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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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Presentation Transcript
slide2

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
slide3

Equlibria

Internal solution

Equilibria

Population change (D P)

P

slide4

General Implications of the Levins’ Model

  • If e>c then P = 0
  • If c>e then there is a stable internal equilibrium point
slide5

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!

slide6

With increasing occupation of patches

“carrying capacity”

This looks like logistic growth!

slide7

Impact of Change in Model Structure

Fewer Patches

Smaller Patches

rate

P

P

slide9

Mainland-Island Metapopulations

Propagule rain

if cm>>c

slide10

Mainland-Island Model Equilibrium

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

slide11

The Rescue Effect

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

eP e(1-P)P

Extinction rate

slide12

Mainland-Island Model with Rescue Effect

equilibrium:

If cm > e then

vs

slide13

Levins’ Model with Rescue Effect

equilibria:

(1) If c> e then

(2) If c < e then

(3) If c = e then neutral equilibrium

slide14

Impact of Habitat Loss

(1- h = 0.25)

Let 1-h = represent the fraction of habitat destroyed

slide15

Levins’ Model Incorporating Habitat Loss

Equilibrium:

Metapopulation extinct

is the fraction of occupied patches at steady state

slide16

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)

slide17

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

slide18

Problems with the Levins’ Rule

vis-á-vis the Rescue Effect

For high migration rates

slide19

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

slide21

Evidence for a Rescue Effect

With rescue effect

No rescue effect

slide22

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)

slide23

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)

slide24

Minimum Viable Metapopulation Size

Island type migration

Distant dependent migration

slide25

Structured Metapopulation Models

Or

Internal Dynamics Matter

slide26

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

slide27

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:
slide28

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)

slide29

A “Simple” Structured Model

Equilibrium conditions

Yuuuch!

B=4e/bK

slide30

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

slide31

A “Simple” Structured Model

Graphical interpretation

r = 1 a’ = 0.5 bK = 1

slide35

A Spatially Explicit Metapopulation Model

in Continuous Time

Rates are patch specific

(Hanski and Gyllenberg 1997)

slide36

Expected Equilibrium

Probabilities of Patch Occupation

where ei = 1/Ai ,

Ai

is the area of patch i, and

(Hanski and Gyllenberg 1997)

slide37

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)

slide38

Solving for

Begin by substituting

into

yielding

(Hanski and Gyllenberg 1997)

slide39

Solving for

is solved by interation

to obtain values for

allowing us to determine

(Hanski and Gyllenberg 1997)

slide40

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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