Galapagos Islands

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. 2008 Am. Nat.)

Galapagos Islands

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

# sp

ecie

s

Species richness increases with island AREA

MacArthur and Wilson (1967)

Rich Glor

Log

scal

e

Log scale

Galapagos land plants

S = c · Az

c is constant of spp./ areaz is the slope

z = 0.32 (~0.3 for most islands)

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)

Species richness decreases with isolation

More isolated islands less likely to receive colonists (immigration low)

MAIN

LAND

large

small

near

far

Extinction more likely for small populations (small islands)

Colonization more likely for closer islands

Island biogeography

MacArthur and Wilson 1967

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?

Simberloff and Wilson’s experiment Florida Keys

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

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

And now for a big mental leap...from diversity to individual population dynamics

Rana cascadae

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

• A system of populations that is linked by occasional dispersal (Levins 1969).

Levins, R. 1969. Bull. Ent. Soc. Am. 15:237-240.

Shaded areas provide an excess of individuals which emigrate to and colonize sink habitats (open).

Metapopulation dynamics

Anthropogenic subpopulations

And here…

Islands?

Classic metapopulation• Deterministic, but based on

stochastic events—extn, colonization

• Subpopulations have independent dynamics and are connected by dispersal

• All patches of identical quality

Classic metapopulationGeneral model:

dp/dt = C – E

p = proportion of sites occupied

C = total colonization rate

E = total extinction rate

Classic metapopulationLevins model (insect pests in fields)

dp/dt = cip (1 –p) - ep


ci = rate of propagule production (internal)

e = local extinction rate (constant)

Available for colonization

Total colonization rate Total extinction rate

Classic metapopulationFinding 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 > ep* = 1 only when e = 0

Gotelli’s model (also called Island-mainland)

dp/dt = ce (1 –p) - ep


ce = rate of propagule rain (external—doesn’t depend on p)

e = local extinction rate (constant—mediated only by p)

‘Propagule rain’ metapopulation

‘Propagule rain’ metapopulation

Finding the equilibrium:

0 = dp/dt = ce (1 –p) – ep

0 = ce – cep – ep

Solve for p

p* = ce /(ce + e)

Habitat loss models

X

Karieva & Wennegren’s model

dp/dt = ci p (1 - D - p) - ep

D = fraction of total habitat destroyed(D=0 is Levins)


ci = rate of propagule production (internal—does depend on p)

e = local extinction rate (constant—mediated only by p)

Habitat loss models

X

Karieva & Wennegren’s model


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

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

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)

• Persistence of some local populations (sinks) depends on some migration from nearby populations (sources).

• Empty patches susceptible to colonization.

Hanski (1998) proposed 4 conditions before assuming that metapopulation dynamics explain spp. persistence

sinksource

Hanski, I. 1998. Nature 396:41−49.

i.e. persistence depends on supply from source population, some fragments may rarely receive or supply migrants

1) 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

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


Hanski’s model

dp/dt = cip (1 - p) - ep (1-p)


ci = rate of propagule production (internal—does depend on p)

e = local extinction rate (constant)

Total extinction rate

Total extinction declines as p ~ 1, propagules likely to land at all sites


Hanski’s model

dp/dt = cip (1 - p) - ep (1-p)


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)

Classic metapopulations

Adding area and isolation

Hanski (1991)Hanski and Ovaskanen (2003)

Incidence Function ModelsMore spatially realistic (uses real data)Incorporate patch size, distance matrixStochastic

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)

Incidence Function ModelsAssumes 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

Galapagos Islands

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