Neanderthals were the original inhabitants of Europe and their species was very stable for more than 60,000 years. Forty thousand years ago the Neanderthals were replaced by our ancestors, the early humans (Cro-Magnon), who came to Europe from Africa much later. There is evidence that the two species coexisted in some parts of Europe. The mass extinction of the Neanderthals was rapid, in 5,000 to 10,000 years. Theories for the demise of the Neanderthals include genocide by the early humans and competition for resources with the humans. (a) Consider the case that Neanderthals N (with population density n) and human H (with population density h) are competing for the same food source C (suppose with population density c). Reproduction is only possible when food is acquired. The associated chemical reactions are N+C2N and H+C 2H. Suppose N and H become food after they died NC and HC. Write down the 3-D ODE model for n, h, and c by the law of mass action. Show that n(t) + h(t)+c(t) is a conserved quantity. Suppose that n(0) +h(0) + c(0) = K and use this conserved quantity to reduce the model into a 2-D model of n and h. What is the state space (phase space) of the 2-D model? (b) Let's define the effective death rate dr and 6= Assume that 6, 5, 5 for this part (b), we want to sketch the state space (phase space) in this problem. By doing that, we will show that the two can coexist. 1. Show that for all points in the state space: (a) If n+h 2. Find all the fixed points. You should be able to notice that the fixed points are all stable and "semi-"attracting (attracting in one direction) based on point 1. 3. Based on the ODE of n' and h', derive the orbit formula that pass through n = no and h = ho- and find the orbit 4. Assume r>rn, sketch the phase portrait. Note that point 3 would allow you to see whether the trajectories should curve upward or downward.