Two neighboring cities, Mistral (M) and Norland (N)
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Two neighboring cities, Mistral (M) and Norland (N), each have populations that follow logistic growth models.
City M: dtdP=0.06P(900−P), where P(t) is the population (in thousands) of Mistral t years after the present.
City N: dtdQ=0.04Q(1200−Q), where Q(t) is the population (in thousands) of Norland t years after the present.
At the present time (t=0), Mistral has P=450 and Norland has Q=400.
(a) Calculate the instantaneous growth rate, in thousands of people per year, for each city at t=0.
(b) Compute the relative growth rate P1dtdP for Mistral and Q1dtdQ for Norland at t=0. Interpret the meaning of each value in context.
(c) For each city, determine the population size at which its instantaneous growth rate is maximized. Show analytical work leading to your results.
(d) Based on your answers to part (c), decide which city will reach the point of its fastest growth first, assuming present trends continue. Provide a justification that references both the model parameters and initial populations.