Logistic Equation – Explanation & Examples

The definition of the logistic equation is:

“The logistic equation is a sigmoid function, which takes any real number and outputs a value between zero and certain positive number.”

In this topic, we will discuss the logistic equation from the following aspects:

  1. What is the logistic equation?
  2. Logistic equation formula.
  3. How to solve the logistic equation?
  4. Application of logistic function in ecology.
  5. Application of logistic function in statistics.
  6. Practice questions.
  7. Answer key.

1. What is the logistic equation?

The logistic equation is a sigmoid function, which takes any real number from negative infinity -∞ to positive infinity +∞ and outputs a value between zero and a certain positive number.

Logistic equation formula

The equation is :

f(x)=L/(1+e^(-k(x-x_0)) )

where:

f(x) is the logistic equation or function.

L is the logistic function or curve maximum value.

e is a mathematical constant approximately equal to 2.71828.

k is the logistic growth rate or steepness of the curve.

x_0 is the value of x at the sigmoid curve midpoint.

For values of x between -∞ to +∞, the logistic equation draws an S-curve with the curve f(x) approaching L as x approaches +∞ and approaching zero as x approaches -∞.

– Example 1

For the x values:

-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Draw the logistic curve when L = 1, k = 1, and x_0=0.

We follow these steps:

1. plot a table of values.

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

2. Using the above formula, calculate the logistic function for each value.

x

f(x)
-6

0.002472623

-5

0.006692851
-4

0.017986210

-3

0.047425873

-2

0.119202922
-1

0.268941421

0

0.500000000

1

0.731058579

2

0.880797078
3

0.952574127

4

0.982013790
5

0.993307149

6

0.997527377

3. Plot the x values on the x-axis and the logistic function value on the y-axis.

Connect the intersecting points with a line to draw the sigmoid curve.

For comparison, we can add two other equations with the same parameters except that L = 5 and L =10 respectively.

We update the table.

x

f(x)_1f(x)_5f(x)_10
-60.0024726230.01236312

0.02472623

-5

0.0066928510.033464250.06692851
-40.0179862100.08993105

0.17986210

-3

0.0474258730.23712937

0.47425873

-2

0.1192029220.596014611.19202922
-10.2689414211.34470711

2.68941421

0

0.5000000002.50000000

5.00000000

1

0.7310585793.655292897.31058579
20.8807970784.40398539

8.80797078

3

0.9525741274.762870639.52574127
40.9820137904.91006895

9.82013790

5

0.9933071494.966535759.93307149
60.9975273774.98763688

9.97527377

Where f(x)_1 is the logistic function with L = 1, f(x)_5 is the logistic function with L = 5, and f(x)_10 is the logistic function with L = 10.

and plot the 3 different sigmoid curves.

We see that the maximum of each logistic function is its L value.

– Example 2

For the x values:

-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Draw the logistic curve when L = 1, k = 1, 2, or 3, and x_0=0.

We follow these steps:

1. plot a table of values.

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

2. Using the above formula, calculate the logistic function for each value and each k value.

We add 3 other columns, one for each logistic function.

x

f(x)_1f(x)_2f(x)_3
-60.0024726236.144175e-06

1.522998e-08

-5

0.0066928514.539787e-053.059022e-07
-40.0179862103.353501e-04

6.144175e-06

-3

0.0474258732.472623e-03

1.233946e-04

-2

0.1192029221.798621e-022.472623e-03
-10.268941421

1.192029e-01

4.742587e-02

0

0.5000000005.000000e-01

5.000000e-01

1

0.7310585798.807971e-019.525741e-01
20.8807970789.820138e-01

9.975274e-01

3

0.9525741279.975274e-01

9.998766e-01

4

0.9820137909.996646e-019.999939e-01
50.9933071499.999546e-01

9.999997e-01

6

0.9975273779.999939e-01

1.000000e+00

Where f(x)_1 is the logistic function with k = 1, f(x)_2 is the logistic function with L = 2, and f(x)_3 is the logistic function with L = 3.

and plot the 3 different sigmoid curves.

