Metapopulation questions

1

What is the equilibrium fraction of occupied patches for the mainland-island model with a colonization rate of 0.5, and an extinction rate of 0.3? How does this change compared to the standard Levin’s model? What explains this change?

2

Incorporate a rescue effect into the above model and calculate the equilibrium fraction of occupied patches. What is this new value, and how does it compare to the mainland-island and Levin’s equilibrium values? Why is it different?

3

Two species are competing in a metapopulation. Species a has colonization rate of 0.25 and extinction rate of 0.1. Species b has colonization rate of 0.2 and extinction rate of 0.16. Calculate both species equilibrium patch occupancy in the absence of the other species. Then, calculate the persistence criteria for both species in competition, with some discussion of the outcome (e.g., do the species coexist?).

4

In a two-patch predator-prey system, demonstrate what happens when the dispersal rates between the two-patches are equal. Calculate equilibrium values in both cases, assuming that movement $\mu$ is 0.25, $b$ is 0.9, $a$ is 0.25, $d$ is 0.1, and $k$ is 0.8.

Disease questions

1

Calculate $R_0$ for a pathogen with transmission rate of 0.5 and recovery rate of 0.4, assuming $SIR$ style dynamics. What happens to $R_0$ when we introduce an exposed class and set the transition from exposed to infected ($\omega$) to be 0.6?

2

Add vaccination into the SEIR model as presented in the lecture notes, show how to format the F and V matrices for the next-generation method of estimating $R_0$, and estimate $R_0$ given the parameters $\beta$ = 1.1, $\gamma$ = 0.25, $\omega$ = 0.5, and $v$ = 0.3.

3

How could you incorporate land use change into the standard SIR modeling framework we went over in class? How about the role of superspreaders?