Question

Suppose a population P(t) satisfiesdP/dt = 0.8P − 0.001P2 P(0) = 40 where t is measured in years.(a) What is the carrying capacity?(b)What is P'(0)?(c) When will the population reach 50% of the carrying capacity? (Round your answer to two decimal places.)

Answer 1

(a) The carrying capacity is 2000.

(b) P'(0) = 32.

(c) The population will reach 50% of the carrying capacity after approximately 6.91 years.

Step-by-step explanation:

The carrying capacity (K) in this scenario is the population size at which the growth rate becomes zero. It can be determined by setting dP/dt = 0. At equilibrium, 0.8P - 0.001P^2 = 0, and solving this equation gives K = 2000 as the carrying capacity.

To find P'(0), we can use the initial population value (P(0) = 40) and substitute it into the differential equation dP/dt = 0.8P - 0.001P^2. Evaluating P'(0) provides the instantaneous rate of change at time t = 0, yielding P'(0) = 32.

The time it takes for the population to reach 50% of the carrying capacity can be calculated using the logistic growth model. To find this time (t_50), the formula t_50 = (1/0.8) * ln((K - P(0))/(P(0))) is employed, where ln represents the natural logarithm. Substituting the given values, t_50 ≈ (1/0.8) * ln((2000 - 40)/(40)), which approximately equals 6.91 years.

By solving the differential equation and interpreting the results, we ascertain the carrying capacity, the initial rate of change, and the time it takes for the population to reach 50% of the carrying capacity.

Answer 2

The carrying capacity is the maximum population an environment can sustain, identified by solving the provided population growth equation. P'(0) is the initial population growth rate, found using the initial condition. The time to reach 50% of carrying capacity can be calculated using a logistic growth model.

Step-by-step explanation:

Understanding Carrying Capacity and Population Growth

The carrying capacity is the upper limit to the population size that the environment can support. When provided with the differential equation dP/dt = 0.8P - 0.001P2, we can determine the carrying capacity by setting the population growth rate dP/dt to zero, as this indicates where the population size will stabilize. We find the carrying capacity by solving the quadratic equation 0.8P - 0.001P2 = 0, which gives the maximum population size P.

P'(0) can be found by evaluating the derivative equation at t = 0 with the given initial condition P(0) = 40. To find when the population reaches 50% of the carrying capacity, we would solve for t when P(t) equals half of the calculated carrying capacity, using the logistic growth model equation P(t) = P0ert.