Levins model: patch occupancy as colonization–extinction balance
0.30
0.15
0.10
50
0.50
Time: 0
Occupancy: 0.00
Equilibrium p*: —
Status: —
Levins metapopulation model (1969): dp/dt = cp(1−p) − ep. The fraction of occupied patches p* = 1 − e/c at equilibrium (metapopulation persists iff c > e). The rescue effect (Brown & Kodric-Brown 1977) reduces local extinction in proportion to occupancy: extinction rate becomes e(1−γp), making persistence easier. The mainland-island model has a constant propagule source — colonization doesn't require occupied patches, so there's no threshold and extinction risk is simply e/(c+e). The spatial network view (Hanski 1994) extends this with explicit patch geometry; this visualization shows patch occupancy dynamics in discrete stochastic simulations.