A population (in hundreds of organisms) evolves according to the differential equation

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

A population P(t)P(t) (in hundreds of organisms) evolves according to the differential equation dPdt=0.1P(1−P500)\dfrac{dP}{dt}=0.1P\bigl(1-\tfrac{P}{500}\bigr), where tt is measured in years. Initially, P(0)=50P(0)=50. (a) Identify all values of PP for which dPdt=0\dfrac{dP}{dt}=0. Interpret these values in terms of the direction field for P≥0P\ge0. (b) On axes with 0≤t≤500\le t\le50 and 0≤P≤6000\le P\le600, sketch a direction field consistent with the differential equation and draw an approximate solution curve that satisfies the initial condition. (c) Using your sketch, estimate the population at t=50t=50 years. Express your answer to the nearest whole organism and explain the reasoning behind your estimate. (d) Discuss the long–term behavior of the population for large tt. In your discussion, indicate whether the limiting value of PP depends on the initial condition. Support your conclusions with evidence from the direction field.

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