Bio 481 - Problem Set 9 - Competition and SIR Models

Lucky you! This problems set is a short one so that you can use some time to work on your projects. Please finish the problem set first though.

1. Take the Lotka-Volterra competition system. Graph the isoclines for the four possible outcomes. Also look at the time series graphs to prove that you see the predicted dynamics.
   
   
2. Add a non-linearity to the competition equations. Calculate the isoclines now and graph them. Analyze the dynamics of this system. What was the overall effect of the nonlinearity you added?
   
   
3. Use a basic SIR model and start your population with a very low level of initial infection. Try various parameter values. How does infection run its course? What is necessary for infection levels to increase initially? Does the disease survive indefinitely?
   
   
4. Add a term that allows individuals to lose their immunity (the disease doesn't cause death and there are no background deaths at this time scale) at a certain rate. How does this change the dynamics?
   
   
5. Use a type II functional response in a classic Lotka-Volterra system. How have the isoclines changed? What is the dynamics now in phase-space? Can you see these dynamics easily in a time series plot?