Logistic Population Model Problem 4
Tutorial to help us answer problem 4.
If we expand the logistic equation as,
we can see ΔN is graphically represented by a parabola opening downward
since the leading coefficient is negative (recall r > 0 and
α = 1/K > 0).
Thus, the vertex of this parabola represents the maximum instantaneous
rate of change of the population. The N-coordinate
(x-coordinate) of the
vertex is given by,
where ΔN = aN2 +bN + c. Therefore, the instantaneous rate of change of the population is maximized when N = K/2 . We interpret this to mean the population is changing (growing) fastest when the population has reached half the size of the carrying capacity of the environment.