# Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution

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?

• This problem might get pretty funny if it was translated into French. – Tom Goodwillie Jan 14 '14 at 17:04
• @TomGoodwillie Poisson by a Poisson process...? – MS26 Jan 14 '14 at 17:07

To know the total number of fishes in the pond at time $t$, one needs to know the lifetimes of the $N$ initial fishes. The number $N_t$ of fishes in the pond at time $t$ not in the pond at time $0$ is $$N_t=\int_0^tB_{t-s}(s)\,\mathrm ds,$$ where, for every $t$ and $u$, $B_t(u)\mathrm d t$ denotes the number of fishes introduced in the pond in the time interval $(t,t+\mathrm dt)$ with an initial lifetime at least $u$. Thus, $E(B_t(u))=\lambda P(Z\geqslant u)$, where $Z$ has distribution $G$, and, for every $t$, $$E(N_t)=\lambda\, E(\min(Z,t)),$$ which converges to the stationary value $E(N_\infty)=\lambda\,E(Z)$, as was to be expected. Likewise, the number $N_t(u)$ of fishes in the pond at time $t$ not in the pond at time $0$ and whose residual lifetime at time $t$ is at least $u$ is $$N_t(u)=\int_0^tB_{t-s}(s+u)\,\mathrm ds,$$ hence $$E(N_t(u))=\lambda\, E(\min((Z-u)^+,t)),$$ which converges to the stationary value $$E(N_\infty(u))=\lambda\,E((Z-u)^+).$$ In other words, the distribution of the residual lifetime $R$ of a typical fish in the pond at time $t$ when $t$ and $\lambda$ are large is characterized by the identities $$P(R\geqslant u)=\frac{E((Z-u)^+)}{E(Z)},$$ hence the density $f_R$ of $R$ is such that, for every $u\geqslant0$, $$f_R(u)=\frac{P(Z\geqslant u)}{E(Z)},$$ and the moments of $R$ are given by $$E(R^n)=\frac{E(Z^{n+1})}{(n+1)E(Z)}.$$ In the other direction, every density $f_R$ is nonincreasing and finite everywhere, and the distribution of $Z$ can de deduced from $f_R$ thanks to the identities $$P(Z\geqslant u)=\frac{f_R(u)}{f_R(0)}.$$