Question

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 = 9, b = 1, and h = 81 4 . Determine whether the population becomes extinct in finite time. The population becomes extinct in finite time if P0 > 9 2 . The population becomes extinct in finite time if P0 = 9 2 . The population becomes extinct in finite time for all values of P0. The population becomes extinct in finite time if P0 < 9 2 . The population does not become extinct in finite time. If so, find that time. (If not, enter NONE.)

Answer 1

Therefore, the equilibrium point of the population model is P = 9/2.

Explanation:

To determine whether the population becomes extinct in finite time, we need to analyze the given harvesting model and the value of P0.

The population model is given by the differential equation dP/dt = P(a - bP) - h, with the initial condition P(0) = P0.

In this case, a = 9, b = 1, and h = 81/4.

To find out if the population becomes extinct in finite time, we need to consider the equilibrium points of the population model. The equilibrium points occur when dP/dt = 0.

Setting dP/dt = 0, we have:

P(a - bP) - h = 0

Substituting the given values of a, b, and h, we get:

P(9 - P) - 81/4 = 0

Expanding and rearranging the equation, we have:

9P - P^2 - 81/4 = 0

Multiplying through by 4 to clear the fraction, we get:

36P - 4P^2 - 81 = 0

Rearranging the equation, we have:

4P^2 - 36P + 81 = 0

Factoring the equation, we find:

(2P - 9)^2 = 0

Solving for P, we get:

2P - 9 = 0

2P = 9

P = 9/2