SCIENCE · INVESTIGATE AND EXPLAIN

Test patch persistence without equating occupancy and abundance

A metapopulation perspective follows occupancy across patches, with local extinction and colonization linking habitat units. Occupancy is the fraction of patches containing a population; it is not the number of individuals within an occupied patch. Patch quality, connectivity and sampling detection affect how an observed occupancy pattern should be interpreted.

Learning outcome: Compare c = 0.2, 0.4 and 0.8 at e = 0.4. Retain rate and equilibrium separately, then explain why shared disturbances or patch-quality differences could invalidate the homogeneous model.

Starting recap

Connect behavioral mechanisms, demographic stages, community interactions and ecosystem budgets while preserving denominators, causal alternatives and external fieldwork gates.

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Try

At c = 0.6 and e = 0.2, initial colonization gain is 0.6×0.4×0.6 = 0.144 and extinction loss 0.08, giving dp/dt = 0.064. Positive equilibrium is 2/3, distinct from the current 0.4 and from the initial rate.

Open the experiment

Predict

Can the equilibrium occupancy be positive while most patches are initially empty?

Your prediction and reason

Experiment

Compare c = 0.2, 0.4 and 0.8 at e = 0.4. Retain rate and equilibrium separately, then explain why shared disturbances or patch-quality differences could invalidate the homogeneous model.

Test a colonization–extinction threshold

Worked teaching model · synthetic values · Identical habitat patches, occupancy p, dp/dt=c p(1−p)−e p, initial p=0.4. Positive equilibrium 1−e/c only if c>e; otherwise extinction equilibrium. This mean-field model omits rescue heterogeneity, correlated disturbances and source–sink quality.

Colonization coefficient: 0.6 per time Local extinction coefficient: 0.2 per time

Reset keeps your prediction, captures and practice history.

Colonization can sustain a positive equilibrium under the supplied homogeneous-patch assumptions.

Equilibrium patch occupancy is not population abundance within a patch. Spatial connectivity, patch quality, rescue effects and correlated disturbances can change the model; no organism movement or habitat intervention is prescribed.

Calculated quantities under stated assumptions
Quantity	Value	Unit
Colonization gain at initial occupancy	0.1440000	occupancy/time
Local extinction loss at initial occupancy	0.08000000	occupancy/time
Initial occupancy rate	0.06400000	occupancy/time
Nonnegative equilibrium occupancy	0.6666667	fraction
Low-occupancy per-capita rate	0.4000000	per time

Retain prior predictions, settings and quantities in the notebook. These are original supplied/calculated comparisons, not physical specimen observations or laboratory participation.

Colonization can sustain a positive equilibrium under the supplied homogeneous-patch assumptions.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Frame the biological question

2. Connect mechanisms and evidence

The original homogeneous-patch equation is dp/dt = cp(1−p)−ep. The colonization term requires occupied source patches and empty recipient patches; local extinction removes occupied patches. The supplied initial occupancy is 0.4. Coefficients c and e are rates, not per-step probabilities unless a discretization and interval are supplied.

When c exceeds e, the positive equilibrium is 1−e/c. If c is no greater than e, there is no positive persistence equilibrium in this simple model. The low-occupancy per-capita rate is c−e. Equilibrium fraction and initial occupancy rate are separate quantities; a presently increasing fraction is not the same as full occupancy.

Real patches are heterogeneous, dispersal can rescue populations, environmental events can affect many patches together, and some occupied sites may be demographic sinks. Connectivity can have both beneficial and harmful effects. This mean-field threshold does not prescribe organism releases, corridor placement or a conservation intervention without independent evidence and ethical review.

3. Reconstruct a worked comparison

4. Predict and investigate

Predict first: Can the equilibrium occupancy be positive while most patches are initially empty?

Compare c = 0.2, 0.4 and 0.8 at e = 0.4. Retain rate and equilibrium separately, then explain why shared disturbances or patch-quality differences could invalidate the homogeneous model.

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 c = 0.2 and e = 0.4, calculate initial rate and long-run nonnegative equilibrium.

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

Compare a worked response

Gain 0.048 minus loss 0.16 gives −0.112 occupancy per time. Since c does not exceed e, the only nonnegative equilibrium is extinction, p = 0. This is a conditional mean-field result, not a measured extinction forecast.

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.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At c = 0.2 and e = 0.4, calculate initial rate and long-run nonnegative equilibrium.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Close energy pathways while distinguishing cycling elements Next: Evaluate disturbance and recovery with a reference comparison

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