Stochastic Process Simulation

Question

I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation, $$ P(t+1) = R P(t) $$ In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.

Answer 1

Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.