A closed population of river otters is initially counted at 250 individuals at noon…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A closed population of river otters is initially counted at 250 individuals at noon (time t=0 measured in days). While food is abundant, biologists model the population growth rate as directly proportional to the current population with a constant relative growth rate of 0.12 day−1. Exactly one day later, a sudden drought reduces available food, and the relative growth rate instantly drops to 0.06 day−1 for all subsequent times. Biologists will not intervene for the first two days.
For all calculations, use a constant step size of Δt=0.5 day and report every population value to the nearest whole otter.
(a) Write an explicit differential equation for the population P(t) on each of the two time intervals 0≤t≤1 and 1<t≤2. Clearly state the initial condition.
(b) Starting from t=0, approximate the population at t=1 day using the given step size. Show the population estimate after each half-day increment.
(c) Using your approximation from part (b) as the initial value at t=1 day, continue the numerical procedure to estimate the population at t=2 days. Again report the estimate after each half-day increment.
(d) Interpret your final numerical result in context. Include appropriate units, and describe one biological implication of the drought during the two-day period.