of this patchiness, as a population of fragmented subpopulations occupying spatially separate habitat patches in a fragmented landscape of unsuitable habitat
Metapopulation structure
Shaded areas provide an excess of individuals which emigrate to and colonize sink habitats (open).
A system of populations that is linked by occasional dispersal (Levins 1969).
Levins, R Bull. Ent. Soc. Am. 15:

17. Metapopulation dynamics

18. Metapopulation dynamics

19. Metapopulation dynamics

20. Metapopulation dynamics

21. Metapopulation dynamics

22. Metapopulation dynamics

23. Metapopulation dynamics

24. Anthropogenic subpopulations
And here…
Anthropogenic subpopulations
Islands?

25. Classic metapopulation
Deterministic, but based on stochastic events—extn, colonization
Subpopulations have independent dynamics and are connected by dispersal
All patches of identical quality

26. Classic metapopulation
General model:
dp/dt = C – E
p = proportion of sites occupied
C = total colonization rate
E = total extinction rate

27. Classic metapopulation
Levins model (insect pests in fields)
dp/dt = cip (1 –p) - ep
p = proportion of sites occupied
ci = rate of propagule production (internal)
e = local extinction rate (constant)
Available for colonization
Total colonization rate
Total extinction rate

28. Classic metapopulation
Finding the equilibrium:
0 = dp/dt = cip (1 –p) – ep
0 = cip – cip2 – ep
Solve for p
p* = 1- e/ci
Lesson:
p* > 0 when ci > e
p* = 1 only when e = 0

29. ‘Propagule rain’ metapopulation
Gotelli’s model (also called Island-mainland)
dp/dt = ce (1 –p) - ep
p = proportion of sites occupied
ce = rate of propagule rain (external—doesn’t depend on p)
e = local extinction rate (constant—mediated only by p)

30. ‘Propagule rain’ metapopulation
Finding the equilibrium:
0 = dp/dt = ce (1 –p) – ep
0 = ce – cep – ep
Solve for p
p* = ce /(ce + e)

31. X Habitat loss models Karieva & Wennegren’s model
dp/dt = ci p (1 - D - p) - ep
D = fraction of total habitat destroyed
(D=0 is Levins)
p = proportion of sites occupied
ci = rate of propagule production (internal—does depend on p)
e = local extinction rate (constant—mediated only by p) X

32. X Habitat loss models Karieva & Wennegren’s model
Finding the equilibrium:
0 = dp/dt = ci p (1 - D - p) – ep
Solve for p
p* = 1- e/ci – D
Simple direct effect of D on metapopulation dynamics X

33. Assumptions of classic models:
Identical patch ‘quality’
Extinctions occur independently, local dynamics are asynchronous
Colonization spreads across patch network and all equally likely
All patches equally connected to all other patches (not spatially explicit)
No population counts or size. Only suitable vs. unsuitable, occupied or unoccupied

34. Classic metapopulations
Unoccupied patches common and necessary for metapopulation persistence
Local dynamics asynchronous
Subpopulations connected by high colonization  single population (synchronized dynamics)
Matrix between sites is homogeneous
Parallel to logistic model of pop growth (filling available ‘space’ up to K)

35. Empty patches susceptible to colonization.
Persistence of some local populations (sinks) depends on some migration from nearby populations (sources).
Empty patches susceptible to colonization.
source
sink
Hanski (1998) proposed 4 conditions before assuming that metapopulation dynamics explain spp. persistence
Hanski, I Nature 396:41−49.

36. 2) No single population is large enough to ensure long-term survival.
Patches should be discrete habitat areas of equal quality i.e. homogeneous.
2) No single population is large enough to ensure long-term survival.
3) Patches must be isolated but not to the extent of preventing re-colonization from adjacent patches.
4) Local population dynamics must be sufficiently asynchronous that simultaneous extinction of all local populations is unlikely.
Without 2 & 3: we have a ‘mainland-island’ metapopulation
i.e. persistence depends on supply from source population, some fragments may rarely receive or supply migrants

37. Core-satellite metapopulation
Observation that high rates of colonization can ‘rescue’ pops nearing extinction
RESCUE EFFECT
Satellites persist because they are resupplied with individuals from occupied sites
Effectively differences in patch quality

38. Core-satellite metapopulation
Hanski’s model
dp/dt = cip (1 - p) - ep (1-p)
p = proportion of sites occupied
ci = rate of propagule production (internal—does depend on p)
e = local extinction rate (constant)
Total extinction declines as p ~ 1, propagules likely to land at all sites
Total extinction rate

39. Core-satellite metapopulation
Hanski’s model
dp/dt = cip (1 - p) - ep (1-p)
Finding the equilibrium:
dp/dt = (ci – e) p(1 - p)
Notice:
If ci > e , sign is positive ( λ > 1), p  1 (stable)
If e > ci , sign is negative ( λ < 1), p  0 (extinct)
If e = ci , λ = 1 (neutral equilibrium)

40. Classic metapopulations

41. Adding area and isolation
Hanski (1991)
Hanski and Ovaskanen (2003)

42. Incidence Function Models
More spatially realistic (uses real data)
Incorporate patch size, distance matrix
Stochastic
Ei = e/Ax
(estimated from field data)
e = probability of extinction for a patch of size A (per unit time)
A = patch Area
x = demographic & environmental stochasticity (if large, E decreases fast with A)

43. Incidence Function Models
Assumes that internal dynamics are ‘fast’ relative to among patch dynamics
Populations within patches ‘reach’ carrying capacity almost instantly (Ei, Ci constant)
Immigration to patch I proportional to connectance (proportional contribution from all occupied sites)
Can evaluate the importance of patch loss on overall stability