A harvested population of fish in a lake is modeled by the differential equation dy…

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

Problem

A harvested population of fish in a lake is modeled by the differential equation

\frac{dy}{dt}=y\Bigl(1-\frac{y}{5}\Bigr)-h,\quad h\ge 0,

where

y

is measured in hundreds of fish and

t

in months. The constant

h

represents a uniform harvesting rate (in hundreds of fish per month). (a) For a fixed value of

h

, write an equation whose real zeros give the equilibrium population levels. Solve this equation in terms of

h

. (b) Determine the critical value

h_{\text{crit}}

at which the number of distinct equilibrium solutions changes. Show analytical work leading to your result. (c) For

0<h<h_{\text{crit}}

, classify each equilibrium solution as stable or unstable. Provide justification that analyzes the sign of

\dfrac{dy}{dt}

on intervals between equilibria. (d) Sketch a bifurcation diagram that displays equilibrium population levels as functions of

h

for

h\ge 0

. Clearly indicate which branches are stable and which are unstable. (e) Interpret the practical meaning of your diagram for fish-stock management. In particular, discuss how changing

h

affects long-term population outcomes and identify safe versus risky harvesting levels.

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