2022 · Problem A — The Need for Bees
Compartmental ODEs Sensitivity Capacity sizingThe problem in one paragraph
(1) Build a model for honeybee colony population over time. (2) Sensitivity-analyze the parameters (lifespan, egg-laying, fertilized/unfertilized ratio). (3) Determine how many hives are needed to pollinate a 20-acre (81,000 m²) crop. (4) Produce a one-page infographic / blog.
Solution outline
Population model
Compartmental ODE across life stages:
$\dfrac{dL}{dt} = \mu_E E - \mu_L L$ (larvae/pupae)
$\dfrac{dW}{dt} = \mu_L L - \mu_W(t, L_W) W$ (workers)
Where $\beta(t)$ is the queen's daily egg-laying rate (seasonal — peaks early summer), $\mu_E, \mu_L$ are stage-transition rates, $\mu_W$ is worker mortality (increases with workload $L_W$).
Add CCD (colony collapse) stress factors as multipliers on $\mu_W$: pesticides, varroa mites, pathogens, food scarcity. The problem hints at this — engage with it.
Sensitivity
Vary one parameter at a time by ±20%:
| Parameter | Effect on steady-state colony size |
|---|---|
| $\beta$ (egg-laying) | Very large — typically the dominant lever |
| $\mu_W$ (worker mortality) | Large, especially in summer |
| $\mu_E, \mu_L$ (development times) | Smaller |
| Sex ratio (fertilized eggs → workers, unfertilized → drones) | Small unless extreme |
Pollination demand
For a 20-acre crop with pollination requirement $p$ (visits per flower per day) and flower density $\rho$ (flowers/m²), total daily visits required = $A \cdot \rho \cdot p$. A single worker visits ~2000 flowers/day; an active colony has, say, 30,000 foragers (~half of all workers). Daily-visit capacity per hive ≈ $30{,}000 \times 2000 = 6 \times 10^7$. Divide demand by capacity.
Account for: bloom duration (weeks, not days), travel distance from hive (effective foraging radius ~6 km, so 20 acres is well within range), and weather (rain days reduce foraging).
Pitfalls
- Constant $\beta$ — bees don't lay year-round. Make it seasonal.
- Ignoring CCD entirely — the problem's framing is about decline, engage with it.
- Pollination part: forgetting that one hive's foraging area greatly exceeds 20 acres.
- Infographic that's just text in a colored box. Make it a real infographic.