A butterfly is an insect with complete metamorphosis: egg, caterpillar (larva), chrysalis (pupa), adult. The caterpillar does almost all the growing; the adult mostly flies, drinks nectar and reproduces.
Growth. A caterpillar sheds its skin (molts) 4 times, so it has 5 growth stages called instars. Here its mass grows about 1000 times, from 0.5 mg to 500 mg, as m = m0 · 1000f where f goes from 0 to 1 over the larval stage (exponential growth, so it adds the most mass in the last days). Its length grows more slowly, about L ∝ m0.37 (about 13 times). Real species differ: a monarch grows roughly 2000 to 3000 times.
Warmth sets the clock. Insects are cold-blooded, so they develop faster when it is warm. The simple degree-day model says development needs a fixed amount of heat: DD = (T − T0) · days, with a base temperature T0 = 10 °C. We use 60 DD for the egg, 190 DD for the caterpillar and 120 DD for the chrysalis (370 DD total), so at constant T the whole life cycle takes 370 / (T − 10) days: about 25 days at 25 °C, 74 days at 15 °C. Below 10 °C nothing happens. Above 32 °C the rate stops rising and falls to zero at 38 °C.
Food. We assume 5 mg of leaf per 1 mg of body mass gained. Without leaf the caterpillar stops growing and waiting adds time. Total leaf for one caterpillar here is about 2.5 g, or 5 leaves.
Pollen. Pollen sticks to a butterfly's body when it drinks nectar and rubs off on the next flower. Moving pollen from one flower to a different one is called cross-pollination; the flower can then make seeds. Grain counts shown are only rough, small numbers.
Independent check. The button re-computes a fresh egg-to-butterfly run with a different method (hourly Runge-Kutta steps, unlimited food) and compares it with the formula above.
Simplifications: constant temperature, a straight-line rate between 10 and 32 °C, equal heat needs for all instars, no illness, no predators, drawing not to scale (the egg is really about 1 mm), real caterpillars wander before making a chrysalis. Models built in Blender 5.2, shown with Three.js.