Assume the carrying capacity of the earth is 9 billion. Use the 1960s peak annual growth rate of 2.1% and population of 3 billion to predict the base growth rate and current growth rate with a logistic model. Assume a current population of 6.8 billion. How does the predicted growth rate compare to the actual growth rate of about 1.2% per year?
What is the base growth rate?
To determine the base growth rate using a logistic model, we first need to calculate the intrinsic growth rate. The intrinsic growth rate is the maximum growth rate that can be sustained indefinitely in an idealized environment without any limiting factors.
The formula for calculating the intrinsic growth rate is:
r = ln(R) / T
where r is the intrinsic growth rate, R is the peak growth rate as a decimal (in this case, 2.1% would be 0.021), and T is the time it takes for the population to double (given by the formula T = ln(2) / r).
Using the peak annual growth rate of 2.1% from the 1960s, we can calculate the intrinsic growth rate:
r = ln(0.021) / T
T = ln(2) / r
Now we can plug in the values and solve for r:
r = ln(0.021) / (ln(2) / r)
To simplify the equation, cross-multiply:
r * ln(2) = ln(0.021)
r = ln(0.021) / ln(2)
Using a calculator, we find that r ≈ -0.0652.
The base growth rate is calculated as:
base growth rate = r * (1 - (P / K))
where P is the current population (6.8 billion) and K is the carrying capacity (9 billion).
Plugging in the values:
base growth rate = -0.0652 * (1 - (6.8 / 9))
base growth rate ≈ -0.0652 * 0.2444
base growth rate ≈ -0.0159 or -1.59% (approximately)
So, the base growth rate is approximately -1.59%.
To predict the base growth rate using a logistic model, we can use the formula:
Base growth rate = Peak growth rate / (4 * Population at carrying capacity * (1 - (Population at carrying capacity / Starting population)))
In this case, the peak growth rate is given as 2.1%, the carrying capacity is given as 9 billion, and the starting population is given as 3 billion. Substituting these values into the formula:
Base growth rate = 2.1% / (4 * 9 billion * (1 - (9 billion / 3 billion)))
Simplifying the equation:
Base growth rate = 0.021 / (4 * 9 * (1 - 3))
Base growth rate = 0.021 / (4 * 9 * -2)