SCIENCE · INVESTIGATE AND EXPLAIN

Challenge an unlimited-growth prediction

Compare density-limited and exponential models under changed resource assumptions.

Learning outcome: Interpret carrying capacity as a contextual model parameter and identify when a coarse update gives an implausible forecast.

Starting recap

Population abundance changes through births, deaths and movement. This model combines net demographic effects into a rate rule and excludes migration.

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Try

At N=20 and K=80, the density-limited increment is 0.25×20×(1−20/80)=3.75; exponential growth adds 5.

Open the experiment

Predict

What happens below, at and above K? Is K a permanent property of a species?

Your prediction and reason

Experiment

Compare N=20 with K=80 and K=40. Then set N=80,K=80 and N=100,K=20. Capture the equilibrium and extreme cases, and identify which results need model revision.

Compare density limitation with exponential growth

Worked teaching model · synthetic values · Ten steps compare N_next = N + 0.25N(1 − N/K) with N_next = 1.25N. Values are continuous expected abundance, not literal fractional organisms. K is fixed; no age structure, migration or random environmental change is modeled. Density-limited values are floored at zero if a coarse step overshoots.

Initial population: 20 individuals Carrying-capacity parameter: 80 individuals

Reset keeps your prediction, captures and practice history.

Compare abundance trajectories

Solid: density-limited model. Dashed: exponential model. Both use the same axes; the trace gives exact values when the larger curve compresses the smaller one.

Inspect both population forecasts

Step	Limited	Exponential
0	20.000	20.000
1	23.750	25.000
2	27.925	31.250
3	32.469	39.063
4	37.292	48.828
5	42.269	61.035
6	47.253	76.294
7	52.089	95.367
8	56.632	119.209
9	60.767	149.012
10	64.420	186.265

Calculated values on the same fixed scale
Row	Quantity	Value (individuals)
1	Initial abundance	20
2	Fixed K parameter	80
3	Density-limited final abundance	64.4197
4	Exponential final abundance	186.2645

After 10 steps: limited 64.42, exponential 186.26 expected individuals.

Below K the density term is positive; at K it is zero; above K it is negative. A chosen K is an assumption about these conditions, not a permanent species constant. The coarse discrete model can fail for extreme initial conditions.

Observation to attach to a capture

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

Understand

Start with two competing forecasts

The exponential rule adds 25% of the preceding abundance each step. From 20, it predicts 25 after one step. The density-limited rule adds 0.25N(1 − N/K). With N = 20 and K = 80, it adds 3.75, giving 23.75.

N denotes abundance and K a carrying-capacity parameter for the modeled conditions. Below K, the density term is positive; at K it is zero; above K it is negative. Resources, interactions and environmental conditions can alter carrying capacity, so it is not a permanent species constant. Background: OpenStax, environmental limits.

Inspect an extreme case

At N = 100 and K = 20, the coarse update predicts 100 + 25×(1 − 5) = 0 in one step. That does not establish that a real population would vanish instantly. A finite-step approximation and assumed rate can produce a poor forecast when the starting state is far outside the intended regime.

Fractional results represent continuous expected abundance, not pieces of organisms. The lab omits age structure, random births and deaths, migration and changing resources. The exponential comparison helps expose the consequence of removing density limitation, not identify a universally correct model.

Connect populations to communities

Competition can change access to resources; consumers and predators can alter survival; mutualists can affect reproduction. Those mechanisms should motivate a model or measurement, rather than be hidden inside an unexplained fitted K.

Compare fixed K = 80 and K = 40 as two scenarios. This does not simulate an event halfway through a run or recovery of a habitat. A disturbance study needs timing, before-and-after observations, comparison sites and evidence about mechanisms.

In your reflection, identify which forecast is supported by the available observations and what new data could reveal a failure. A smooth curve alone is not evidence that the model describes the population.

Apply

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

A habitat disturbance reduces resources. Explain why comparing two fixed K scenarios is not a complete simulation of the disturbance, interactions or recovery.

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: Connect roots, leaves and a water budget Next: Keep food-web energy and nutrient accounts separate

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