Each of the sequences from the previous page, that is:

1,2,3,4,5 , ...

1, 2, 4, 6, 8, 10, 12, ....

1, 2, 4, 8, 16, 32, 64, 128, ...

1, 1/2, 1/4, 1/8, ....

can be defined by a recursion process, which is to say, as the orbit of a map as per May's article

Here are the maps:

F(x) = x+1

F(x) = x+2

F(x) = 2x

and

F(x) = (1/2) x

If we seed F with x₁ =1 then we will get the above sequences.

LINEAR RECURSION:

the simplest linear recursion relation is, using May's notation:

X_{t + 1} = r X_t (*)

where r is a constant. We can give its solution with a formula:

X_t = X₀ r^t

The last two sequences above are of this form, with r = 2, and r =1/2.

THEOREM: if the sequence X_t satisfies the linear recurrence relation (*),

then $LaTeX: r> 1 \implies X_t \to \infty$ and $LaTeX: 0 < r < 1 \implies X_t \to 0$ .

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A very famous sequence, perhaps the first model in population dynamics, is Fibonacci's sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Can you see the pattern?

Add the previous two to get the next one: 5+8 = 13, 8+13 =21.

In symbols: $LaTeX: F_{i+2} = F_{i+1} + F_i$

We now need two numbers to seed the sequence, i.e. to get it going. We take these to be F_0 = 1 and F_1 = 1,

leading to F_2 = 1 + 1 = 2, frowm which we can stairstep up using the recursion rule.

Fibonacci sequences frequently arise in biological growth processes, notably in plant leaves, flowers, etc.

See for example here.

For more on Fibonacci's sequence, its history and use in population models, and more on the use of iteration or

mappings in population biology see from the book by Bottsford et al, some pages of which are here.