Math Insight

We will explore three sets of assumptions for how patches are colonized by fruit flies or go extinct, resulting in three metapopulation models, each with different formulas for $C$ and/or $E$.

1. Model 1

   For our first model of Solanum fruit flies, let's assume that people who travel to Hawaii on vacation from the mainland are accidentally bringing fruit flies with them in their luggage. In this case, 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.

   1. 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?
   2. Let $m$ be the rate at which a tourist visits Hawaii. (We use the letter $m$, as it represents the migration of fruit flies.) 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 rate is an expression for $C$, the colonization rate of the metapopulation equation.
   3. Now let's consider the other term in the equation: extinction rates. Let's assume that the fruit fly population on each Hawaiian island goes extinct at rate $x$. (We use the letter $x$ for extinction.) What is the overall rate of which occupied fruit fly populations go extinct? (Your extinction rate should reflect the fact that $p$ is the proportion of patches that could potentially go extinct.) This rate is an expression for $E$, the extinction rate of the metapopulation equation.
   4. Putting this all together, write the metapopulation equation in terms of these specific expressions for $C$ and $E$. This equation is our Model 1.

2. Model 2

   For our second model, we 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 ($1-p$).

   1. 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 $C$.
   2. Write down the metapopulation equation using this expression for $C$ along with the expression for $E$ from part (a). This equation is our Model 2. This model is also called the Levins model after Richard Levins, who developed this metapopulation model in 1969.

3. Model 3

   For our third model, 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 will change the extinction rate so that it decreases when more islands are occupied.

   1. To make the extinction rate decrease as $p$ increases, we will include a factor of $1-p$ in $E$. 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 $E$.
   2. Write down the metapopulation equation using this expression for $C$ from part (b) along with this expression for $E$. This equation is our Model 3. This model is sometimes called the “core-satellite” model.

Now that we have three different possible models for fruit fly metapopulation dynamics, let's find their equilibria. Remember that finding an equilibrium in this model means looking for stable proportion of sites that are occupied ($p$). (We strongly recommend getting confirmation about your step 1 results before going too far with the analysis, as the rest depends on having step 1 correct.)

1. Set $\frac{dp}{dt} = 0$ for each of your 3 models and solve for all equilibria for each model. The equilibria might depend on the parameters $m$ and $x$. Evaluate if each equilibrium is biologically plausible (as $p$ must be between zero and one). If an equilibrium depends on $m$ and $x$, state for which values of these parameters the equilibrium is biologically plausible. (Both $m$ and $x$ are positive numbers.)

   1. Model 1:
   2. Model 2:
   3. Model 3 (assume $m \ne x$):
2. Also analyze each of these models graphically, as follows. 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.

   Since we don't have numbers for $m$ and $x$, the exact plots of $C$ and $E$ are not determined. Instead, just draw representative graphs. For some models, the graphs may look quite different depending on the relationship between $m$ and $x$ (e.g., the number of crossings or their relative positions might change, depending on the parameter values). If, for a model, there is more than one case you need to shown, then draw more than one graph of that model.

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 (to determine by the relative position of $C$ and $E$ where $p$ should be increasing or decreasing).

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, let's assume that a fraction $D$ of the total habitat is destroyed. In the original models, the proportion of available patches to colonize at a particular time was $1-p$. With the habitat loss, what is the proportion of sites that are available to colonize? (This expression is the proportion of the total number of patches before any habitat destruction.)
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 ($S+I+R=1$). Based on this equation, write down an expression for the fraction of susceptible individuals ($S$) in terms of $I$ and $R$.

3. 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: let $V$ represent the fraction of the population that is vaccinated. Assuming that the population size remains constant, write down an expression for the fraction of susceptible individuals ($S$) in terms of $I$, $R$, and $V$.

4. 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?
5. 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?