Question

A species with an initial population of 350

is growing in an environment where the
carrying capacity is 7000. After 5 years
the population is up to 900. Find the
logistic function that models this
population as a function of time.

Answer 1

Explanation:

To find the logistic function that models this population as a function of time, we need to use the logistic equation:

dP/dt = rP(1 - P/K)

where P is the population, t is time, r is the growth rate, and K is the carrying capacity.

At the start, the population is 350, so we have:

P(0) = 350

After 5 years, the population is 900, so we have:

P(5) = 900

The carrying capacity is 7000, so we have:

K = 7000

To find the growth rate r, we can use the formula:

r = (ln(P(5)/P(0))) / 5

r = (ln(900/350)) / 5

r = 0.2928 (rounded to four decimal places)

Now we can write the logistic function as:

P(t) = K / (1 + A * e^(-rt))

where A = (K - P(0)) / P(0) and r is the growth rate we just calculated.

Substituting the values we know:

P(0) = 350

K = 7000

r = 0.2928

A = (K - P(0)) / P(0) = (7000 - 350) / 350 = 19

we get:

P(t) = 7000 / (1 + 19 * e^(-0.2928t))

So the logistic function that models this population as a function of time is P(t) = 7000 / (1 + 19 * e^(-0.2928t)), where P(t) is the population at time t in years.