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.

Question

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.

Answer 1

To find P'(0), we need to take the derivative of the logistic formula with respect to t and then plug in t=0 and P(0)=1000:

dP/dt = 0.04P(1 - P/20000)

P'(0) = dP/dt |t=0 = 0.04(1000)(1 - 1000/20000) = -0.02

This means that at the beginning of 2015, the population is decreasing at a rate of 0.02 individuals per year. In other words, the population is declining.