Consider the following mathematical problem:
Fish Pond:
- Suppose there is a pond that has $n_1$ fish on the first day ($n_1$ is a positive integer greater than 0)
- Generate a random number $x_1$ from ~ Beta ($x_0$, 1-$x_0$) and a random number $y_1$ from ~ Beta ($y_0$, 1-$y_0$)
- On the second day, there is a probability of $x_1$ that the population of the pond can increase by a factor of $x_0$ (i.e. $n_2$ = $n_1$ + $x_0$ * $n_1$ ) and a probability of $y_1$ that the population of the pond can decrease by a factor of $y_0$ (i.e. $n_2$ = $n_1$ - $y_0$ * $n_1$ )
- Next, generate a random number $x_2$ from ~ Beta ($x_1$ , 1-$x_1$ ) and a random number $y_2$ from ~ Beta ($y_1$, 1-$y_1$)
- On the third day, there is a probability of $x_2$ that the population of the pond can increase by a factor of $x_1$ and a probability of $y_2$ that the population of the pond can decrease by a factor of $y_1$
- From here, we can use a recursive formula to define the population growth of the pond
- The population of the pond can never be less than some positive integer $n_{min}$ = $c$. The first time the population of the pond goes below $n_{min}$, the population of the pond automatically becomes $n_{min}$ and stays $n_{min}$ until it eventually increases. However, there is no upper bound on the population.
Fisherman:
- A fisherman comes to the pond on the first day and catches $Z_1$ fish. On the $k^{th}$ day, the fisherman catches $Z_k$ fish. There is a 0.5 probability of catching the same number of fish that he caught on the $k-1^{th}$ day, and a 0.5 probability of catching any other number of fish between 0 and $Z_{max}$. The fisherman catches fish with replacement (i.e. catches fish, tags them, releases them back in the pond) and each fish has an equal probability of being caught:
\begin{equation} P(Z_k = z | Z_{k-1} = z) = 0.5 \end{equation}
\begin{equation} P(Z_k = z^c | Z_{k-1} = z) = 0.5 \end{equation}
\begin{equation} Z^c = Z_{max} \setminus Z = \{0, 1, 2, ..., Z_{max}\} \setminus \{z\} \end{equation}
$$P(Z^c = 0) = P(Z^c = 1) = \ldots = P(Z^c = z_{\text{max}}) = \frac{1}{|Z^c|}$$
General Question : Given some fixed initial conditions $n_{min}$, $n_1$, $x_0$, $y_0$,$z_1$, $Z_{max}$ - On the start of the $k^{th}$ day:
- What is the expected total population of the pond?
- What is the expected number of total fish the fisherman will catch?
- What is the expected number of fish the fisherman will catch that he has never seen before?
- What percent of the pond has the fisherman encountered in the past?
- What percent of the pond has the fisherman not encountered in the past?
- How many days will it take on average for the population of the pond to surpass some number $g$?
Due to the highly complex and stochastic nature of this problem, I suspect that these questions likely do not have closed form/analytical solutions and likely need to be simulated to solve. That is, simulate the behavior of the pond/fisherman from day $n_1$ to day $n_{k}$ and record the results ... repeat the simulation many times ... and then average the results for these simulations.
My Question: But is there some mathematical proof to show that these kinds of questions likely do not have closed form/analytical solutions (e.g. expected time to absorption https://en.wikipedia.org/wiki/Absorbing_Markov_chain) ... and can only be solved through simulation?
Thanks!
- Note: We assume that the pond population (i.e. births and deaths) on the $k^{th}$ day is not affected (i.e. independent) by the number of fish caught by the fisherman from the $1^{st}$ day to the $k-1^{th}$ day.
- Conceptually, this means that overfishing and underfishing does not encourage the fish to reproduce more or reproduce less (i.e. birth rate), as well as the mortality rate, and life expectancy of the fish