SME
Ecology · Problem type
Competing species: do they coexist?
Use this for “K₁ = 500, K₂ = 400, α = 0.5, β = 0.6. Do the species coexist, and at what sizes?”
Diagram
Solid: N₁ stops growing on N₁ + αN₂ = K₁; dashed: N₂ stops on N₂ + βN₁ = K₂
They cross at N₁ₑ = 428.6, N₂ₑ = 142.9: both species coexist
This is an example. Type your own values to start a new problem.
K₁
Carrying capacity 1
example
K₂
Carrying capacity 2
example
α
Effect of species 2 on 1
example
β
Effect of species 1 on 2
example
N₁ₑ
Species 1 at equilibrium
worked out (not typed)
N₂ₑ
Species 2 at equilibrium
worked out (not typed)
Formulas
N₁ₑ
=
K₁
−
α
×
K₂
1 −
α
×
β
428.6
=
500 − 0.5 × 400
1 − 0.5 × 0.6
N₂ₑ
=
K₂
−
β
×
K₁
1 −
α
×
β
142.9
=
400 − 0.6 × 500
1 − 0.5 × 0.6
Assumptions
•
Lotka–Volterra competition: dN₁/dt = r₁N₁(K₁ − N₁ − αN₂) ÷ K₁, and likewise for species 2 with β.
•
Species 1 stops growing on the line N₁ + αN₂ = K₁; species 2 on N₂ + βN₁ = K₂.
•
With αβ < 1 and both N₁ₑ and N₂ₑ above 0, the lines cross at a stable point: the species coexist.
•
If N₁ₑ or N₂ₑ would be 0 or less, its box stays blank and the picture names the winner.
Step-by-step
WE KNOW
K₁ = 500, K₂ = 400, α = 0.5, β = 0.6
FIND
species 1 at equilibrium (N₁ₑ), species 2 at equilibrium (N₂ₑ)
Step 1 · Find species 1 at equilibrium (N₁ₑ)
Use N₁ₑ = (K₁ − α × K₂) ÷ (1 − α × β)
Both species stop growing where the lines cross: solve the two line equations together for N₁.
N₁ₑ
=
500 − 0.5 × 400
1 − 0.5 × 0.6
N₁ₑ
=
500 − 200
1 − 0.3
N₁ₑ
=
300
0.7
N₁ₑ = 428.6
Step 2 · Find species 2 at equilibrium (N₂ₑ)
Use N₂ₑ = (K₂ − β × K₁) ÷ (1 − α × β)
The same crossing, solved for N₂.
N₂ₑ
=
400 − 0.6 × 500
1 − 0.5 × 0.6
N₂ₑ
=
400 − 300
1 − 0.3
N₂ₑ
=
100
0.7
N₂ₑ = 142.9
Check
Put the numbers back in. Both sides should match.
428.6
=
500 − 0.5 × 400
1 − 0.5 × 0.6
✓
142.9
=
400 − 0.6 × 500
1 − 0.5 × 0.6
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.