As stated in the problem. Given a bacterium that at every time step $t$, either divides with a probability $p$ or dies off. What is the expected number of bacteria after $n$ time steps?

Does this use a binomial theorem? Markov Chains? I have no clue..

Experimental results show that with $p = 0.75$, here are the numbers per timestep $t$ (with $10^7$ simulations):

| t | population | |----|------------| | 1 | 1.500407 | | 2 | 2.2507928 | | 3 | 3.376627 | | 4 | 5.0640936 | | 5 | 7.5959494 | | 6 | 11.3945926 | | 7 | 17.0909234 | | 8 | 25.6368554 | | 9 | 38.4575074 | | 10 | 57.6884722 | | 11 | 86.5342376 | | 12 | 129.802634 |

So the experimental results have shown that this follows a trend of $(2p)^i$ where $i$ is the timestep. I am not sure why that is though... any ideas?

  • $\begingroup$ It seems like with $p=0.25$, the population should eventually reach $0$: the bacterium is more likely to die off than to split. (In fact, $\frac34$ of the time, the bacterium dies without ever splitting.) Or am I misunderstanding the setup? $\endgroup$ – Misha Lavrov Oct 4 '18 at 2:29
  • $\begingroup$ Oh sorry, it's the opposite. It's 0.75! Thanks for that :) $\endgroup$ – QuantumHoneybees Oct 4 '18 at 2:30

If there are $n$ bacteria at time $t$, then on average $pn$ of them split and $(1-p)n$ die, giving an expected number of $2pn$ bacteria at time $t+1$.

If we start with one bacterium at time $t=0$, then by induction at time $t$ the expected number of bacteria is $(2p)^t$. For example, when $t=10$ and $p=0.75$, this gives us about $57.665$ bacteria in expectation.

More generally, the distribution of the bacteria follows a branching process. In particular, we can compute the distribution for any $t$ with generating functions: let $f(x) = 2px^2 + (1-p)$. Then the probability that there are $n$ bacteria at time $t$ is the coefficient of $x^n$ in the $t$-fold composition $$ \underbrace{f(f(f(\cdots(f(}_tx))\cdots))). $$ But deriving further properties of this distribution takes some work.

  • $\begingroup$ So I did, I actually have the wikipedia page open on another tab. But that seems to deal with populations where the recurrence relation is described as $Z_{{n+1}}=\sum _{{i=1}}^{{Z_{n}}}X_{{n,i}}$, isn't the relation more of a product and not a sum? $\endgroup$ – QuantumHoneybees Oct 4 '18 at 2:40
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    $\begingroup$ You can think of each bacterium as having either $0$ or $2$ children. In that case, the next generation's population is a sum of the number of children of all the bacteria in the current generation, which is exactly the branching process setup. $\endgroup$ – Misha Lavrov Oct 4 '18 at 2:42
  • $\begingroup$ Oh ok, yeah I reread the Wikipedia page after reading your comment and it made perfect sense. Thanks! $\endgroup$ – QuantumHoneybees Oct 4 '18 at 2:45

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