Editing Exponential growth (section)

== Other growth rates ==
In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the [[Malthusian catastrophe]]) as well as any [[polynomial]] growth, that is, for all {{mvar|α}}:
<math display="block">\lim_{t \to \infty} \frac{t^\alpha}{a e^t} = 0.</math>

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See {{section link|Degree of a polynomial|Computed from the function values}}.

Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called [[hyperbolic growth]]. In between exponential and hyperbolic growth lie more classes of growth behavior, like the [[hyperoperation]]s beginning at [[tetration]], and <math>A(n,n)</math>, the diagonal of the [[Ackermann function]].

=== Logistic growth ===
[[File:Verhulst-Malthus.svg|thumb|The J-shaped exponential growth (left, blue) and the S-shaped logistic growth (right, red).]]
{{main|Logistic curve}}
In reality, initial exponential growth is often not sustained forever. After some period, it will be slowed by external or environmental factors. For example, population growth may reach an upper limit due to resource limitations.<ref>{{cite book| last1=Crauder|first1=Bruce|last2=Evans|first2=Benny|last3=Noell|first3=Alan|title=Functions and Change: A Modeling Approach to College Algebra|url=https://books.google.com/books?id=CZ4EAAAAQBAJ|year=2008|publisher=Houghton Mifflin Harcourt| isbn=978-1-111-78502-4|page=398}}</ref> In 1845, the Belgian mathematician [[Pierre François Verhulst]] first proposed a mathematical model of growth like this, called the "[[logistic curve|logistic growth]]".<ref>{{cite book| last=Bernstein| first=Ruth |title=Population Ecology: An Introduction to Computer Simulations|url=https://books.google.com/books?id=X1FcA0e9Tv8C| year=2003|publisher=John Wiley & Sons|isbn=978-0-470-85148-7|page=37}}</ref>