Question

Suppose a population P(t) satisfies

dP/dt = 0.8P − 0.001P2 P(0) = 50
where t is measured in years.
(a) What is the carrying capacity? ______
(b) What is P'(0)? P'(0) = ______
(c) When will the population reach 50% of the carrying capacity? (Round your answer to two decimal places.)_____yr

Answer 1

The carrying capacity can be found by setting the rate of change equation to 0. P'(0) can be found by differentiating the equation and evaluating at t=0. To find when the population reaches 50% of the carrying capacity, set P=0.5K and solve.

Step-by-step explanation:

(a) The carrying capacity (K) represents the maximum population size that the environment can support. In this scenario, the carrying capacity can be determined by finding the value of P when dP/dt = 0. To find this, set 0.8P - 0.001P^2 = 0 and solve for P. This will give you the carrying capacity.

(b) P'(0) represents the rate of change of the population at time t=0. To find this, differentiate the given equation dP/dt = 0.8P - 0.001P^2 with respect to t and substitute t=0. Simplify the expression and evaluate P'(0).

(c) To determine when the population will reach 50% of the carrying capacity, set P = 0.5K in the original equation dP/dt = 0.8P - 0.001P^2. Solve this equation to find the time t when the population reaches 50% of the carrying capacity.