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Metapopulations or “Populations of populations” (Levins 1970)

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- 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

General Implications of the Levins’ Model

- If e>c then P = 0

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

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!

Mainland-Island Model Equilibrium

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

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

eP e(1-P)P

Extinction rate

Levins’ Model with Rescue Effect

equilibria:

(1) If c> e then

(2) If c < e then

(3) If c = e then neutral equilibrium

Levins’ Model Incorporating Habitat Loss

Equilibrium:

Metapopulation extinct

is the fraction of occupied patches at steady state

“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)

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

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

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)

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)

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

- 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:

Levins’ Model

Strucutured Model

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

b = rate of successful colonization

(Hanski and Zhang 1993)

Graphical interpretation

no stable equilibrium point

2 stable equilibrium points – one positive

1 stable equilibrium point

r = 1 a’ = 0.5 bK = 1

Spatially Explicit Metapopulation Models

Area effects

Spatially Explicit Metapopulation Models

Distance effects

A Spatially Explicit Metapopulation Model

in Continuous Time

Rates are patch specific

(Hanski and Gyllenberg 1997)

Probabilities of Patch Occupation

where ei = 1/Ai ,

Ai

is the area of patch i, and

(Hanski and Gyllenberg 1997)

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)

is solved by interation

to obtain values for

allowing us to determine

(Hanski and Gyllenberg 1997)

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.

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