Fish are introduced into a lake with an abundant food supply

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

Problem

Fish are introduced into a lake with an abundant food supply. The population

P

(in hundreds of fish) satisfies

\frac{dP}{dt}=r\,P\Bigl(1-\frac{P}{K}\Bigr)-h,

where

r

and

K

are positive constants representing intrinsic growth rate and carrying capacity, respectively, and

h

(in hundreds of fish per month) is a constant monthly harvesting rate. Assume

r=0.8

and

K=10

are fixed, while

h

serves as the adjustable parameter. (a) Express the equilibrium population levels

P

in terms of

h

and determine the range of

h

for which at least one non-negative equilibrium exists. (b) For

h=1

, sketch a representative slope field and classify the stability of each equilibrium found in part (a). (c) Determine the critical harvest rate

h_c

at which the number of non-negative equilibria changes. Describe how the slope field transforms as

h

passes through

h_c

. (d) Suppose the initial population is

P(0)=6

. Using your analysis, discuss the long-term outcome for the fish population when

h=1.5

and when

h=2.2

. Support each conclusion by referencing features of the slope field rather than solving the differential equation explicitly.