Logistic and Harvesting Models: Population Dynamics

The logistic model describes a population growing toward a carrying capacity K. A harvesting model adds human removal — fishing, hunting or logging — and the mathematics tells us how much we can take without collapsing the population.

Set the fishing quota too high and the stock crashes, destroying livelihoods for years. Set it wisely and the harvest can continue forever. This is the science behind "sustainable yield".

Key Concepts

• Harvest rate H — individuals removed per unit time. Two types: constant quota (fixed H) or proportional effort (H = qEN, where E is fishing effort).
• Maximum sustainable yield (MSY) — the largest harvest that can be taken indefinitely: rK/4 for the logistic model with constant quota.
• Equilibria — population sizes where growth exactly balances harvest; some are stable, some are tipping points.
• Overharvesting — harvesting above the sustainable level, which can push the population past a tipping point into collapse.
• Effort E — fishing boats, hunting days; proportional harvesting scales with both effort and population.

Formulas

• Constant quota: dN/dt = rN(1 − N/K) − H — the logistic growth minus a fixed harvest H. Sustainable only if H ≤ rK/4.
• MSY = rK/4 at N = K/2 — the peak of the parabola rN(1 − N/K). Assumes logistic growth and constant quota; real ecosystems need safety margins.
• Proportional effort: dN/dt = rN(1 − N/K) − qEN — harvest proportional to population; the equilibrium is N* = K(1 − qE/r), which stays positive only if qE < r.
• Tipping point (constant quota) — if H > rK/4, no positive equilibrium exists and the population declines to extinction from any starting point.

Worked Example

A fish stock: r = 0.4 per year, K = 100,000 fish. A fishery proposes a constant quota H = 8,000 fish per year. Is it sustainable?

1. Step 1: Compute MSY = rK/4 = 0.4 × 100,000 / 4 = 10,000 fish per year.
2. Step 2: Compare — 8,000 < 10,000, so a positive equilibrium exists; the quota is sustainable in the model.
3. Step 3: Find equilibrium: solve rN(1 − N/K) = H → 0.4N(1 − N/100,000) = 8,000. Trying N = 72,360: 0.4 × 72,360 × (1 − 0.7236) = 28,944 × 0.2764 ≈ 8,000. ✓ The stable equilibrium is about 72,000 fish.
4. Step 4: Warning — if the quota were raised to 12,000 (> MSY of 10,000), the model predicts collapse to zero. A 20% safety margin below MSY is standard practice.

Common Misconceptions

• "MSY is a safe target." It is the edge of the cliff, from an idealized model. Real managers target well below it because r and K are uncertain and environments fluctuate.
• "If the population is above K/2, any harvest is fine." A large population can still collapse if the quota exceeds the growth at that size — sustainability depends on the rate, not just current abundance.
• "Proportional harvesting can never cause extinction." It can if effort is too high: when qE ≥ r, the only equilibrium is N = 0.

Real-World Applications

• Fisheries quotas — bodies like ICES use harvested-logistic models to recommend total allowable catches for cod, tuna and other stocks.
• Forestry — sustainable timber yields are computed from growth models minus harvest, exactly this framework.
• Wildlife management — hunting permits for deer and game birds are set from population models with harvesting terms.

Adding a harvest term to the logistic model turns population biology into resource economics. Maximum sustainable yield rK/4 at half the carrying capacity is one of ecology's most applied formulas. But harvest at the model's edge, and reality's uncertainties will push you over it.