Deterministic population dynamics · 4

Lotka–Volterra competition

dN₁/dt = r₁N₁(1 − (N₁ + α₂₁N₂)/K₁)dN₂/dt = r₂N₂(1 − (N₂ + α₁₂N₁)/K₂)

N₁, N₂: the two species (SP1, SP2)
r₁, r₂: growth rate per head when rare and alone
K₁, K₂: carrying capacity of each species alone
α₂₁, α₁₂: α₂₁ — the effect of one SP2 individual on SP1, counted in SP1 individuals; α₁₂ — the effect of SP1 on SP2 (this is the owner's convention; many textbooks swap the indices)

Phase plane

r₁1.00 r₂0.80 K₁10.0 K₂8.0 α₁₂0.40 α₂₁0.60 t_max50

Each line is one possible history. Tap or click to start your own.

Both populations from your last start in the phase plane, or from 10% of each carrying capacity.

Both populations over time, from your last start.

The two competition coefficients decide who wins; the carrying capacities set where the borders are.

Who wins, for every pair of α. Tap to move the dot.

Two logistic species. Each grows logistically alone (r_i, K_i); the other species adds a linear drag.
Competition coefficients. α₂₁ converts SP2 into SP1-equivalents (α₂₁ = 2: one SP2 individual weighs on SP1 like two of its own); (N₁ + α₂₁N₂) is the crowding SP1 feels.
Equilibria. e₀ = (0, 0), e₁ = (K₁, 0), e₂ = (0, K₂), and e₃ = ((K₁ − α₂₁K₂)/(1 − α₁₂α₂₁), (K₂ − α₁₂K₁)/(1 − α₁₂α₂₁)) when it is positive.
Stability comes from the Jacobian's eigenvalues at each equilibrium (the README's table: node, saddle, spiral, centre).
Invasion tests. SP1 can grow when rare in SP2's world iff K₁ > α₂₁K₂; SP2 iff K₂ > α₁₂K₁. Both → coexistence; one → that species wins; neither → bistability (founder control).
Coexistence needs each species to limit itself more than it limits the other.
No spirals. In competition the Jacobian at e₃ has real eigenvalues, so populations approach without oscillating.
Growth rates set the speed and the shape of the paths, not the outcome.
Degenerate cases. With α₁₂ = K₂/K₁ and α₂₁ = K₁/K₂ (so α₁₂α₂₁ = 1) the two nullclines coincide: a whole line of equilibria (neutral); near it the approach is extremely slow.