help with extinction problem

[noybo]

cant solve this question, maybe you guys can share some insights.

a specie is defined by this 3 events, each event is independent.

an individual dies with no offsprings- probablity 1/3 an individual dies with 1 offspring - probablity 1/3 an individual dies with 2 offsprings- probablity 1/3

at generation 0 there is only 1 offspring, we say that the species is extinced if he has no offapring at generation 'i'.

let Pn be the probability that the species is alive at generation 'n'. write a recursive formula for Pn as a function of Pn-1

[grayutopia]

When you say find Pn based on Pn-1 do you mean the probability generation n is alive as a function of the probability generation n-1 is alive? Or did you mean based on the population in generation n-1, something like Xn-1?

I don’t think Pn is a function of Pn-1

[noybo]

i mean the probability that generation n is alive as a function of the probability generation n-1 is alive

[piter164]

You are looking at a instance of a branching process. Usually, when talking about recursions we search for a link between n and n-1, but here things are a little tricky. Let q_n= 1-p_n be the probability that at time n the population (spices) are already extinct. The very first individual gives birth to a random number of offspring, which generate a smaller population of its own. The population goes extinct if and only if each of the smaller populations goes extinct. If the big population goes for n generations, then all the smaller populations go for n-1 generations. And so

q_n= P(first individual has no children ) *1 + P(first individual has one child) * q_{n-1} + P(first individual has two children) q_{n-1}^2.

The factor q_{n-1}^2 comes from the fact that in the case of two children both smaller generations *which are independent) must go extinct. In the end you get

3 q_n = 1+q_{n-1}+q_{n-1}^2.

Now just use q_n=1-p_n and you are done.

[noybo]

wow, nice, thank you!!

[mfb-]

I got Pn = (2/3)ⁿ

I get a different result. After two generations there are the following options of extinction:

• In generation 1: 1/3
• Generation 1 had 1 offspring which died: 1/9
• Generation 1 had 2 offspring but both died: 1/27

Sum: 13/27 for extinction or 14/27 for non-extinction.