Question

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,

Answer 1

The question above is not complete and the question is;

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) = Po, where a,b,h, and Po are positive constants. Suppose a=5,b=1, and h=4. Since the DE is autonomous, use the phase portrait concept to sketch representative solution curves corresponding to the cases P0>4,1<P0<4, and 0<P0<1. Determine the long-term behavior of the population in each case.

Answer:

The long term behavior of the population in each case is;

P(t) → 0 for 0 < Po < 1

P(t) → 4 for 1 < Po < 4 and Po > 4

The phase portrait is attached

Explanation:

First of all from the question, the Differential equation (DE) is autonomous, we'll have to solve dp/dT = 0 to arrive at the equilibrium solution.

Thus;

P(a-bP) - h = 0

Expanding, we have;

aP - bP² - h = 0

From the question, a=5, b=1 and h=1.

Thus,substituting in the equation, we have;

5P - P² - 4 = 0

Rearranging this, we have

-P² + 5P - 4 = 0

This is a quadratic equation, and solving for the roots, we get,

P = 1 and P= 4.

In the phase potrait attached to this explanation, it's seen that lim(t→∞) for P(t) = 4

And so, P(t) → 0 for 0< Po < 1

Also, P(t) → 4 for 1 < Po < 4 and Po > 4

Answer 2

Complete Question

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=5,b=1, and h=4. Since the DE is autonomous, use the phase portrait concept to sketch representative solution curves corresponding to the cases P0>4,1<P0<4, and 0<P0<1. Determine the long-term behavior of the population in each case.

Answer:

See Explanation Below

Step-by-step explanation:

Given.

a = 5

b = 1

h = 4

The DE is autonomous;

That means we'll solve dP/ft = 0 to arrive at an equilibrium solution

i.e.

P(a - bP) - h = 0 --- open the bracket

aP - bP² - h = 0

Substitute in, the values of a b and h

5P - P² - 4 = 0 ---- Solve quadratic equation

First, rearrange

-P² + 5P - 4 = 0

-P² + 4P + P - 4 = 0

-P(P - 4) + 1(P - 4) = 0

(-P + 1)(P - 4) = 0

-P + 1 = 0 or P - 4 = 0

-P = -1 or P - 4 = 0

P = 1 or P = 4

See attachment for phase portrait.

It can be seen from the portrait that

lim t->∞ P(t) = 4 and P(t) -> 0 for 0<Po<1

P(t) -> 4 for 1<Po<4 and Po > 4