The logistic equation is often used to model the density-dependent growth of a population. If the population is being hunted or harvested, we can add a constant harvesting term to the dimensionless version of the model arriving at the following equation: dx dt = f(x) = x(1 x) (1) with 0. For = 0 and = 1, draw the phase portrait f(x) versus x. Then for these two values of sketch solutions x(t) versus t for a variety of initial conditions.

At what value of does a bifurcation occur? What type of bifurcation is it? Sketch the bifurcation diagram.

Problem 1: Simple population growth. Consider a population of organisms that starts out with P0 members at time t = 0. Assuming that in a given time step (say one year), some fraction, fr, of the members will have reproduced and added a single member to the population. Another fraction, fm, will have died.

1a (5 pts): Let Pn+1 denote the population size at time t = n + 1. Write down a single linear difference equation model for Pn+1.

1b (5 pts): Suppose P0 = 80, fr = 0.1, and fm = 0.6. What will be the size of the population at time t = 4?

1c (5 pts): Based on the equation you wrote for part (a), is it possible for your model to exhibit oscillations that are biologically meaningful? Explain why or why not.

Problem 4: Infectious disease. A basic SIR model of disease spreading is given by dS dt = SI dI dt = SI vI dR dt = vI where S, I, and R are the number of people that are susceptible, infective, and recovered (or removed), respectively.

4a (10 pts): Non-dimensionalize the equations and show that the system can be written as dS d = S I dI d = S I I dR d = I Find .

4b (5 pts): In the nondimensionalized system, we have that S + I + R = 1. Given this, what is the interpretation of S , I , and R ? When removed individuals can become susceptible again (i.e. they lose their immunity), we have an SIRS model which can be reduced to the following equations dS dt = SI + (N S I) dI dt = SI vI

4c (5 pts): What is the Jacobian matrix for this system?

4d (5 pts): Let N = 2, = 1, v = 1, and = 1. There are two fixed points; find them.

4e (5 pts): Classify the type and stability of the two fixed points.

4f (5 pts): Draw a phase portrait for the system in part (d) with S on the horizontal axis and I on the vertical axis. Sketch and label the S- and I-nullclines clearly with arrows showing the direction of flow along them.