4First-order differential equations
IA Differential Equations
4.8 Discrete equations (Difference equations)
Since differential equations are approximated numerically by computers with
discrete equations, it is important to study the behaviour of discrete equations,
and their difference with continuous counterparts.
In the logistic equation, the evolution of species may occur discretely (e.g.
births in spring, deaths in winter), and we can consider the population at certain
time intervals (e.g. consider the population at the end of each month). We might
have a model in the form
x
n+1
= λx
n
(1 − x
n
).
This can be derived from the continuous equation with a discrete approximation
y
n+1
− y
n
∆t
= ry
n
1 −
y
n
Y
y
n+1
= y
n
+ r∆ty
n
1 −
y
n
Y
= (1 + r∆t)y
n
−
r∆t
Y
y
2
n
= (1 + r∆t)y
n
1 −
r∆t
1 + r∆t
y
n
Y
Write
λ = 1 + r∆t, x
n
=
r∆t
1 + r∆t
y
n
Y
,
then
x
n+1
= λx
n
(1 − x
n
).
This is the discrete logistic equation or logistic map.
If λ < 1, then deaths exceed births and the population decays to zero.
x
n+1
x
n
x
n+1
= x
n
x
0
We see that x = 0 is a fixed point.
In general, to find fixed points, we solve for
x
n+1
=
x
n
, i.e.
f
(
x
n
) =
x
n
. For
the logistic map, we have
λx
n
(1 − x
n
) = x
n
x
n
[1 − λ(1 − x
n
)] = 0
x
n
= 0 or x
n
= 1 −
1
λ
When 1 < λ < 2, we have
x
n+1
x
n
x
n+1
= x
n
x
0
We see that
x
n
= 0 is an unstable fixed point and
x
n
= 1
−
1
λ
is a stable fixed
point.
When 2 < λ < 3, we have
x
n+1
x
n
x
n+1
= x
n
x
0
There is an oscillatory convergence to x
n
= 1 −
1
λ
.
When
λ >
3, we have a limit cycle, in which
x
n
oscillates between 2 values,
i.e. x
n+2
= x
n
. When λ = 1 +
√
6 ≈ 3.449, we have a 4-cycle, and so on.
x
n+1
x
n
x
n+1
= x
n
x
0
We have the following plot of the stable solutions for different values of
λ
(plotted
as r in the horizontal axis)
Credits: Wikimedia Commons: Jordan Pierce - Public Domain CC0 1.0
Note that the fixed point still exists after
λ
= 3, but is no longer stable. Similarly,
the 2-cycles still exist after λ = 1 +
√
6, but it is not stable.