Logistic growth of a population size occurs when resources are limited, thereby setting a maximum number an environment can support.
Exponential growth is possible when infinite natural resources are available, which is not the case in the real world. To model the reality of limited resources, population ecologists developed the logistic growth model. As population size increases and resources become more limited, intra specific competition occurs. Individuals within a population who are more or less adapted for the environment compete for survival. The population levels off when the carrying capacity of the environment is reached.
The logistic model assumes that every individual within a population will have equal access to resources and thus an equal chance for survival.
Yeast, a microscopic fungus, exhibits the classical logistic growth when grown in a test tube. It's growth levels off as the population depletes the nutrients that are necessary for its growth.