Distinguish current growth from coexistence under competition

1. Frame the biological question

Competition is a negative effect associated with shared limiting resources, but its strength can differ in each direction. Resource partitioning changes overlap, while predation, facilitation and environmental variation can modify outcomes. An ecological niche includes conditions and resource use; it is not simply a geographic address.

2. Connect mechanisms and evidence

The original continuous model uses carrying scales K1 = 100 and K2 = 80, intrinsic rate 0.2 and competition coefficients alpha and beta. At current abundances 40 and 30, per-capita rates are 0.2[1−(40+alpha×30)/100] and 0.2[1−(30+beta×40)/80]. Multiplying by each population gives its current rate, not its next census count.

To ask whether a species can invade when rare, evaluate it near the other species’ single-species equilibrium. Species 1 has rare-invasion rate 0.2(1−alpha×80/100); species 2 has 0.2(1−beta×100/80). Both positive signs support stable coexistence in this fixed two-species competition model, whereas both negative signs create dependence on initial conditions.

Current positive growth of both populations is insufficient to establish coexistence. A field pattern can also reflect habitat sorting, transient recovery or changing resources. Boundary cases with a zero invasion rate require separate analysis, and these coefficients are supplied rather than measured evidence of a real species interaction.

3. Reconstruct a worked comparison

At alpha = beta = 0.5, current population rates are 3.6 and 2.25 individuals per time. Rare-invasion rates are 0.12 and 0.075 per time, both positive. The current and invasion comparisons answer different questions.

4. Predict and investigate

Predict first: Can both species currently grow even if one cannot invade the other’s established population?

Compare coefficients 0.5/0.5 with 1/1. Reconstruct current and rare-invasion rates, then explain why one census pair cannot identify long-run coexistence or a competition coefficient.

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 alpha = beta = 1, calculate current rates and both rare-invasion signs.

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

Compare a worked response

Current population rates are 2.4 and 0.75, both positive. Species-1 invasion rate is +0.04 and species-2 is −0.05 per time, so the model does not predict stable coexistence despite simultaneous current growth.

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.