Compare exponential and density-limited growth models

Mechanism before a label

A population changes through births, deaths, immigration and emigration. A net increase alone does not reveal which contribution changed. Age structure and life history influence future births and survival even for populations with equal present size.

The continuous exponential model dN/dt=rN uses per-capita net rate r and abundance N. The logistic model dN/dt=rN(1−N/K) adds a density term with carrying capacity K. These are instantaneous rates, not automatically exact next-year counts.

Carrying capacity is a model parameter tied to resources and conditions, not a permanent ceiling imposed on every population. Delays, seasonal variation and species interactions can produce overshoot or fluctuations. A model fit must be compared with observations and residual patterns.

Predict, compare and calculate

The following record is an original synthetic teaching dataset. Its assigned conditions are not a claim that an actual organism was measured. Write your prediction before revealing the changed condition.

Low-density model	Value (displayed model values)
N	100
K	1000
r times100	20

r=0.2 per year; N and K are individuals. The scaled r row is explicitly20=100r, not abundance.

Your prediction before comparison

This scratch prediction is not saved automatically. Copy it to the lesson notebook to retain it.

Evidence conditionReference comparisonChanged condition / follow-up

Compare exponential and density-limited growth models

Supplied synthetic evidence: r=0.2 per year; N and K are individuals. The scaled r row is explicitly20=100r, not abundance.

Low-density model: reference condition

Quantity	Value	Unit
N	100	displayed model values
K	1000	displayed model values
r times100	20	displayed model values

Task: For r=0.2/year,K=1000,N=1200, find the logistic instantaneous rate and interpret its sign.

These rows have different biological meanings; use the stated formulas, not sums across rows.

Compare with the conditional interpretation

Exponential rate20 individuals/year; logistic rate18 individuals/year.

At N=100, rN=0.2×100=20 and1−N/K=0.9, so logistic growth rate=18. At N=800 the multiplier is0.2, yielding0.2×800×0.2=32.

Case index for this investigation

0. Distinguish directional responses, activity and fitness
1. Track energy flow and distinguish production from standing biomass
2. Estimate abundance with sampling assumptions and uncertainty
3. Compare exponential and density-limited growth models
4. Use per-capita evidence to identify density dependence
5. Infer direct and indirect interactions from controlled contrasts
6. Separate richness, evenness and sampling completeness
7. Connect disturbance, nutrient cycles and evidence-based intervention

No numerical confidence, laboratory credit or independent mastery is inferred from opening a supplied case. Save your first prediction and comparison in the notebook.

Reset this unsaved comparison

Change one ecological assumption at a time

Bounded synthetic arithmetic practice, not a fitted ecosystem model or physical investigation. Scratch controls are not saved or graded. Record your first prediction and selected outputs in the lesson notebook.

N (individuals)K (individuals)r (per year)

rN=20.000 individuals/year. Density multiplier1−N/K=0.900. Logistic instantaneous rate=18.000 individuals/year.

Try N=500,1000,1200 with fixed K=1000. Predict the largest positive rate, zero rate and negative rate. These are instantaneous continuous-model rates, not exact next-year counts. Resources, age structure, delays and stochasticity are omitted.

Reset practice

Worked reasoning

At N=100, rN=0.2×100=20 and1−N/K=0.9, so logistic growth rate=18. At N=800 the multiplier is0.2, yielding0.2×800×0.2=32.

Exponential rate20 individuals/year; logistic rate18 individuals/year.

Apply to a new case

For r=0.2/year,K=1000,N=1200, find the logistic instantaneous rate and interpret its sign.

Compare your reasoning with the worked response

0.2×1200×(1−1.2)=−48 individuals/year. The model predicts decline above its stipulated K, not instantaneous disappearance or a universally fixed capacity.

This public response supplies feedback, not a score for independent reasoning. Record which part of your prediction changed and why.

Evidence and investigation boundary

These rows have different biological meanings; use the stated formulas, not sums across rows. A one-year Euler step is only an approximation and may fail for large rates or time steps.

For a supervised investigation, predeclare the question, biological unit of replication, treatment/comparison assignment, quantities and units, exclusion rules, graph and uncertainty method. Keep the raw records and the first prediction separate from later revisions. A simulated case does not supply laboratory hours or establish AP authorization. No physical manipulation is required by this lesson.

Source and continuation

Framework topics: 8.3–8.4. Current AP Biology framework; supporting scientific reference. Sources checked 30 October 2026; all teaching wording, questions and numbers here are original. Return to AP Biology to compare this mechanism with another scale of organization.