Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.

However, realistic environmental conditions limit the number of individuals that can occupy a habitat. This limit is known as the habitat’s carrying capacity. Because of carrying capacities, population growth is generally better represented by S-shaped (logistic) curves. The population initially increases exponentially until the carrying capacity is reached, at which point resource limitations cause growth to level off or fluctuate around the carrying capacity, producing an S-shaped curve.

The per capita rate of population increase, r, is the change in population size (calculated as the present population size minus the initial population size) divided by the initial population size. When there are no environmental limits and immigration and emigration are assumed to be equal, the population can grow at its maximal rate, known as its biotic potential, or r_{max}. Therefore, the per capita rate of increase under unlimited conditions is r_{max} * N (population size). When the population size is plotted over time, this produces a J-shaped growth rate curve, representing an unchecked population explosion. For example, bacteria, like *E. coli*, reproduce by fission, doubling their population size every generation after being placed in a new medium. Exponential growth will continue to occur until the cells are no longer viable.

Realistic populations in nature are limited by various environmental factors including predators, prey, space, water, and other resources. Therefore, exponential growth cannot continue indefinitely. The number of individuals of a population that a habitat can support is called the carrying capacity, or K. A population’s potential rate of growth is proportional to (K-N)/K. Here, K-N is the number of individuals that can be added to the populations before it reaches capacity. Thus, (K-N)/K represents the fraction of available carrying capacity. Therefore, as the number of individuals in the population gets closer to the carrying capacity, the rate of population growth decreases. This plots as an S-shaped logistic growth rate curve characterized by initial exponential growth (due to low population size and ample resources), followed by a decrease in growth as resources become more limited. For example, populations of *Anolis* lizards on islands have lower growth rates than their mainland counterparts as a result of decreased access to resources and space, which lowers the carrying capacity on islands.

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