- #1

Cryg9

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## Homework Statement

500 blue and 500 orange bacteria are placed in a growth medium. Each bacterium divides every hour. A predator eats exactly 1000 bacteria per hour irrespective of color.

a) What is the ultimate probability distribution for the colors of bacteria in the growth medium?

b) How long will it take to reach this equilibrium state?

c) If the predator has a 1% preference to blue bacteria how does this change the final distribution?

## Homework Equations

N

_{b}(t)=2

^{t}N

_{b0}- number consumed...

(Not sure how to express that probabilistically)

same for N

_{o}

I also thought about using the different ways the 1000 consumed bacteria could be chosen from each of the orange and blue but they are indistinguishable (right?) so that does not matter.

## The Attempt at a Solution

a) It seems the most probable solution is that the distribution would remain equal but this is an unstable equilibrium, a small perturbation in either direction will be exponentiated through the growth process. Consequently I think that the ultimate probability will be all of one type, either blue or orange, equal probability for each option. I am not sure how to show this rigorously though.

Except if it takes an hour before the bacteria divides into 2, it seems the predator would consume all 1000 (500 orange and 500 blue) in the first hour before any are able to divide. I figured this was an error in the phrasing of the question but is this significant?

b) A long time... Not sure how to rigorously determine this.

c) Now the final distribution will be all orange because the number of blue will asymptotically approach 0 as time progresses.