Question

. A model that describes the population p(t ) of a fishery in which harvesting takes place at

a constant rate is given by

dp/dt=kp-h

where k and h are positive constants.

(a) Solve the DE subject to P(0) = p

(b) Describe the behavior of the population P(t) for increasing time in the three cases for p=h/k , p=h/k , and 0<p<h/k

(c) Use the results from part (b) to determine whether the fish population will ever go extinct

in finite time, that is, whether there exists a time T>0 such that p(t )=0 . If the population

goes extinct, then find T

Answer 1

a)

$dP/dt=kP-h$

$\frac{dP}{kP-h}=dt$

$\frac{1}{k}ln(kP-h)+lnc_1=t$

$c_2(kP-h)=e^{kt}$

$P(t)=\frac{ce^{kt}+h}{k}$

$P(0)=\frac{c+h}{k}=p$

$c=pk-h$

$P(t)=\frac{(pk-h)e^{kt}+h}{k}$

b)

for $p=h/k$ :

$c=0$

$P(t)=\frac{h}{k}$

for $p>h/k$ :

$c>0$

$P(t)>\frac{h}{k}$

for $0<p<h/k$ :

$0<P(t)<\frac{h}{k}$

c)

$P(t)=\frac{(pk-h)e^{kT}+h}{k}=0$

$e^{kT}=\frac{h}{h-pk}$

$T=\frac{ln(h/(h-pk))}{k}$

$T>0$ if $h/(h-pk)>1$