**Exponential Growth and Carrying Capacity**

Exponential growth happens when the species in a population duplicate at a steady rate. At a steady rate of reproduction, the growth rate increases in increasingly large increments due to the increase in breeders until it approaches an infinitely vast size. Of course, this infinite population does not actually occur due to a variety of limiting factors. Populations just starting out in a new region may experience a quick exponential increment in numbers to create a J formed curve, but this curve can't maintain its shape in that environment. The new population will eventually face one or more scarcities due to it's increased size, and the rate of growth will slow, stop, or even fall into decline. When this happens, either a partial or full migration may occur, in which either the entire population moves or a small group of the population goes to colonize another area.

**Evaluation of Exponential Growth Rate**

The growth rate and size of population can be related through the following equation:

__dN /dt__* = rN*

where *N* = number of individuals, *r* = biotic potential and t = time

*Biotic Potential*

This is the extreme reproduction limit of a living being under ideal natural scenario. It is frequently communicated as a corresponding rate, or shown in terms of percentage increment every year. Full articulation of the biotic capacity of a specific species is confined by ecological resistance, or any element that represses the expansion in population size. These elements incorporate unfavorable seasonal scenarios such as absence of space, light, or a reasonable substitute, mineral insufficiency, and the repressing impacts of predators.

The exponential growth model can also be presented as:

y=ae^{bt}.

Using this particular formula b represents a constant. In this case, If b > 0, the function presents the span of a growing establishment. If b < 0, the function presents the span of a decaying establishment. The a is the size of the establishment at initial time when t = 0 and y demonstrates the size at any later time t.

**Carrying Capacity**

Carrying capacity represents the manageable extreme limit of population from a specific group of species under given circumstances of the residing area, considering unlimited resources introduced in that environment. For exponential increase in population, development begins gradually, enters a quick incremental stage and after that settles down when the carrying capacity for that species has been achieved. The population span then varies above or below that limit. Reproductive lag time is the period for the birth rate to fall and the death rate to rise corresponding to the resource limits. This time span may also lead the population to pass beyond the carrying capacity for the time being.

**Understanding Carrying Capacity and it's Effect on Exponential Growth**

To understand the concept of carrying capacity, suppose for pond that the carrying capacity for fish is 800. In this scenario, we can say that the pond can support 800 fish indefinitely with present resources. Now consider that the fish population starts at 600 fish. These 600 fish have everything they need to maintain their population. Imagine now that another 500 fish migrate from another pond connected by a stream, but the other factors that determine carrying capacity do not change. This is a carrying capacity of only 800, but an actual population of 1,100. The entire fish population would immediately begin suffering the limiting factor of scarcity. Scarcity of food, space, and potentially even mates would drive reproduction down while eliminating adults through starvation and young through increased predation by starving adults.