Persistence in Metapopulations

Patches/Islands: Open or occupied

Consider single patch: Open or occupied

Occupation number 0: Extinct, 1: Extant

Single Deme: Year t to (t + 1)

Time t: x = 1 (occupied patch)

Time (t+1) x = 0 with Probability = qx = 1 with Probability = 1 – q

q Annual Extinction Probability

q Annual Extinction Probability

Persistence of Deme

Time t: x = 1

Pr[x = 1 at time (t+1)] = 1 – qPr[x =1 at (t+1) and (t+2)] =

Generalize

q Annual Extinction Probability

Time t: x = 1

Prob. deme persists at least n consecutive years

Average duration of deme = years

Two Demes: Year t to (t + 1)

Time t: Demes a and bOccupation numbers: xa = 1, xb = 1Both demes extant

Annual Extinction Probability = q

Demes independent wrt extinction

Two Demes, Annual Extinction Pr = q

Metapopulation Persistence

Pr[1 or Both Persist 1 Year] = 1 – Pr[Both Extinct]

Pr[Both Extinct at (t + 1)] = Pr[Persistence] =

Generalize

Two Demes, Annual Extinction Pr = q

Consider m extant demes at time tExtinctions independent

Pr[Persistence] = Pr[1 or more demes persist] =

Larger m often implies greater persistence

Colonization-Extinction Equilibrium

p(t) Occupation frequency, time t

Colonization rate C[p(t)] = Extinction rate E[p(t)] =

Which model?

Colonization-Extinction Equilibrium

C = E at equilibrium;

Colonization-Extinction Equilibrium

p(t) Occupation frequency, time t

Colonization rate C[p(t)] = Extinction rate E[p(t)] =

Which model?

Colonization-Extinction Equilibrium

C = E at equilibrium;

Persistence

Text 4.2

Manage Endangered Frog Population

1 Deme, 100 IndividualsPr[Annual Extinction] = q = 0.1

3 Demes, 33 Individuals/DemePr[Deme, Annual Extinction] = q = 0.5

Greater Annual Persistence?

Persistence

1 DemePr[Persists 1 Year] =

3 DemesPr[Persistence for 1 Year] = 1 – Pr[All Extinct in 1 Year]

Single Deme (Larger Population) “Better”