Investigate the following harvesting model both qualitatively and analytically. If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by:

dP/dt = P(a − bP) − h, P(0) = P0,
where a, b, h, and P0 are positive constants. Suppose a = 7, b = 1, and h = 49 4 .

Determine whether the population becomes extinct in finite time.

a. The population does not become extinct in finite time.
b. The population becomes extinct in finite time for all values of Po.
c. The population becomes extinct in finite time if Po > 7/2
d. The population becomes extinct in finite time if Po < 7/2
e. The population becomes extinct in finite time if Po= 7/2

If so, find that time. (If not, enter NONE.)
t=

a. The population does not become extinct in finite time.

Step-by-step explanation:

The model for the population of the fishery is

If we rearrange and replace the constants we have:

Now we can calculate if the population become 0 in any finite time

To be a finite time, t>0

We can conclude that the only finite time in which P=0 is when the initial population is 0.

Because P0 is a positive constant, we can say that the population does not become extint in finite time.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is