€

E = eP

C = mP(1− P)

Metapopulation dynamics

Rate at which subpopulations go extinctProbability of each subpopulation going extinct

Proportion of patches that are occupied Rate of colonization of empty patches

Rate of colonization of empty patchesProbability of colonization (constant that reflects rate of dispersal between patches)

Proportion of patches that are occupied

Metapopulation dynamics

€

ΔP /Δt = C − E

ΔP /Δt = [mP(1− P)] − eP

Proportion of patches that are occupied

Rate of colonization of empty patches

Rate at which subpopulations go extinct

dNdt

=rNK −NK

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Lotka-Volterra models of interspecific competition

Population growth with carrying capacity:

Competitive impact of species 2 on species 1 = N2

= Impact per individual of species 2 on 1

Competitive impact of species 1 on species 2 = N1

dN1

dt=r1N1

K1 −N1 −αN2

K1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

dN2

dt=r2 N2

K2 −N2 −βN1

K2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

N1 + αN2 = K1

€

N1 + N2 = K2

For species 1

For species 2

€

N1 =K2 /β

€

N2 =K1 /α

>> species 1 can’t grow

>> species 2 can’t grow

Competition between two diatoms, Asterionella formosa and Synedra ulna for silica.