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.
(a) CalculateP′(0)
(b) Use the number from the previous part to estimate the population in the middle of 2015. That is, estimate P (0.5)
(c) What assumption is made in the computation in the previous part? Use the formula given for P ′ to see whether or not the assumption is true, to within 1%
(a) We have dP/dt = 0.04P(1- P/20000). To find P'(0), we need to evaluate this expression at t=0 and P=1000, and then differentiate with respect to t:
dP/dt = 0.04(1000)(1-1000/20000) = -0.16
P'(0) = dP/dt evaluated at t=0 and P=1000 = -0.16
(b) To estimate P(0.5), we can use the midpoint method: we assume that the population changes linearly between P(0) and P(1), so we use the average rate of change over that interval to estimate the population at the midpoint:
P(0.5) ≈ P(0) + 0.5P'(0)(1-0)
P(0.5) ≈ 1000 + 0.5(-0.16)(1)
P(0.5) ≈ 992
(c) The assumption made in part (b) is that the population changes linearly over the interval [0,1]. This assumption may not be very accurate if the population changes rapidly or erratically during that time. To check how accurate our estimate is, we can use the formula for P':
dP/dt = 0.04P(1- P/20000)
dP/dt = 0.04(992)(1-992/20000)
dP/dt ≈ -0.146
This means that the population is decreasing at a rate of about 0.146 per year at t=0.5 and P≈992. To estimate the actual population at t=0.5, we can integrate this expression over the interval [0,0.5]:
ΔP = ∫0.5dP/dt dt
ΔP ≈ ∫0.5-0.146 dt
ΔP ≈ -0.073
This means that the population decreases by about 0.073 between t=0 and t=0.5. Therefore:
P(0.5) ≈ P(0) - ΔP ≈ 1000 - 0.073 ≈ 927
So our estimate from part (b) is off by about 6%.