The Logistic Equation
Robert M. Hayes
2003
Overview
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Historical Context
Summary of Relevant Models
Logistic Difference Growth Model
Linear Growth
The Logistic Equation
Historical Context
 The starting point for population growth models is The
Principle of Population, published in 1798 by Thomas R.
Malthus (1766-1834). In it he presented his theories of
human population growth and relationships between
over-population and misery. The model he used is now
called the exponential model of population growth.
 In 1846, Pierre Francois Verhulst, a Belgian scientist,
proposed that population growth depends not only on
the population size but also on the effect of a “carrying
capacity” that would limit growth. His formula is now
called the "logistic model" or the Verhulst model.
Recent Developments
 Most recently, the logistic equation has been used as
part of exploration of what is called "chaos theory".
Most of this work was collected for the first time by
Robert May in a classic article published in Nature in
June of 1976. Robert May started his career as a
physicist but then did his post-doctoral work in applied
mathematics. He became very interested in the
mathematical explanations of what enables competing
species to coexist and then in the mathematics behind
populations growth.
Summary of Relevant Models
Population Difference growth Population growth
Exponential Growth
pt+1 = pt + r*pt
Logistic Growth
pt+1 = pt + ((K – pt)/K)*r *pt pt+1 = ((K – pt)/K)*(1 + r)*pt
Linear Growth
pt+1 = pt + C
pt = (1 + r)*pt
pt+1 = pt + C*(K - pt)/K
 Logistic growth models are derived from the exponential
models by multiplying the respective factors r and (1+r) in
the exponential models by (K – pt)/K.
 Note that the two models for exponential growth are
identical but the two for logistic growth are different.
 The linear growth model is important both in itself and as
a part of the logistic models.
Logistic Difference Growth Model
 The logistic difference growth model will be
considered in two contexts:


The Incremental Context, in which growth takes
place at discrete points in time
The Continuous Context
The Incremental Context - I
 Verhulst modified the exponential growth model to
reflect the effect of a maximum for the size of the
population. He denoted the carrying capacity as K and
multiplied the ratio r in the difference exponential
model by the factor (K - pt)/K to represent the effect
of the maximum limit:
(1) pt+1 = pt + ((K – pt)/K)*r*pt
 Equivalently,
(2) pt+1 – pt = ((K – pt)/K)*r*pt
The Incremental Context - II
 Equation (2) will be called the logistic difference
equation. The term "difference" emphasizes that the
left hand side of the equation is the difference between
successive values.
 The following chart illustrates logistic difference
growth, assuming a carrying capacity K = 1000, a
growth rate r = 0.3, and a starting population p0 = 1.
 At the start, with a small value of population, the
factor (K – pt)/K will be very close to one, and the
growth will be nearly exponential.
 As the population value grows and gets closer to K, the
factor will limit the population growth.
Illustrative Logistic Difference Growth
K = 1000, p0 = 1, r = 0.3
1200
1000
800
600
400
200
0
1
11
21
31
41
51
 As this shows, the curve produced by the logistic
difference equation is S-shaped. Initially there is an
exponential growth phase, but as growth gets closer to
the carrying capacity (more or less at time step 37 in
this case), the growth slows down and the population
asymptotically approaches capacity.
The Continuous Context
 The differential counterpart to equation (6) is given by
(3) dp = r*p(t)*(K – p(t))/K dt
 There is a closed solution to this equation:
(4) p(t) = K/(1 + ((K – p(0))/p(0))*e-r*t)
Linear Growth
 Note that, qualitatively, there are three main sections of
the logistic curve. The first has exponential growth and
the third has asymptotic growth to the limit. But
between those two is the third segment, in which the
growth is virtually linear.
The Logistic Equation

Turning from the to the logistic population model to the
"logistic equation":
(7) pt+1 = ((K – pt)/K)*(1 + r)*pt = ((K – pt)/K)*s*pt,
where s = 1 + r.

