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Metapopulations or “Populations of populations” (Levins 1970). pop. pop. pop. pop. pop. pop. pop. Assumptions. Homogeneous patches. No spatial structure. No time lags. Constant c and e. Amount of colonization and extinction is a function of P.

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Metapopulations or populations of populations levins 1970 l.jpg

Metapopulations or “Populations of populations”(Levins 1970)

pop

pop

pop

pop

pop

pop

pop


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Assumptions

  • Homogeneous patches

  • No spatial structure

  • No time lags

  • Constant c and e

  • Amount of colonization and extinction is a function of P

  • Large number of patches (essentially ∞)

  • No internal dynamics


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Equlibria

Internal solution

Equilibria

Population change (D P)

P


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General Implications of the Levins’ Model

  • If e>c then P = 0

  • If c>e then there is a stable internal equilibrium point


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Growth of a newly colonized metapopulation

For a very small value of P,

the term cP2 in

can be ignored leading to an early growth rate of

“intrinsic rate of increase”

This looks like exponential growth!


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With increasing occupation of patches

“carrying capacity”

This looks like logistic growth!


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Impact of Change in Model Structure

Fewer Patches

Smaller Patches

rate

P

P



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Mainland-Island Metapopulations

Propagule rain

if cm>>c


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Mainland-Island Model Equilibrium

Note: the multispecies extension forms the basis of the theory of island biogeography.


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The Rescue Effect

“Reduction in the probability of extinction when more population sites are occupied”(Gotelli 1995)

eP e(1-P)P

Extinction rate


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Mainland-Island Model with Rescue Effect

equilibrium:

If cm > e then

vs


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Levins’ Model with Rescue Effect

equilibria:

(1) If c> e then

(2) If c < e then

(3) If c = e then neutral equilibrium


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Impact of Habitat Loss

(1- h = 0.25)

Let 1-h = represent the fraction of habitat destroyed


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Levins’ Model Incorporating Habitat Loss

Equilibrium:

Metapopulation extinct

is the fraction of occupied patches at steady state


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The Levins’ Rule

“A sufficient condition for meta-population survival is that the number of patches remaining after habitat destruction exceeds the number of empty, but suitable, patches prior to patch destruction”

(Hanski et al. 1996)


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Problems with the Levins’ Rule

vis-á-vis the Rescue Effect

For high migration rates

where

Fis the probability of a subpopulation

surviving for one time interval and

G is the probability of recolonization when all patches are occupied


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Problems with the Levins’ Rule

vis-á-vis the Rescue Effect

For high migration rates


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Problems with the Levins’ Rule

vis-á-vis the Rescue Effect

“A sufficient condition for meta-population survival is that the number of patches remaining after habitat destruction exceeds the number of empty, but suitable, patches prior to patch destruction”

(Hanski et al. 1996)

e/c



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Evidence for a Rescue Effect

With rescue effect

No rescue effect


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Metapopulation Persistence as a Function of Patch Number

From a stochastic version of the Levins’ model

TM expected time to metapopulation extinction

TL expected time to local extinction

H number of suitable patches

(Gurney and Nisbet 1978)


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Minimum Viable Metapopulation Size

If we define long term persistence to mean

then

insures long term persistence, where

H is the number of available patches.

(Gurney and Nisbet 1978)


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Minimum Viable Metapopulation Size

Island type migration

Distant dependent migration


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Structured Metapopulation Models

Or

Internal Dynamics Matter


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Structured Metapopulation Models

Structured Model

Unstructured Model

  • Slow local dynamics

  • Fast local dynamics

  • Higher migration rates

  • Low migration rates

(1) Local dynamics can be ignored

(2) Focus is on extinction rates

(1) Migration rates may impact local dynamics

(2) Local dynamics cannot be ignored


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A “Simple” Structured Model

  • An extension of the basic Levins’ model

  • Equilibrium state depends on emigration (m) and extinction (e) rates

  • All populations have the same internal dynamics

  • characterized by logistic growth:


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A “Simple” Structured Model

Levins’ Model

Strucutured Model

a’ = fraction of migrants surviving and landing in a patch

b = rate of successful colonization

(Hanski and Zhang 1993)


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A “Simple” Structured Model

Equilibrium conditions

Yuuuch!

B=4e/bK


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A “Simple” Structured Model

Graphical interpretation

no stable equilibrium point

2 stable equilibrium points – one positive

1 stable equilibrium point

r = 1 a’ = 0.5 bK = 1


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A “Simple” Structured Model

Graphical interpretation

r = 1 a’ = 0.5 bK = 1





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A Spatially Explicit Metapopulation Model

in Continuous Time

Rates are patch specific

(Hanski and Gyllenberg 1997)


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Expected Equilibrium

Probabilities of Patch Occupation

where ei = 1/Ai ,

Ai

is the area of patch i, and

(Hanski and Gyllenberg 1997)


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A Closer Look at the Function Ci(t)

Ci(t)is the amount of colonization into patch i at time t from all other patches j in the metapopulation

c = colonization rate

a = distance decay rate for migration

dij = distance between patches i and j

R = the number of patches in the metapopulation

(Hanski and Gyllenberg 1997)


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Solving for

Begin by substituting

into

yielding

(Hanski and Gyllenberg 1997)


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Solving for

is solved by interation

to obtain values for

allowing us to determine

(Hanski and Gyllenberg 1997)


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The Incidence Function Model of Hanski

The incidence

is the long term probability that

patch i will be occupied.

This is a discrete time stochastic model similar in design to the previous, continuous time model. However, it is more flexible in how spatial interactions are entered into the model.