The Logistic Model
If the population of a species depends only on the population P, then P may be modelled by a first order autonomous system
We may assume the population growth rate to be proportional tosince asincreases, mates become easier to find and mating becomes easier to find. We may writeshould be a decreasing function offor the reason that the environment is a finite resource. here is the difference between the birth rate per unit population and the death rate per unit population.
The simplest example of a decreasing function ofis a linear one. We can writeto give
We can separate the variables to giveand integrate by expressing as partial fractions: