A certain population is known to be growing at a rate given by the logistic equation dx/dt=x(b-ax) Show that the minimum rate of growth will occur when occur when the population is equal to half the equilibrium size,that is,when the population is b/2a?

1 Answer
Jul 1, 2018

To optimise growth rate, take its derivative and set to zero:

# x'=x(b-ax) = xb - ax^2#

#:. qquad x'' = b - 2ax = 0 qquad implies bb( x = b/(2a))#

The next derivative confirms the nature of the critical point:

#x^((3)) = - 2a < 0 qquad a gt 0#