# Exponential Growth & Carrying Capacity

Exponential Growth

Exponents attached to the cats ears = Exponential Growth

Logistic

Log or stick = Logistic

# 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=aebt.

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.

# Exponential Growth and Carrying Capacity

1. Zoom: whole scene

Hot Spot: Ring carrying the cab from Cab City – Carrying Capacity

Information Mode Script: This CoursePic describes the term carrying capacity, which is the maximum stable population size that can be sustained in an environment over a long period. Here, we see a cab labeled “Cab City,” suggestive of the word capacity. The name of the cab driver, Poppy, is a reminder of the word population. To illustrate the word carrying, we see there is a large ring under the cab, carrying it. Thus, “carry” “ring” “cab” “city” reminds us of carrying capacity.

Story Mode Script: Poppy has been working for Cab City for a while now, but business has dropped off lately. The problem is the new taxi alternative, Uber. Poppy decides he needs to find a new marketing strategy to combat the competition. Since he’s always been a cat person, he decides to market his rides as being “pet friendly.” Unfortunately, the local cats take him too literally and start hitching rides even without their owners! Here, we see his cab is already full, yet many other cats are vying to jump on board. Trying to avoid the cats, he takes a shortcut through the high-end jewelry district. As he speeds down the street, a giant ring used as an advertisement outside one of the jewelry stores rolls into the street and becomes lodged under Poppy’s cab!

2. Zoom: Close-up of Cats

Hot Spot: Exponents attached to the cats’ ears – Exponential Growth

Information Mode Script: The other, related term illustrated in this Coursepic is exponential growth. This is the growth of a population where the number of individuals multiplies with every successive generation. In favorable circumstances, a population can be expected to grow rapidly with few limitations. Here, we see the cats have small numbers attached to their ears, depicting the term “exponent” as a word association.

Story Mode Script: To keep track of all the cats, the owners have labelled their ears with numbers. Unfortunately, there are so many cats and they are reproducing so rapidly that some numbers are duplicated and, frankly, Poppy is growing very frustrated. He can’t keep track of where the cats need to be taken and he constantly rejects new riders until he drops off those he already has in the cab.

3. Zoom: Close-up of Poppy and stick

Hot Spot: Log or stick – Logistic

Information Mode Script: Although populations may be expected to grow undeterred in exceptional environments, in most cases there are many factors limiting growth. In this case, we typically see logistic population growth, where the growth rate slows as the population reaches its carrying capacity.  We see Poppy waiving a log or stick, reminding us of the term “logistic.”

Story Mode Script: Poppy’s good nature has finally cracked. His cab is likely damaged by the giant ring and he fears he’ll never get all the cat hair out of his upholstery. He stops briefly to pick up a log to use as a stick to waive off the cat hoards. Poppy’s convinced he may need to re-think this new marketing strategy!

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