Assumptions for reaching the logistic map equation
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
Conjecture $3$: There is only one dominant influence carried on by an individual required which must extend
Population Dynamics
Let the total population of the $i$'th generation be $T_i$ and the maximum of the distribution be $x_{max}$.
Let's presume we do not know which generation we are in ($i$ or $i+1$):
Probability of belonging to the $i$'th generation = $ \frac{T_{i}}{T_i + T_{i+1}}$
Probability of an individual to influencing a generation = $ \frac{1}{T_i + T_{i+1}}$
Probability of finding the maximum of the distribution = $ 1/ x_{max}$
Consider,
$$ \underbrace{\frac{T_{i}}{T_i + T_{i+1}} \times \frac{1}{T_i + T_{i+1}}}_{\text{probability of individual influencing current generation}}$$
and, $$ \underbrace{\frac{T_{i}}{T_i + T_{i+1}} \times \frac{T_{i}}{T_i + T_{i+1}}}_{\text{probability of all members influencing each other}} \times \underbrace{\frac{1}{x_{max}}}_{\text{probability of finding global distribution maxima}}$$
and,
$$ \underbrace{\frac{T_{i+1}}{T_i + T_{i+1}} \times \frac{1}{T_i + T_{i+1}}}_{\text{probability of individual influencing future generation}}$$
For continuity of the influence from one generation to another we have:
$$ \frac{1}{4} \times \frac{T_{i+1}}{T_i + T_{i+1}} \times \frac{1}{T_i + T_{i+1}} + \frac{T_{i}}{T_i + T_{i+1}} \times \frac{T_{i}}{T_i + T_{i+1}} \times \frac{1}{x_{max}} = \frac{T_{i}}{T_i + T_{i+1}} \times ( \frac{1}{T_i + T_{i+1}})$$
We introduce the parameter $a$ with $T_i$ the total population in the i'th iteration. Hence:
$$x_i = T_i/(a/4)$$
With:
$$x_{max} = a/4$$
Upon simplyfication:
$$ T_{i+1} = 4 T_i \times (1- \frac{T_i}{x_{max}})$$
We arrive upon the logistics equation which describes the population dynamics equation:
$$x_{i+1} = a x_i (1 - x_i) $$
Question
What is the correct interpretation of probability in this situation? The logistic's equation has many uses in many systems does the interpretation change?