Guys, could you please help me with this problem? I'm a bit lost.
Consider a population that begins growing exponentially at a rate of 5% per year and then follows a logistic growth pattern. If the carrying capacity is 100 million, find the actual growth rate when the population is 10 million, 50 million and 90 million.
Thank you..
The equations you need can be found here:
http://mathworld.wolfram.com/LogisticEquation.html
Of course, I'd be happy to help you with this problem!
To find the actual growth rate at different population values, we first need to understand the formula for logistic growth. The logistic growth model is given by the equation:
dP/dt = rP * (1 - P/K)
Where:
dP/dt represents the rate of change of the population with respect to time
r represents the growth rate
P represents the population at a given time
K represents the carrying capacity (maximum population size the environment can sustain)
In this case, the population starts growing exponentially at a rate of 5% per year. This implies that the initial growth rate (r) is 0.05. The carrying capacity (K) is given as 100 million.
Now, to find the actual growth rate when the population is at various values (10 million, 50 million, and 90 million), we need to rearrange the logistic growth equation to solve for the growth rate (r):
r = (dP/dt) / (P * (1 - P/K))
Let's calculate the growth rates for each population value you provided:
1. When the population is 10 million (P = 10 million):
r = (dP/dt) / (P * (1 - P/K))
Since the population is growing exponentially at a rate of 5% per year initially, we can assume that (dP/dt) is equal to 0.05 * P.
Substituting the values, we get:
r = (0.05 * 10 million) / (10 million * (1 - 10 million/100 million))
2. When the population is 50 million (P = 50 million):
r = (dP/dt) / (P * (1 - P/K))
Substituting the values, we get:
r = (0.05 * 50 million) / (50 million * (1 - 50 million/100 million))
3. When the population is 90 million (P = 90 million):
r = (dP/dt) / (P * (1 - P/K))
Substituting the values, we get:
r = (0.05 * 90 million) / (90 million * (1 - 90 million/100 million))
Now, you can calculate the actual growth rate at each population value by simplifying the expressions and doing the calculations.