A wildlife management agency stocks a large lake with a species of fish
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A wildlife management agency stocks a large lake with a species of fish. Let F(t) denote the number of these fish in thousands after t months. The population satisfies the differential equation
dtdF=rF(K−F),
where r and K are positive constants. Biologists measure the instantaneous change in population at two different population sizes and obtain the following data:
• When F=2 (i.e., 2000 fish), the observed rate of change is 300 fish per month.
• When F=4, the observed rate of change is 200 fish per month.
Respond to the prompts below.
(a) Translate the two observations into a system of equations involving the parameters r and K.
(b) Solve the system you wrote in part (a) to obtain numerical values for r and K.
(c) State the value approached by F(t) as t→∞. Include units.
(d) Discuss one practical implication of this long-term value for the lake’s ecosystem management plan.