Which of the following is the correct equation for Verhulst-Pearl logistic growth?

The formula for exponential population growth is

Sigmoid growth curve is represented by

Read the following statements and determine how many of them are correct:
$(1)$ Population is the total number of individuals of a species in a given area at a given time.
$(2)$ Developed countries show a low birth rate.
$(3)$ An equal proportion of reproductive and post-reproductive age individuals indicates a stable population.
$(4)$ In the presence of limited resources,the growth curve is sigmoid ($S$-shaped).

The human population growth graph is:

Under which of the following conditions is the logistic population growth curve obtained?