1. What are the units of the parameters r and L? 2. Find the equilibrium solutions of (1) (i.e. the equation without harvesting). Then use separation of variables to nd the exact (non-equilibrium) solution to (1) (your answer should be reported in terms of the unspecied variables and parameters x, t, r, L, and some unknown initial population x(0) = x0). 3. Suppose that the mountain lion population has a natural growth rate of r = 0:65 and a carrying capacity of LM = 5:4. (a) If there are initially 6 mountain lions in the population2, use Euler's method over the interval t 2 [0; 25] with step sizes h1 = 0:5, h2 = 0:1, and h3 = 0:01 to numerically solve the logistic equation (1). Plot the numerical solutions against the exact solution found in Question 2 above. (The plot should contain three numerical solution curves, the exact solution curve, and a legend). Discuss the plot. Specically address how the value of the step size h in uences the accuracy of the numerical solution. (b) The absolute error of a numerical approximation is dened by Abs. Error = jExact Solution Approximate Solutionj Use semilogy to plot the absolute errors of all numerical solutions found in part (a) (plot these error curves together on one plot. Be sure to include a legend). Describe the plot. In particular, speculate about why the error curves contain downward \spikes" around time t = 7. (c) Whenever a problem is solved numerically, it is important to make sure that the calculation carefully balances numerical \accuracy" { the correctness of the approximation obtained { with numerical \eciency," which is the amount of time needed to complete the calculation. For the calculations in part (a) above, which step size might give the best balance of numerical accuracy and eciency? Justify your response (the result of part (b) may help)