A population of squirrels lives in a forest with a carrying capacity of 2800. Assume logistic growth with growth constant k=0.5.Find a formula for the squirrel population P(t), assuming an initial population of 700 squirrels.
Question
A population of squirrels lives in a forest with a carrying capacity of 2800. Assume logistic growth with growth constant k=0.5.Find a formula for the squirrel population P(t), assuming an initial population of 700 squirrels.
Solution
The logistic growth model is given by the differential equation dP/dt = kP(1 - P/K), where P is the population size, t is time, k is the growth constant, and K is the carrying capacity of the environment.
The solution to this differential equation is given by the formula:
P(t) = K / (1 + (K/P0 - 1) * e^(-kt))
where P0 is the initial population size.
Given that K = 2800, k = 0.5, and P0 = 700, we can substitute these values into the formula to get:
P(t) = 2800 / (1 + (2800/700 - 1) * e^(-0.5t))
Simplifying the equation gives:
P(t) = 2800 / (1 + 3 * e^(-0.5t))
So, the population of squirrels at any time t is given by P(t) = 2800 / (1 + 3 * e^(-0.5t)).
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