A territorial hummingbird, bees and rivals share one flower patch. Hover power, nectar chemistry, foraging rules and a pollination network, all from equations.
Tick species to remove them and watch the mutualistic network re-equilibrate.
The science
Hover power. Momentum theory: the wings accelerate air through a disc of area A; induced velocity v_i = sqrt(mg / 2ρA), induced power P_ind = mg·v_i; profile power taken as 0.8 P_ind; muscle efficiency 12 % gives metabolic power (about 0.9 W for a 4 g bird). A blade-element estimate of lift gives a wingbeat frequency of the right order (25-55 Hz). Wings are drawn slowed to a few Hz.
Nectar. Sucrose solution viscosity rises steeply with concentration; a suction feeder's flow scales as 1/μ, energy per microlitre as c·ρ·16.5 kJ/g. Intake power ∝ c/μ(c) peaks at moderate concentration (a closed form c* = 1/k exists for μ = μ₀e^{kc}, used as a check). Viscosity fit and flow constant are illustrative.
Foraging. Marginal value theorem: leave a patch when the instantaneous gain rate falls to the long-run average, g'(t*) = g(t*)/(T + t*); checked against a Monte-Carlo forager. The owner bird feeds in bouts between a lower and an upper energy-reserve threshold.
Territory. Income = flowers × production × hours × energy per µL; defence cost grows with perimeter (intruders per metre of border × chase time × chase power). The smallest radius whose net income meets the daily need is the economic territory (Gill and Wolf style); all coefficients illustrative.
Network. Linear mutualism with intra-guild competition (Bastolla-type): dN_i/dt = N_i (r_i − Σ_j M_ij N_j), interaction strengths from visit traits. Equilibria solved by RK4 and by Gaussian elimination (check). Species with no partners die out; the cascade after removals is qualitative.
Animals are drawn 5× larger than life by default. Daily energy numbers are order-of-magnitude; do not use for animal care decisions (feeders: real sugar water is about 20-25 %, no dye, cleaned often).