a wild preserve can support no more than 250 gorillas.
how long will it take for the gorilla population to reach the carrying capacity of the preserve
dp/dt = 0.0004P(250-P) and
p = (250)/(1+7.929e^-.1t)
show that the function is a solution of a logistic differential equation. identify k and the carrying capacity. estimate p(0).
a population of rabbits is given by the formula
P(t) = 1000/(1+e^(4.8-0.7t))
To show that the function P(t) = 250/(1+7.929e^(-0.1t)) is a solution of a logistic differential equation, we need to compare it with the general form of a logistic differential equation.
A logistic differential equation is given by dp/dt = kP(1 - P/C), where:
- dp/dt represents the rate of change of the population P with respect to time t,
- k is a constant representing the intrinsic growth rate of the population, and
- C is the carrying capacity of the environment.
Comparing this with the given function dp/dt = 0.0004P(250 - P), we can see that k = 0.0004 and C = 250. Therefore, the given function is a solution to the logistic differential equation.
To estimate P(0), we need to substitute t = 0 into the function P(t) and calculate the value.
P(0) = 250/(1+7.929e^(-0.1*0))
= 250/(1+7.929e^(0))
= 250/(1 + 7.929)
= 250/8.929
≈ 28.02 (rounded to two decimal places)
Therefore, based on the given formula, the estimated initial population of gorillas is approximately 28.02.