Logistic growth is a form of population growth first described by Pierre Verhulst in 1845. It can be illustrated by a graph that has time on the horizontal, or "x" axis, and population on the vertical, or "y" axis. The exact shape of the curve depends on the carrying capacity and the maximum rate of growth, but all logistic growth models are sshaped.
Parameters of a Logistic Growth Model

A logistic growth model depends on the initial population, the carrying capacity and the maximum rate of population growth. The initial population is self explanatory; the carrying capacity is the maximum size of the population that can live in the environment; and the maximum rate of growth is how fast the population can grow, if there are no constraints (for example, a rabbit population can grow a lot faster than a human population).
Initial Phase of Logistic Growth

The initial phase of a logistic growth model is relatively stable, or flat over time.
Intermediate Phase of Logistic Growth

After the initial period, the rate of growth may change, depending on the relationship between the initial population and the carrying capacity. If the initial population is much less than the carrying capacity, the population rises rapidly. If the initial population is much larger than the carrying capacity, then the population shrinks rapidly (this could happen, for example, after some ecological devastation reduces the carrying capacity). If the initial population is close to the carrying capacity, then the population will be stable.
Final Phase of Logistic Growth

The final phase of logistic growth begins when the population is at or near the carrying capacity. At this point, the population stabilizes, until or unless the carrying capacity changes.