logistic equation

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The Logistic Equation

Robert M. Hayes

2003



Overview

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 betweenover-population and misery. The model he used is nowcalled the exponential model of population growth.

In 1846, Pierre Francois Verhulst, a Belgian scientist,proposed that population growth depends not only onthe population size but also on the effect of a ―carryingcapacity‖ 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 aspart of exploration of what is called "chaos theory".Most of this work was collected for the first time byRobert May in a classic article published in Nature in

June of 1976. Robert May started his career as aphysicist but then did his post-doctoral work in appliedmathematics. He became very interested in themathematical explanations of what enables competingspecies to coexist and then in the mathematics behind

populations growth.



Summary of Relevant Models

Logistic growth models are derived from the exponentialmodels by multiplying the respective factors r and (1+r) inthe exponential models by (K – p

t

)/K.

Note that the two models for exponential growth areidentical but the two for logistic growth are different.

The linear growth model is important both in itself and asa part of the logistic models.

Exponential Growth

Logistic Growth

Linear Growth

Population Difference growth Population growth

pt+1 = pt + r*pt pt = (1 + r)*pt

pt+1 = pt + ((K – pt)/K)*r *pt pt+1 = ((K – pt)/K)*(1 + r)*pt

pt+1 = pt + C pt+1 = pt + C*(K - pt)/K



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 toreflect the effect of a maximum for the size of thepopulation. He denoted the carrying capacity as K andmultiplied the ratio r in the difference exponentialmodel by the factor (K - p

t)/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 differenceequation. The term "difference" emphasizes that theleft hand side of the equation is the difference betweensuccessive values.

The following chart illustrates logistic difference

growth, assuming a carrying capacity K = 1000, agrowth rate r = 0.3, and a starting population p0 = 1.

At the start, with a small value of population, thefactor (K – p

t)/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

As this shows, the curve produced by the logisticdifference equation is S-shaped. Initially there is anexponential growth phase, but as growth gets closer tothe carrying capacity (more or less at time step 37 inthis case), the growth slows down and the population

asymptotically approaches capacity.

K = 1000, p0 = 1, r = 0.3

0

200

400

600

800

1000

1200

1 11 21 31 41 51



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 onthe value of s = (1+r). We will illustrate the behavior withdifferent values of s.



Some simple mathematical properties

First, though, there are two simple mathematicalproperties that will be of importance. To identify them,simplify the equation by letting p

t= p and p

t+1= f(p)

The fixed points for f(p) occur when f(p) = p, and that iseither when p = 0 or when p = K – K/s = K*(s - 1)/s

The maximum value for pt occurs when the derivativeof 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) orwhen p = K/2. At that value, p

max= 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) , thepopulation 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

0

100

200

300

400

500

1 6 11 16 21 26

(1+r) = 0.25 (1+r) = 0.60 (1+r) = 0.95



Normal Growth (1 < s < 3) - I

When the value of s is between 1 and 3, the populationwill 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

0

200

400

600

800

1 11 21 31 41

(1+r) = 1.25 (1+r) = 2.00 (1+r) = 2.75



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 isthat 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

0

200

400

600

800

1000

1 11 21 31 41

(1+r) = 1.25 (1+r) = 2.00 (1+r) = 2.75



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 andthen 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

0

500

1000

1500

1 11 21

(1+r) = 3.00 (1+r) = 3.55



K = 1000, p0 = 1, (1+r) = 3.56

0

200

400

600

800

1000

1 11 21 31 41 51

At s = 3.56, the oscillation is among eight values:



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

0

200

400

600

800

1000

1 21 41 61 81 101



Another example of Chaos

K = 1000, p0 = 1, (1 + r) = 3.90

0

200

400

600

800

1000

1200

1 21 41 61 81 101



Solutions to the Logistic Equation

There are only three values of s for which there areclosed form solutions to the logistic equation. Tosimplify, 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

logistic equation

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