the population typically decreases. Thus, the carrying capacity is the maximum number of individuals of a species that an environment can support. Population size decreases above carrying capacity due to a range of factors depending on the species concerned, but can include insufficient space, food supply, or sunlight. The carrying capacity of an environment varies for different species. In the standard ecological algebra as illustrated in the simplified Verhulst model of population dynamics, carrying capacity is represented by the constant K: where is the population size, is the intrinsic growth rate is the carrying capacity of the local environment, and , the derivative of with respect to time , is the rate of change in population with time. Thus, the equation relates the growth rate of the population to the current population size, incorporating the effect of the two constant parameters and . (Note that decrease is negative growth.) The choice of the letter came from the German Kapazitätsgrenze (capacity limit). This equation is a modification of the original Verhulst model: In this equation, the carrying capacity , , is When the Verhulst model is plotted into a graph, the population change over time takes the form of a sigmoid curve, reaching its highest level at . This is the logistic growth curve and it is calculated with: where is the natural logarithm base (also known as Euler's number), is the value of the sigmoid's midpoint, is the curve's maximum value, is the logistic growth rate or steepness of the curve and 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 . What determines a specific system's carrying capacity involves a limiting factor; this may be available supplies of food or water, nesting areas, space, or the amount of waste that can be absorbed without degrading the environment and decreasing carrying capacity. Where resources are finite, such as for a population of Osedax on a whale fall or bacteria in a petridish, the population will curve back down to zero after the resources have been exhausted, with the curve reaching its apogee at . In systems in which resources are constantly replenished, the population will reach its equilibrium at . Software is available