Natural growth examples
Many things grow, but not all things grow in the same way. There are two basic ways for something to grow: linearly, and exponentially. When something grows linearly it grows repeatedly by the same amount; when something grows exponentially it grows repeatedly by the same proportion.
In the real world, pure examples of linear growth and exponential growth don't really happen; growth rates vary, usually because the resources that feed the growth are not steadily available, and growth usually comes to an end because the resources run out, or are unable to grow fast enough themselves. Nevertheless, the growth of many real world things and systems are either linear or exponential in character.
On the 12th of May, 2008 the Sichuan earthquake in China caused landslides that dammed several rivers, and lakes grew behind these natural dams. The largest lake was Lake Tangjiashan. Lake Tangjiashan grew because the water from the Jian River ran into it at a more or less steady rate that was five times the rate that the water was able to run out of the lake. Between the 12th and the 22nd of May the lake grew to contain 200 million cubic metres of water, and threatened to overtop the dam and drown over one-million people.
The growth of lake Tangjiashan was linear in character because the water that flowed into the lake didn't grow itself when it got there; the lake only grew because of the more or less steady inflow of water; the volume of the lake got bigger by the same amount every hour, or every day.
The growth of many real world things and systems are exponential in character; that is, the things that grow from the original stock also grow even as the original stock continues to grow.
For example, a species may enter a new environment that is suitable for it and that provides an abundance of resources for it. If resources limitations don't initially place a limit on that specie's growth it will be able to expand exponentially for a while, until its resources can no longer keep up with its growth, or some other species finds it useful as a resource and is able to grow even faster and overtakes it.
In the earlier part of the 19th century a cactus called prickly pear (Opuntia species) was introduced to Australia. It was cultivated for stock fodder near Sydney prior to 1840. Inland Australia's climate suited it, and nothing in Australia ate it or otherwise used it.
By 1900, prickly pear covered four-million hectares (ten-million acres) of land, and by 1920 it covered 24 million hectares (58 million acres) of land. This is an area equivalent to the US state of Wyoming, or the entire United Kingdom.
Prickly pear took approximately sixty years to cover its first four-million hectares (ten-million acres), and only twenty years to cover the next twenty-million hectares (58 million acres), indicating clearly that it grew exponentially. Just as the old growth was able to continue to grow and reproduce, so was the new growth that grew from it.
The exponential expansion of prickly pear was eventually halted by introducing the cactoblastis moth from the prickly pear's homeland, the Americas, which was able to grow faster on this huge food resource and eventually overwhelm it.
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