Imagine that each individual dies with a probability of $\eta=0.3$ per day. (The symbol $\eta$ is the Greek letter “eta”.) For simplicity, we will imagine that, at the same moment, another individual is born so that the population size stays exactly at $M=70$. Rather than tracking the number of individuals (as that stays constant at $M$), we will track the number of copies of the $A$ allele. Let $N(t)$ be the number of copies of the $A$ allele at time $t$, where $t$ is measured in days. At each replacement event (i.e., where one individual dies and another is born to replace it), $N(t)$ could increase or decrease, depending on the allele of the individual who dies and the individual who was born. Our goal is to develop a model for the dynamics of $N(t)$.
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First, given that there are $M=70$ individuals and each is replaced with a probability per day of $\eta = 0.3$, what is the total rate of the replacement events?
per day
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If at a moment in time, the number of copies of the $A$ allele is $N(t)=n$, how many copies are there of the $a$ allele?
(Remember, the total population size is $M=70$, so the total number of copies of all alleles must be $70$.) For a given replacement event, what is the probability that the individual who died had the $A$ allele?
What is the probability that the individual who died had the $a$ allele?
We will assume that the individual who is born has the allele of a randomly chosen individual (chosen randomly from all $70$ individual before any individual died). Therefore, what is the probability that the individual who was born had the $A$ allele?
What is the probability that the individual who was born had the $a$ allele?
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Since we assume that the allele for the born individual was chosen independently of the individual who died, we can multiply the probabilities to determine the probability of every combination for the replacement event.
- The probability that the individual who died had the $A$ allele and the individual who was born had the $A$ allele is
- The probability that the individual who died had the $A$ allele and the individual who was born had the $a$ allele is
- The probability that the individual who died had the $a$ allele and the individual who was born had the $A$ allele is
- The probability that the individual who died had the $a$ allele and the individual who was born had the $a$ allele is
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What could cause $N(t)$ to increase? If the individual who died has allele
and the individual who was born had allele
, then $N(t)$ would increase by
at the time of the replacement event. If $N(t)=n$, then the probability that a given replacement event causes $N(t)$ to increase by $_$ is
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Recall that each individual is replaced with probability per day of $\nu=0.3$ so that the total rate of the replacement events is $_$. Given that each of these events causes $N(t)$ to increase from $n$ to $n + _$, the probability that the number of copies of the $A$ allele increases in a short time window $\Delta t$ is
$P(N(t+\Delta t) = n+1 \,|\, N(t)=n) =$
.
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What could cause $N(t)$ to decrease? If the individual who died has allele
and the individual who was born had allele
, then $n(t)$ would decrease by
at the time of the replacement event. If $N(t)=n$, then the probability that a given replacement event causes $N(t)$ to decrease by $_$ is
.
The probability that the number of copies of the $A$ allele decreases in a short time window $\Delta t$ is
$P(N(t+\Delta t) = n-1 \,|\, N(t)=n) =$
.
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In a short time window of length $\Delta t$, we will assume that it is impossible for $N(t)$ to change by more than
. Therefore, what is the probability that the number of copies of the $A$ allele does not change in a short time window $\Delta t$?
$P(N(t+\Delta t) = n \,|\, N(t)=n) =$
.