A population (in hundreds) evolves according to for

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

Problem

A population y(t)y(t) (in hundreds) evolves according to dydt=y(1−y5)\dfrac{dy}{dt}=y\left(1-\dfrac{y}{5}\right) for t≥0t\ge0, where tt is measured in months. (a) Compute the slope indicated by the differential equation at (t,y)=(0,0)(t,y)=(0,0), (0,2)(0,2), (0,5)(0,5), and (0,8)(0,8). (b) Sketch a slope field for 0≤t≤60\le t\le6 and 0≤y≤80\le y\le8 that includes enough information to reveal the qualitative behavior of solution curves. (c) Identify all equilibrium solutions and classify each as stable or unstable, citing evidence from your slope field. (d) Using your sketch, predict the long-term behavior of any solution that begins at y(0)=1.5y(0)=1.5. Explain your reasoning in terms of what the slope field suggests.

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