Question 1. In class we derived a susceptible-infected model of host-pathogen interactions. We focused primarily on the dynamics of the infected class, lumping the loss of infecteds due to recovery and death into a single parameter (m). For this question, you will build upon the model we developed in class.

(a) Write the full expressions for the dynamics of susceptible and infected hosts, assuming that some infected hosts die from infection while others recover, becoming susceptible again. Note that you will now need to represent death and recovery processes independently. Please use the symbols 'c' for the per-capita recovery rate and 'd' for the per-capita death rate of infected hosts due to infection. Label each term in your equations, using arrows or brackets, with the biological interpretation of the term (e.g., "loss of susceptibles to infection" and "gain of new infections"). (8 points)

(b) Using your response to part (a), derive the threshold density of susceptible hosts that is required for the pathogen to spread through the host population. (4 points)

(c) Imagine that you designing a management plan to prevent disease outbreaks in a herd of elk, and that you are limited to one of these three options:

1. Increase the recovery rate so that infected animals clear the pathogen more quickly.
2. Decrease the death rate so that infect animals survive longer, but they do not clear the pathogen.
3. Both: Increase recovery rate AND decrease the death rate.

Which of these options would be most likely to limit the spread of a disease in the elk herd? Your answer should be based on your response to part (b), and please explain your logic in words. (4 points)

(d) Now consider a disease (like chicken pox) in which hosts that recover cannot become infected again. Write out the full expressions for the dynamics of susceptible (S), infected (I), and recovered (R) hosts. As in (a), use the symbol 'c' for the per-capita recovery rate, and please label your terms with their biological meaning. (8 points)

(e) Would your responses to parts (b) and (c), which were based on the model from part (a), be different for the model you developed in part (d)? Why or why not? (2 points)

Question 2. On Owlspace you will find a .csv file that contains data on Carabid beetle (ground beetle) communities in southern Finland. These are real data that come from this study:

Niemelä J., Haila Y., Halme E., Lahti T., Pajunen T. & Punttila
P. (1988): The distribution of carabid beetles in fragments of old coniferous taiga and adjacent managed forest. Annales Zoologici Fennici25: 107-199

These authors sampled beetles in five different habitats: wet primeval forests, dry primeval forests, managed forests, old plantations, and young plantations. Each row in the data file is a beetle species and the cell entries are counts of each species in each habitat type (columns). Using these data, and in a software platform of your choosing, do the following.

(a) Draw the rank-abundance curve for each habitat type. The x-axis should be species rank and the y-axis should be proportional abundance. You may plot all habitats on the same or different axes, whichever you think best facilitates interpretation.

(b) Calculate the Shannon H-index for diversity of each habitat type, and rank the habitats by diversity.

(c) How are wet and dry primeval forests similar? How do they differ? Consider species richness, evenness, diversity, and composition in your answer.

(d) How are old and young plantations similar? How do they differ? Consider species richness, evenness, diversity, and composition in your answer.

Attachment:- carabids_finland.xlsx