This equation exhibits fascinating behavior depending on
the value of s = (1+r). We will illustrate the behavior with
different values of s.
Some simple mathematical properties
 First, though, there are two simple mathematical
properties that will be of importance. To identify them,
simplify the equation by letting pt = p and pt+1 = f(p)
 The fixed points for f(p) occur when f(p) = p, and that is
either when p = 0 or when p = K – K/s = K*(s - 1)/s
 The maximum value for pt occurs when the derivative
of equation (10) is set to zero:
(8) d f(p)/dp = s*(1 – 2*p/K) = 0,
which can occur only if s = 0 (which is the minimum) or
when p = K/2. At that value, pmax = s*K/4.
 Since pmax  K, s  4.
Decline to Zero (0 < s < 1)
 When the value of s is between 0 and 1 (r  0) , the
population will eventually decrease to zero. This is
illustrated in the following graph, with K = 1000 and an
initial population p0 = 500 individuals:
K = 1000, p0 = 500
500
400
300
200
100
0
1
6
(1+r) = 0.25
11
16
(1+r) = 0.60
21
26
(1+r) = 0.95
Normal Growth (1 < s < 3) - I
 When the value of s is between 1 and 3, the population
will increase towards a stable value. The following
graph illustrates with three values of s (in each case,
K = 1000 and p0 = 1.00):
K = 1000, p0 = 1.00
800
600
400
200
0
1
11
(1+r) = 1.25
21
(1+r) = 2.00
31
(1+r) = 2.75
41
Normal Growth (1 < s < 3) - II
 Recall that the fixed points for f(p) occur at K*(1 – 1/s).
For these three values of s, the fixed points therefore
are at 200, 500, and 636. The three cases show very
different speeds towards achieving their stable values.
 One more thing to notice is that when the value of s is
larger than 2.4, the equation shows an oscillation which
is larger as it gets closer to 3.0, but in all cases the
oscillation dies down and the population value settles
down to its steady value.
Normal Growth (1 < s < 3) - III
 One very interesting aspect of the logistic equation is
that the long-term value of the population will be the
same regardless of where it starts:
K = 1000 and p0 = 400, p0 = 200, and p0 = 800
1000
800
600
400
200
0
1
11
(1+r) = 1.25
21
(1+r) = 2.00
31
(1+r) = 2.75
41
Multiple Stable Values (3 < s < 3.7) - I
 When s reaches the value of 3, the population oscillates
between two steady values, and at 3.4495 the
population switches among four values! This effect
continues, with oscillation among eight values when
s = 3.56, sixteen values when s = 3.596, etc.
 Note that for s > 3, there is truly a spectacular rate of
growth—more than tripling in each time period. It is
therefore not surprising that there should be a rebound
as the population bounces against its upper limit and
then recovers rapidly only to rebound again.
 The following charts show three graphs of this
behavior:
At s = 3.00, the oscillation is among two values and at
3.55, among four values.
K = 800, K = 1600 and p0 = 1
1500
1000
500
0
1
11
(1+r) = 3.00
21
(1+r) = 3.55
At s = 3.56, the oscillation is among eight values:
K = 1000, p0 = 1, (1+r) = 3.56
1000
800
600
400
200
0
1
11
21
31
41
51
Chaos (3.7 < s < 4)
 When s reaches a value of 3.7, the population jumps in
what appears to be a random, unpredictable way, and
it behaves so until s reaches a value 4.0. While this
behavior looks random, it really isn't.
 This phenomenon has been called "chaos“, first used to
describe this phenomenon by James A. Yorke and Tien
Yien Li in their classic paper "Period Three Implies
Chaos" [American Mathematical Monthly 82, no. 10,
pp. 985-992, 1975]
 The following graph uses a value of r = 3.75.
An example of Chaos
K = 1000, p0 = 1, (r+1) = 3.75
1000
800
600
400
200
0
1
21
41
61
81
101
Another example of Chaos
K = 1000, p0 = 1, (1 + r) = 3.90
1200
1000
800
600
400
200
0
1
21
41
61
81
101
Solutions to the Logistic Equation
 There are only three values of s for which there are
closed form solutions to the logistic equation. To
simplify, let K = 1, so that f(p) = s*p*(1 – p)
 For s = 4,
f(pn) = (1 – cos(2n*g))/2, where
g = cos-1(1 – 2 p0)
 For s = 2,
f(pn) = (1 – exp(2n*g), where
exp(x) = ex and g = log(1 – 2*p0)
 For s = – 2,
f(pn) = 1/2 – cos( + (-2)n*g), where
g = cos-1(1 – 2 p0)
THE END