Mathematics

Mathematics

Problem

A biological population satisfies dydt=k y (a−y)\dfrac{dy}{dt}=k\,y\,(a-y). Measurements show that when the population is 5050 individuals, its growth rate is 7575 individuals per month, and when the population reaches 200200 individuals, its growth rate is 00. Which ordered pair (k,a)(k,a) is consistent with these observations? a) (0.0075,200)(0.0075,200) b) (0.02,150)(0.02,150) c) (0.01,200)(0.01,200) d) (0.015,200)(0.015,200)
Intuitive explanation

This is a logistic growth model, where the population grows fastest at intermediate sizes and slows as it approaches a carrying capacity. The key insight is that the two measurements give you two equations: plug in the known values of yy and dydt\frac{dy}{dt} to create a system you can solve for the unknown parameters kk and aa. The second measurement—where growth rate is zero—is especially powerful because it immediately reveals the carrying capacity aa.

Intuitive explanation

This is a logistic growth model, where the population grows fastest at intermediate sizes and slows as it approaches a carrying capacity. The key insight is that the two measurements give you two equations: plug in the known values of yy and dydt\frac{dy}{dt} to create a system you can solve for the unknown parameters kk and aa. The second measurement—where growth rate is zero—is especially powerful because it immediately reveals the carrying capacity aa.

Browse the problem library