Math Insight

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 $N(t)$.

1. 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

2. 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 $M=70$ and the individuals are haploid, 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?
   

3. 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.

   1. The probability that the individual who died had the $A$ allele and the individual who was born had the $A$ allele is
      
   2. The probability that the individual who died had the $A$ allele and the individual who was born had the $a$ allele is
      
   3. The probability that the individual who died had the $a$ allele and the individual who was born had the $A$ allele is
      
   4. The probability that the individual who died had the $a$ allele and the individual who was born had the $a$ allele is
      
4. 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
   .

   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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   Hint

   The probability that there is a replacement event in a time interval of width $\Delta t$ is the product of probability of

   • the probability per day that a given individual is replaced,
   • the number of individuals, and
   • the length of the time interval (measured in days to match the probability per day, but we're not worrying about that detail).

   To obtain the probability that $N$ increases by 1 during the time interval, you simply need to multiply this product by the probability that any given replacement event increases the number of $A$ alleles by 1, which you just calculated.

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5. 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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   Hint

   Follow same procedure as for previous part. The only difference is that the last factor is the probability that any given replacement event decreases the number of $A$ alleles by 1, which you just calculated.

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6. 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 $\Delta t$?
   $P(N(t+\Delta t) = n \,|\, N(t)=n) =$
   .
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   Hint

   The probabilities of all possible events should sum to one.

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The birth rate $\lambda_n$ of an $A$ allele is depends on number of $A$ alleles according to the following expression: $$\lambda_n =\frac{n}{M} \left(1 - \frac{n}{M}\right) M \gamma \Delta t$$ The first two factors are the probability that a replacement event replaces an $a$ allele with an $A$ allele. The factor $M\gamma$ is the rate at which the replacement events occur, where $\gamma=0.2$ is the probability per day that an individual dies and is replaced. (The symbol $\gamma$ is the Greek letter “gamma”.)

The death rate $\mu_n$ of an $A$ allele is the same expression $$\mu_n = \frac{n}{M} \left(1 - \frac{n}{M}\right) M \gamma \Delta t.$$ Since we haven't included in our model any selection bias, the probability that an $A$ allele is replaced by an $a$ allele is the same as the probability that an $a$ allele is replaced by an $A$ allele. We assume individuals are equally fit no matter which allele $A$ or $a$ they have.

We are interested in exploring the importance of using a stochastic model for the dynamics of $N(t)$.

1. Write down a deterministic approximation for the birth-death process. The deterministic model should be an equation for the derivative $dN/dt$.
   
   
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   Hint

   The net growth rate of the deterministic model should be the net birth rate, calculated by subtracting off the death rate.

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2. If we start at the initial condition of $N(0) = M/2 = 25$, solve the deterministic equation for $N(t)$.
   $N(t) = $
   
3. In the stochastic model, if at some point, the number of $A$ alleles reached $N(t) = M = 50$, what would happened to $N(t)$ for times? From that point onward, $N(t)=$
   .

   In the stochastic model, if at some point, the number of $a$ alleles reached $M = 50$ so that the number of $A$ alleles reached $N(t) =$
   , what would happened to $N(t)$ for times? From that point onward, $N(t)=$
   .

4. If we start at the initial condition of $N(0) = M/2 = 25$, is it possible that $N(t)$ might reach $M=50$? yes no
   If that happened, then the number of $A$ alleles would be fixed at 0 continue to fluctuate be fixed at M
   .

   If we start at the initial condition of $N(0) = M/2 = 25$, is it possible that $N(t)$ might reach $0$? yes no
   If that happened, then the number of $A$ alleles would continue to fluctuate be fixed at M be fixed at 0
   .

5. What is the likelihood that $N(t)$ would eventually be fixed at either $0$ or $M=50$? $N(t)$ will eventually get fixed at other $0$ or $M=50$ with probability
   

   If you like, you can simulate the process this R script. You may want to slowly increase the simulation length tmax even as large as 2500 days to make sure you are seeing what happens in the long term.

6. How well are the dynamics of $N(t)$ captured by the deterministic model? If you cared about the long term fate of the number of $A$ alleles in the population, the deterministic model would give you a good idea of the result be a reasonable starting place tell you absolutely nothing