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A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with equation where , the value of the sigmoid's midpoint; , the supremum of the values of the function; , the logistic growth rate or steepness of the curve. 
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows.
The logistic growth curve depicts how population growth rate and carrying capacity are inter-connected. As illustrated in the logistic growth curve model, when the population size is small, the population increases exponentially. However, as population size nears carrying capacity, the growth decreases and reaches zero at K. 
The logistic model (or logistic function) is a function that is used to describe bounded population growth under the previous two assumptions. The logistic function is bounded at both extremes: when there are not individuals to reproduce, and when there is an equilibrium number of individuals (i.e., at carrying capacity ).
The logistic model takes the shape of a sigmoid curve and describes the growth of a population as exponential, followed by a decrease in growth, and bound by a carrying capacity due to environmental pressures. 
The generalized logistic functionor curveis an extension of the logisticor sigmoidfunctions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curveafter F. J. Richards, who proposed the general form for the family of models in 1959. Definition
The logistic growth equation is an effective tool for modelling intraspecific competition despite its simplicity, and has been used to model many real biological systems. At low population densities, N (t) is much smaller than K and so the main determinant for population growth is just the per capita growth rate.