The rate of change of a population P of an environment is determined by the logistic formula dP/dt= 0.04P(1−P/20000) where t is in years since the beginning of 2015. So P(1) is the population at the beginning of 2016. Suppose P(0) = 1000. (e) Using time increments of 1 month, use Euler’s method to estimate the population at the beginning of 2019, that is P(4). [Use a spreadsheet or something similar.]
Using Euler's method, we have:
Δt = 1/12 (since we are using monthly time increments)
t(0) = 0 (beginning of 2015)
t(4) = 4 (beginning of 2019)
P(0) = 1000
We can use the formula:
P(i+1) = P(i) + Δt * dP/dt(i)
where i represents the ith time increment.
First, we need to find dP/dt(i), which is given by:
dP/dt = 0.04P(1−P/20000)
At i=0, P(0) = 1000, so:
dP/dt(0) = 0.04(1000)(1−1000/20000) = -16
Now we can use Euler's method to estimate P(4):
P(1) = P(0) + Δt * dP/dt(0) = 1000 + (1/12)(-16) = 998.67
dP/dt(1) = 0.04(998.67)(1−998.67/20000) = -15.95
P(2) = P(1) + Δt * dP/dt(1) = 998.67 + (1/12)(-15.95) = 997.34
dP/dt(2) = 0.04(997.34)(1−997.34/20000) = -15.91
P(3) = P(2) + Δt * dP/dt(2) = 997.34 + (1/12)(-15.91) = 996.01
dP/dt(3) = 0.04(996.01)(1−996.01/20000) = -15.87
P(4) = P(3) + Δt * dP/dt(3) = 996.01 + (1/12)(-15.87) = 994.68
Therefore, using Euler's method, we estimate that the population at the beginning of 2019 (t=4) is approximately 994.68.