Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total number of fish), and survive for a time given by some general distribution $G$. Starting from a state where there are $N$ fish in the pond, what do we need to know about $G$ in order to calculate the fish population dynamics over time (the rate of increase or decrease over time)? Consider that if $G$ is bimodal (early age / child mortality often does this to lifetime probability distributions), the mean lifetime expectation may not be very helpful.

Is the median lifetime more useful in this regard? Or is the answer always going to be that "it depends"?

Edit: I suppose there are two parts to this question, the second one conditional on the first: (1) Is there some simple or minimal metric that we can measure about $G$ (mean, median, etc.) that serves the purpose of allowing us to determine fish population dynamics over time? (2) If not, and given perfect information about the probability distribution $G$ (or at least a good enough approximation from experimental / simulation data), we must necessarily have enough information to determine the fish population dynamics, but how do we do this in practice?

For a fun example, let's imagine the lifetime probability distribution $G$ looks like the positive component of some damped harmonic oscillator (after appropriate scaling, etc.): http://tinyurl.com/o5o8p9l (WolframAlpha query "Plot[(Re[Sqrt[e^(-t)*Sin[2*pi*t]]])^2, {t, 0, 3.5}]" --- the full URL failed to hyperlink properly due to the presence of special characters). One could also imagine those peaks having arbitrary or random heights. This kind of scenario could occur if the primary mechanism for fish mortality was fishing, and a set of catch or kill rules were put in place based on a fish's age (maybe it has to be a prime number, etc). I'd argue that the mean or median value for random variates sampled from $G$ might not be too useful in estimating the fish population dynamics (though I could certainly be wrong). Are there more "robust" metrics?