Log-normal distribution of an exponentially growing population

[allison_k]

Homework Statement

Consider an exponentially growing bacterial population in a fluctuating environment.
The bacterial population can be modeled as a multiplicative random walk, starting at x = x0.
x evolves in time according to the following rules

$x(t + 1) = x(t)*(1 + ε):$ probability p;
$x(t + 1) = x(t)*(1 + δ):$ probability q;

where |ε|<< 1 and |δ|<< 1, but  they can be positive or negative.
Let y = ln x. For large t, what is the probability distribution P(y)?

What are the mean and variance of this distribution?

Now we want to know the probability distribution of x, Q(x)dx. If x is a monotonically varying function of y, then

$Q(x) dx = P(y) \left| \frac{∂y}{∂x}\right|dx$

What is the probability distribution of x for large t?
Calculate the mean of this distribution. Feel free to express your answers in terms of the mean and variance of P(y).

Finally, on average, which grows faster: the bacteria of the sloppy grad student who
sometimes forgets to feed the bacteria, and then overcompensates by giving
them extra food at later times, or the bacteria of the fastidious grad student
who maintains the same average growth rate at all times?

Homework Equations

The Attempt at a Solution

Really unsure how to proceed. I know I can simplify y to

y(t+1) = ln(x) + ε; probability p
y(t+1) = ln(x) + δ; probability q

But I'm no really sure how to proceed

 

[lippyka]

from there. For the last part, I think it depends on the data given, but I'm not sure how to proceed.