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

The logistic equation

What is the logistic equation?

• The standard logistic equation for a population N is of the form

  
  
  

  • N represents the size of the population at time t

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

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

    • For the models dealt with here it is most common to have k > 0

    • A larger value of k represents 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 N₀ > a, the population N will shrink and will converge to the value a as time t increases

Why are logistic equations used in modelling?

• The differential equation   is a very simple example of a model where the rate of change of a population is dependent on the size of the population (N) at a particular time t

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

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

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

• 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