The Logistic Equation (DP IB Analysis & Approaches (AA)): Revision Note

The Logistic Equation

What is the logistic equation?

• The differential equation   is a very simple example of a model in which the rate of change of a population at any moment in time is dependent on the size of the population (N) at that time

  • The solution is  (where A > 0 is a constant)

  • If k > 0, this represents unlimited exponential growth of the variable N

• In many real-world contexts (for example when considering populations of living organisms), unlimited growth is not a realistic modelling assumption

  • For reproducing populations it is logical to assume that the rate of change of the population will be dependent on the size of the population (more rabbits means more production of baby rabbits!)

  • But there are generally limiting factors on populations that prevent them from growing without limits

    • For example, availability of food or other resources, or the presence of predators or other threats, may limit the population that can exist in a given area

• A logistic equation incorporates such limiting factors into the model, and therefore can provide a more realistic model for real-world populations

• The standard logistic equation is of the form

  
  
  

  • t represents the time (since the moment defined as t = 0) that the population has been growing

  • N represents the size of the population at time t

  • ​ is a constant determining the relative rate of population growth

    • For the models dealt with here it is most common to have k > 0, with a larger value of k representing a faster rate of change

  •  is a constant that places a limit on the maximum size to which the population N can grow

    • For a population model it can be assumed that a > 0

    • For k > 0 and an initial population N₀ such that 0 < N₀ < a, the population N will grow and will converge to the value a as time t increases

    • For k < 0 and an initial population N₀ such that 0 < N₀ < a, the population N will shrink and will converge to the value a as time t increases

• There are other forms of logistic equation

  • The exact form of the logistic equation you are to use will always be given in an exam question

How do I solve problems that involve a logistic equation model?

• Solving the differential equation will generally involve the technique of separation of variables

  • Usually this will also involve rearranging one of the integrals using partial fractions (see the worked example below for an example)

• You will usually be given ‘boundary conditions’ specific to the context of the problem

  • For example, you may be told the initial population at time t = 0

  • These conditions will allow you to work out the exact value of any integrating constants that occur while solving the differential equation

• You will need to take account of the context of the question in answering the question or in commenting on the model used