Let's start by ignoring competition. We'll add that effect later.
What part of the current generation will be present in the next generation? Some will die, so the population decreases by its death rate D.
In addition, some will be born, so the population increases by its birth rate B.That is,
Pn+1 = (1 + B - D)*Pn.
(The 1 in the parentheses is there to account for the fact that the population will not change if there are no births and no deaths.)
This is just Malthus' model, and the future is easy to predict.
the model loses its validity, for example, the competition factors cannot be ignored, or
(according to the gloomy Malthus) war, disease, and famine bring the growth within bounds.
Return to the Logistic Map.