# Fibonacci like monkey population problem ( population dynamics )

Assume a female and male monkey.

Lets call this generation $1$.

All males and females have one and one only partner for breeding in their entire life assuming there is at least $1$ available.

And they only breed with others of their own generation.

All females give birth to all their boys (if any) before all their girls(if any).

All females and males live for $1$ generation and they never try to kill eachother nor have enemies and there is sufficient food.

The probability for birth of a boy is independant of that for a girl and vice versa.

Males do not get pregnant.

All Females that find a Male get pregnant in their lifetime but not all give succesful birth.

Lets denote the probability for birth from a female -that finds a male - in her entire lifespan by $P(x)$.

$P(0$ $sons) = X$
$P(1$ $son) = A$
$P(2$ $sons) = B$
$P(3$ $sons) = C$
$P(4$ $sons) = D$

note again that the probability of birth for boys and girls is unrelated !

Thus $X+A+B+C+D = 1$

For daughters we have

$P(0$ $daughters) = Y$
$P(1$ $daughter) = A^*$
$P(2$ $daughters) = B^*$
$P(3$ $daughthers) = C^*$

( And here also $Y+A^*+B^*+C^*=1$ ofcourse )

(Note that there is no change per generation for the birth probabilities per pregnant(!) female.)

Let generation be $G$, the amount of sons of $G$ be $M$ and the amount of daughters of $G$ be $F$.

Let $P_M(G,X,A,B,C,D,Y,A^*,B^*,C^*)$ be the probability for $M$ sons in generation $G$ with the given birth probabilities.

Let $P_F(G,X,A,B,C,D,Y,A^*,B^*,C^*)$ be the probability for $F$ daughters in generation $G$ with the given birth probabilities.

Let $exp(G,X,A,B,C,D,Y,A^*,B^*,C^*)$ be the expectation value for children in generation $G$.

Find expressions for $'P_M'$ , $'P_F'$ and $'exp'$.

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"All females give birth to all the boys (if any) before all the girls(if any)" does not look consistent with "All females and males live for 1 generation" – Henry Oct 24 '12 at 20:13
I edited. Guess it it clear now. Thanks. – mick Oct 24 '12 at 20:18
Why do you believe there should be closed-form expressions? If I understand correctly, any excess males/females that don't have a partner don't produce offspring? Then the offspring is determined by the minimum of the numbers of males and females, and the non-linearity of the minimum is going to make it hard to obtain a closed form. – joriki Oct 24 '12 at 21:19
Depends what you mean with closed form. I do not expect elementary solutions ( for the general case ). But a sum or integral or a hypergeometric should be attainable. There are probably many expressions in fact. Also special cases for some fixed values are intresting. – mick Oct 24 '12 at 21:25
The solution might contain a floor function as such problems often do. although there might be a way around that. – mick Oct 24 '12 at 21:28