Math Insight

1. In the example of Solanum fruit flies, let's start by assuming that people who travel to Hawaii on vacation from the mainland are accidentally bringing fruit flies with them in their luggage.

   A new colonization event will occur whenever a tourist carrying a fruit fly visits an island in Hawaii that was previously empty (had no fruit flies). Suppose each tourist picks a Hawaiian island to visit at random. If $p$ is the proportion of Hawaiian islands already occupied by fruit flies what is the probability that a tourist picks an island that is currently unoccupied by fruit flies?

   Let $m$ be the rate at which a tourist visits Hawaii. What is the rate at which a new colonization event occurs (i.e. the rate at which a tourist visits Hawaii and also picks an island that is currently unoccupied by fruit flies)?

   This should give you an expression for $C$, the colonization rate, for the equation above.

   Now lets consider the other term in the equation: extinction rates. Lets assume that the fruit fly population on each Hawaiian island goes extinct at rate $x$. What is the overall rate of which occupied fruit fly populations ($p$) go extinct?

   This should give you an expression for $E$, the colonization rate, for the equation above.

   Write down the equation above ($\frac{dp}{dt} = C – E$) in terms of these specific expressions for $C$ and $E$. Let's call this Model 1.

2. Alternatively, we could assume that Hawaiian islands are colonized by fruit flies not from the mainland, but from other islands in Hawaii.

   Here, a new colonization event will occur whenever a fruit fly leaves an occupied island ($p$) and travels to an unoccupied island ($1-p$). If we assume that fruit flies travel between islands at migration rate m, what is the rate at which islands become colonized by fruit flies?

   This should give you an alternative expression for $C$.

   Write down the $\frac{dp}{dt}$ equation using this expression for $C$ along with the expression for $E$ from part (a). Let's call this Model 2. This is also called the Levins model after Richard Levins, who developed this metapopulation model in 1969.

3. Let's consider one more alternative: that even if fruit fly populations are dying within one island, they could be replaced by new fruit flies migrating from other occupied islands (this is called the “rescue effect”). To include this effect in our model, we assume that the extinction rate decreases when more islands are occupied. Take the expression for $E$ that you wrote down in part (a) above and replace '$x$' with '$x(1-p)$'. What is the overall rate of which occupied fruit fly populations go extinct?

   This should give you an alternative expression for $E$.

   Write down the $\frac{dp}{dt}$ equation using this expression for $C$ from part (b) along with this expression for $E$. Let's call this Model 3. This sometimes gets called the ''core-satellite'' model.

Now that we have three different possible models for fruit fly metapopulation dynamics, lets find their equilibrium. Remember that finding an equilibrium in this model means looking for stable proportion of sites that are occupied ($p$).

1. Set $\frac{dp}{dt} = 0$ for each of your 3 models and solve for all equilibria for each model.

   Model 1:

   Model 2:

   Model 3:

2. Let's also analyze each of these models graphically. For each model, plot the colonization rate ($C$) as a function of the fraction of occupied sites ($p$). On the same graph, plot the extinction rate ($E$) as a function of the fraction of occupied sites ($p$). Be sure to label your lines. Indicate all the equilibria on each graph. Note: you may need more than one graph for each model if there is more than one case you need to show.
3. Determine the stability of each of the equilibria for each model. (You can do this by either evaluating the sign of $\frac{dp}{dt}$ or by examining the graphs.)

1. If the assumptions in Model 1 were true, what fraction of Hawaiian islands would you expect to see occupied by fruit flies? How does this depend on the model parameters ($m$ and $x$)? Is it possible to have the fraction of sites occupied be 0? If so, how – what management strategies would you need to use?
2. If the assumptions in Model 2 were true, what fraction of Hawaiian islands would you expect to see occupied by fruit flies? How does this depend on the model parameters? Is it possible to have the fraction of sites occupied be 0? If so, how – what management strategies would you need to use? How do these results differ from part (a)?
3. If the assumptions in Model 3 were true, what fraction of Hawaiian islands would you expect to see occupied by fruit flies? How does this depend on the model parameters? Is it possible to have the fraction of sites occupied be 0? If so, how – what management strategies would you need to use? How do these results differ from part (a)?

So far in our model we have assumed that the amount of habitat available for fruit flies to colonize remains constant over time. In this section we will explore what happens under habitat loss, where the amount of habitat available to fruit flies decreases over time.

1. To include habitat loss, lets assume that a fraction $D$ of the total habitat is destroyed. This means that instead of there being $1-p$ available sites to colonize per time, there are how many sites available to colonize per time?
2. Take your Model 2 above (the Levins model) and replace the term ($1-p$) with the term you just wrote down.
3. Calculate the equilibrium fraction of sites colonized. How does habitat loss affect the equilibrium fraction of sites colonized?

In the second module of this class we built a model to explore influenza disease dynamics (Influenza Project).

1. Write down an equation for the rate of change of the number of infected individuals ($\frac{dI}{dt}$), assuming that the rate of infection is $\beta S I$ and the rate of recovery is $\gamma I$.

2. In the Influenza project, we used our model to describe the total number of individuals in each of the susceptible ($S$), infected ($I$) and recovered ($R$) classes, and we assumed that the total population size remained constant ($S+I+R=N$). For this section, we will instead use our model to describe the fraction of individuals in each class, assuming the population size remains constant ($S+I+R=1$). Write down an expression for the fraction of susceptible individuals ($S$) in terms of $I$ and $R$.

   
   
   

   As in the Influenza project, lets assume that we have the ability to give a fraction of the population a flu shot, vaccinating them against influenza. This introduces a new class of individuals, $V$. Assuming that the population size remains constant, write down an expression for the fraction of susceptible individuals ($S$) in terms of $I$, $R$, and $V$.

3. Substitute this expression for $S$ into your equation for the rate of change of the number of infected individuals ($\frac{dI}{dt}$). Compare this equation to the equation you wrote down above for a Levins metapopulation with habitat loss. What are the similarities?
4. Based on the parallels between these two equations, and the effect of habitat loss that you described above, how do you expect that vaccination would affect the equilibrium fraction of individuals infected in a population? How does this relate to your findings of the effect of vaccination from the Influenza Project?