A fish population (in thousands of fish) living in a protected lake satisfies the…

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

Problem

A fish population

P(t)

(in thousands of fish) living in a protected lake satisfies the differential equation

\frac{dP}{dt}=0.2P\,(15-P),\quad t\ge 0.

Assume

P

is a differentiable function of time measured in months. (a) Interpret the numerical value

15

that appears in the differential equation within the biological context of the lake. (b) Suppose

P(0)=8

. Explain, without solving the differential equation, whether the population will ultimately increase or decrease, and determine its limiting value as

t\to\infty

. (c) Suppose instead that

P(0)=18

. Explain, without solving the differential equation, whether the population will ultimately increase or decrease, and determine its limiting value as

t\to\infty

. (d) For each of the three population sizes

P=7

P=15

, and

P=22

, determine the sign of

\dfrac{dP}{dt}

and explain what that sign indicates about the instantaneous change in population at that size.

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