III.3 Logistic Growth

III.3.1 Malthus revisited

Although according to Malthus populations tend to increase exponentially, resources that support populations (e.g. food) are fixed.  As the exponential growth of the population outstrips the linear growth of the resources, the resulting constraints would damp the population growth.

Taking the population  growth  differential  equation developed in the previous section, one can add a dampening effect  to account for excessive growth  in the system.  Instead of having linear  in , we will make P' quadratic in . This approach leads to the so-called continuous logistic equation.  It was introduced by the Belgian statistician Pierre-Francois Verhulst (1804-1849).

With this model, the rate of change of will be when and when . Thus we can view as the fixed  maximum number of animals or plants that a one-species ecosystem can support. The population dependence of the birth and death factors is usually referred to as the crowding effect. Before we actually solve this ODE, it is instructive to analyze it graphically using vector fields.

Note that does not depend at all on .  It only depends on .  So all the columns of vectors in the above vector field are identical.

REQUIRED
III.3.1a Change the factor from 2.0 to 0.2, explain the observed behaviour.

OPTIONAL
III.3.1b Explain the behaviour of the plot created above.

III.3.2 Going Cubic

If, instead of setting to be quadratic in , one can set cubic in . A model can be built which has three stationary points:  .  

P = T: Is an equilibrium point that represents the threshold of a population. That is the number below which the population  becomes extinct.  
In the range   We set .
In the range we set .  
For , we want .  

A cubic function with these properties is easily constructed:

ADVANCED
III.3.2a Explain the behaviour of this plot in terms of population dynamics. Give an interpretation of the distinct area's in the curve. Experiment with different values of the parameters.

This cubic function to assign values to in a differential equation.  Then solution  curves are plotted  for various values of the initial  condition.

OPTIONAL
III.3.2b Vary the value of (e.g. from 0.2 to 2.0) and interpret the figures you obtain.