With increasing the k value, the sigmoid curve becomes steeper in its growth.

2. How to solve the logistic equation?

The logistic function finds applications in many fields, including ecology, chemistry, economics, sociology, political science, linguistics, and statistics.

We will focus on the application and the solving of logistic function in ecology and statistics.

– Application of logistic function in ecology

A typical application of the logistic equation is to model population growth, where the rate of reproduction is proportional to both the existing population and the amount of available resources.

The logistic differential equation for the population growth is:

dP/dt=rP(1-P/K)

Where:

P is the population size.

t is the time. The units of time can be hours, days, weeks, months, or years.

dP/dt is the instantaneous rate of change of the population as a function of time.
r is the growth rate.

K is the carrying capacity. The carrying capacity of an organism in a given environment is defined to be the maximum population of that organism that the environment can sustain indefinitely. It has the same unit as the population size.

The population growth rate changes over time. Biologists have found that in many biological systems, the population grows until a certain steady-state population is reached.

The concept of carrying capacity allows for the possibility that in a given area, only a certain number of a given organism or animal can thrive without running into resource issues.

Suppose that the initial population is small relative to the carrying capacity. Then P/K is small, possibly close to zero. Thus, the quantity in parentheses on the right-hand side of the logistic equation is close to 1, and the right-hand side of this equation is close to rP. The value of the rate r represents the proportional increase of the population P in one unit of time. So, the population grows rapidly.

However, as the population grows, some members of the population interfere with each other by competing for some critical resource, such as food or living space. The ratio P/K also grows, because K is constant. If the population remains below the carrying capacity, then P/K is less than 1, so 1-(P/K)>0 but less than 1. Therefore the growth rate decreases as a result.

If P=K then the right-hand side is equal to zero, and the population does not change (this is called maturity of the population).

The solution to the equation, with P_0 being the initial population is:

P(t)=K/(1+((K-P_0)/P_0 )e^(-rt) )

Note that K is the limiting value of P:

If the P_0 < K, then population grows till reaching K.

If P_0 >K, then population decreases till approach K.

– Example 1

A population of rabbits in a meadow is observed to be 200 rabbits at time t=0. Using an initial population of 200 and a growth rate of 0.04 per month, with a carrying capacity of 750 rabbits.

Draw the logistic curve of growth for this population.

We follow these steps:

1. The growth rate is per month so the x-axis will be in months. The 0 value will represent the current month, and 1 is the next month, and so on.

We can also plot negative values on the x-axis to represent the previous month and so on.

In a table, we write the next 12 values (next year) and the previous 12 values (previous year).

So the x values will range from -12 to 12.

In a table:

t_months

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

2. We know that the population at the current month or t = 0 is 200. We use the above equation to know the population size for all these months:

P(t)=K/(1+((K-P_0)/P_0 )e^(-rt) )=750/(1+((750-200)/200)e^(-0.04t) )

For example, at time = 0:

The population size at time = 0 = P(0) = 750/1+(750-200/200)Xe^(-0.04X0) = 750/(1+((750-200)/200))= 200.

Using the above formula, calculate the population size for each time value and update the table.

t_months

population

-12

137.7612

-11

142.3165

-10

146.9863
-9

151.7710

-8

156.6710

-7

161.6865

-6

166.8174

-5

172.0634
-4

177.4243

-3

182.8993
-2

188.4877

-1

194.1884
0

200.0000

1

205.9211
2

211.9500

3

218.0847

4

224.3229
5

230.6621

6

237.0997

7

243.6326
8

250.2578

9

256.9716

10

263.7706

11

270.6506
12

277.6077

3. Plot the t values on the x-axis and the population on the y-axis.

Connect the intersecting points with a line to draw the sigmoid curve.