Separate stage-structured transients from long-run growth

1. Frame the biological question

Individuals at different stages contribute differently to future population size. A population with many juveniles can change differently from one with many adults even when both share the same vital-rate parameters. A single total-count ratio therefore need not identify the long-run growth tendency.

2. Connect mechanisms and evidence

The original recurrence defines the census explicitly: next juveniles J′ = F A, and next adults A′ = sJ+0.5A. F is juvenile production per adult per interval, s the juvenile transition fraction, and one-half of existing adults persist. Initial counts are J = 100 and A = 50. The coefficients are supplied and are not estimated from the displayed sequence.

For constant positive coefficients, the dominant multiplier solves λ²−0.5λ−Fs = 0, so λ = [0.5+√(0.25+4Fs)]/2. The corresponding stable juvenile/adult ratio is F/λ. The first total-census ratio, however, is (50F+100s+25)/150 and depends on the initial stage composition.

The model is density independent and deterministic. Expected counts may be noninteger; real demographic variation, environmental correlation, migration and resource limits are omitted. Neither exponential continuation nor a stable stage ratio should be assumed over unlimited time in a real habitat.

3. Reconstruct a worked comparison

At F = 2 and s = 0.4, the next census is 100 juveniles and 65 adults, a total ratio 1.1. The dominant multiplier is about 1.17871, with stable juvenile/adult ratio about 1.6967. The first ratio is not yet the long-run multiplier.

4. Predict and investigate

Predict first: Can the same fixed vital rates produce changing total census ratios during an initial transient?

Reconstruct the first two transitions independently, then compare the plotted stage series. Explain which stage composition and constant-rate assumptions would be needed to use the dominant multiplier as a forecast.

Retain the first prediction, controlled settings, complete ledgers and one conflicting or null comparison. These are original supplied calculations, not observed populations or completed fieldwork.

5. Transfer to a changed case

Apply the mechanism to a changed case

At F = 1 and s = 0.2, find the next stages, first total ratio and dominant multiplier.

Retain your calculation, units and assumptions in the lesson notebook before opening this public practice response.

Compare a worked response

Next juveniles 50 and adults 45 give total ratio 95/150 ≈ 0.63333. The dominant multiplier is [0.5+√1.05]/2 ≈ 0.76235; both indicate decline, but they are not the same quantity.

6. Defend a conditional conclusion

Separate mechanism, population pattern and causal evidence. Identify a missing biological process and an independent observation that could discriminate alternatives. Preserve first reasoning and substantive human critique in the General Biology II workspace. Use supplied or ethically archived information only; no animal manipulation, habitat intervention or organism release is requested.

Supporting scientific reference, checked 30 October 2026. Original prose, models and packets; no source exercise or figure reproduced. Actual field sampling and independently assessed transfer remain separate evidence gates.

USGS stage-matrix modeling context, checked 30 October 2026. The supplied numerical model is original, not an extracted field dataset.