The Logistic Equation

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When considering a possible model for a population that would take into consideration the competition between individuals we presented the following model

P'[t] = a P[t] - s (P[t])^2

where t denotes time and P[t] the number of individuals in the population. We called this the logistic equation.
The term a P[t], gives the rate of growth of the population (birhts and natural deaths included) and the term s (P[t])^2 gives the deaths due to encounters of competing individuals.

We claimed that this equation provides a good model for studying a population with competition. Lets analyze some examples and see why.

First Example

Second Example

To Do On Paper After completing the notebook.

General Solution in terms of the parameters a and s.