Treat a renewal maximum as a model property, not a harvest instruction

Connect the Earth-system mechanism

A renewable resource can replenish on a decision timescale, but its replenishment rate may depend on the stock and environment. A nonrenewable stock has negligible replacement over that same timescale, even if geological formation occurs over much longer periods. Classifying a resource as renewable does not mean any extraction rate is sustainable.

Build the declared budget

The supplied biomass renewal curve is rS(1−S/K), with r 0.2/year and K 1000 tonnes. It is low near small stock and near the modeled capacity, and maximal at S=K/2. This mathematical result follows by completing the square: rK/4−(r/K)(S−K/2)². The maximum is 50 tonnes/year under the assigned coefficients.

Separate the competing quantities

Net instantaneous stock rate is renewal minus extraction. An extraction below the maximum can still exceed current renewal at a lower stock. The rate is not an exact end-of-year stock forecast unless the changing stock dependence is integrated. A catch or harvest target based solely on the maximum ignores uncertainty, age structure, habitat, ecological interactions and management error.

Challenge the inference

A defensible comparison considers replenishment evidence, stock measurement, alternative models, precautionary limits and the distribution of benefits and burdens. Monitoring should specify an action trigger before decline is observed, but no real extraction prescription is supplied here. The conditional curve is used to expose the danger of confusing potential maximum production with safe use.

Worked original packet

At stock 500 tonnes, renewal 50 and extraction 45 give net instantaneous+5 tonnes/year. At stock 100, renewal 18 and the same extraction 45 give−27. The extraction remains below the curve maximum but exceeds renewal at the depleted stock.

Investigation

Compare stocks 100, 500 and 900 at extraction 45. Explain the symmetry of the assigned curve and why a real resource need not have symmetric dynamics.

Changed-case transfer

At stock 300 and extraction 45, calculate renewal and net rate. What is the net rate for a nonrenewable stock with negligible replenishment at the same extraction?

Feedback

Renewal 0.2×300×0.7=42; net−3 tonnes/year. With negligible replenishment, net−45. Neither is an exact long-term depletion date without changing-stock, demand and intervention assumptions.

Sources and evidence boundary

Original prose, numerical packets and exercises. Mechanism/method context checked 30 October 2026: primary source. These are conditional computational models, not measured field records, carbon credits, safe yields, treatment standards or independent mastery. No physical extraction, chemical addition or environmental release is requested.