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) Calculate P'(0). Explain what this number means.
To calculate P'(0), we need to take the derivative of the logistic formula with respect to t and then evaluate it at t=0.
dP/dt = 0.04P(1−(P/20000))
Using the product rule and chain rule of differentiation, we get:
dP/dt = 0.04(1−(P/20000))P' − 0.0008P^2
At t=0, P(0)=1000. Therefore,
dP/dt = 0.04(1−(1000/20000))P' − 0.0008(1000)^2 = 0.02P' − 640
Simplifying further,
P'(0) = (dP/dt)(0) = 0.02P'(0) − 640
0.98P'(0) = 640
P'(0) ≈ 653.06
This means that at the beginning of 2015, the population was increasing at a rate of approximately 653.06 individuals per year.