Deterministic population dynamics · 3

Harvesting with Holling responses

dN/dt = r_m N(1 − N/K) − H(N)

H(N): the harvest rate (by hunters, fishers or grazers)
θ model: H = h₀N^θ/(N₀^θ + N^θ): it saturates at h₀, half of it at N = N₀; θ = 0 is a constant h₀/2 (Type 0), θ = 1 Type II, θ > 1 Type III (it starts slowly)
Type I: a fixed share h₀ of the population, H = h₀N: no θ gives it, because the θ model never takes more than h₀
y, k, h, t′: y = N/N₀, k = K/N₀, h = h₀/(r_mN₀), t′ = r_mt — the rescaled form the README uses for the θ model

Growth against harvest

r_m1.50 h₀1.00 k1.80 h0.800 θ1.00 start N(0)/K0.90 t′_max100

Where growth is above harvest the population rises; it settles where they cross.

Runs from several starts

The same harvest, eight different starting populations and yours (thick white): yellow ends at the upper state, blue at a lower one, pink collapses.

Eight starts and yours: where each one ends up.

Bifurcation diagram

Every equilibrium for every harvest level: the yellow line is the harvest you chose.

Equilibria against the harvest: solid stable, dashed unstable.

Tipping point: a slow ramp

The harvest rises slowly from 0 to past the threshold over half the time T, then falls back to 0. The run is computed in full; Play replays it.

T800

The harvest rises slowly past the threshold, then falls back.

The theory, in short

Harvest on top of growth. Without harvest the population grows logistically; the harvest H(N) is subtracted: dN/dt = r_m N(1 − N/K) − H(N).
Holling types. Type 0: a fixed catch c. Type I: a fixed share h₀N. Type II: h₀N/(N₀ + N), which saturates at h₀ (harvesters can only take so much). Type III: h₀N^θ/(N₀^θ + N^θ), slow at low density (harvesters switch to other prey), with shape θ.
One θ model for three of them. In Type III's formula θ = 1 is Type II, and θ = 0 makes N^θ = N₀^θ = 1, so H = h₀/2 for every N > 0: a constant catch, Type 0 with c = h₀/2. Type I is not in the family: H never exceeds h₀, while h₀N grows without limit; it is only what Type II looks like far below the half-saturation density (N ≪ N₀: H ≈ (h₀/N₀)N).
Rescaling. For the θ model, y = N/N₀, k = K/N₀, h = h₀/(r_mN₀), t′ = r_mt give dy/dt′ = y(1 − y/k) − h y^θ/(1 + y^θ): only k, h, θ matter.
Equilibria are where growth equals harvest; one is stable when harvest overtakes growth there (f ′(u*) < 0, linear stability test).
Fold (saddle-node) bifurcations. When the harvest rises past a critical value, a stable and an unstable equilibrium meet and vanish: the population collapses. Type 0 (θ = 0): h_c = k/2, which is c_c = r_m/4 in the README's units. Type II (θ = 1): h_c = (k + 1)²/(4k) for k > 1; for k ≤ 1 there is no fold, the population fades out at h = 1. Type I has no fold either: x* = 1 − h₀/r_m reaches 0 at h₀ = r_m.
Hysteresis. After a collapse, lowering the harvest back to h_c is not enough: Type II needs h < 1, Type III needs the lower fold. Type 0 (θ = 0) never recovers: extinction is absorbing.
The shape θ. θ > 1: extinction is impossible (f ′(0) = 1), but the population can be pushed down to a low refuge. θ < 1: the per-head harvest grows without limit as N → 0, so small populations are always harvested out. θ = 0: the constant catch of Type 0.
Critical slowing down. Near a fold, f ′(u*) → 0, so the relaxation time τ = 1/|f ′(u*)| grows without limit: the population recovers from small pushes ever more slowly, an early warning of collapse.
Maximum sustainable yield is the peak growth rate, r_mK/4 at N = K/2: Type I reaches it at h₀ = r_m/2, and a constant catch (θ = 0) can take up to it, h/2 = k/4 in rescaled units.

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