University of California, Los Angeles LIFESCIENC 30A. Homework 6 Solutions. All Exercises.

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Homework 6 LIFESCIENC 30A. Homework 6 Solutions. A FE 4: Mary is going to have an outdoor party in 10 days. She wants to have her backyard pond covered in water lilies before the party, so she goe... s to the nursery to buy some water lilies. Mary gives the clerk the dimensions of her pond and the clerk, knowing the growth rate of the water lilies that he stocks, calculates 3.2. When the derivative is zero 141 that if she purchases a single water lily, it will produce a population of 10,000 lilies that will completely cover the surface of the pond in 20 days. Mary reasons that if she buys two water lilies instead of one, she can meet her goal of having pond surface covered in 10 days. Is there anything wrong with Marys reasoning? How many water lilies will Mary need to buy to meet her goal? The spread of a genetic mutation in a population of mice can be modeled by the differential equation P0 = 2P(1 - P)(1 - 3P) where P is the fraction of the mice that have the new gene. (So this means that 0 ≤ P ≤ 1 always.) (a) Find the equilibrium points of this model, and determine the stability of each one. (b) If 10% of the mice have the new gene (so P = 0.1) initially, what fraction of the population will have the new gene in the long run? (c) What if the initial fraction is 90% of the mice? FE 6: The method of over and under can be applied whenever we have one curve representing an inflow and one representing an outflow. Sketch three sets of such curves. For each set, mark the equilibria on the horizontal axis and find their stability. No equations are necessary FE 3: Consider the following Romeo and Juliet model: R0 = J - 0:25R2 J0 = R + J (a) Plot the nullclines of this system. negative!) (Remember that both R and J can be (b) Use the nullclines and/or algebra to find the equilibrium points of the system. (c) Sketch the direction of the change vectors along each nullcline. Then, fill in the change vectors in the rest of the vector field. (d) Use your sketch of the vector field to determine the type of each equilibrium point. FE 7: Let D be the size of a population of deer, and M the population of moose in the same area. The LotkaVolterra competition model for these species might look like the following: D0 = 0:3D - 0:02D2 - 0:05DM M0 = 0:2M - 0:04M2 - 0:02DM (It wouldnt hurt to refresh your memory about what each of the six terms in the equations above represent.) (a) Plot the nullclines of this system. (b) Use the nullclines and/or algebra to find the equilibrium points of the system (c) Sketch the direction of the change vectors along each nullcline. Then fill in the change vectors in the rest of the vector field. (d) Use your sketch of the vector field to determine the type of each equilibrium point. (e) What will happen to these two populations in the long run? Can they coexist? Exercise 1: Use graphical stability analysis to confirm what was said about the stabilities of A and K, both when A < K and when A > K Exercise 4: Does the pair of equilibria produced by a saddle-node bifurcation have to consist of one that is stable and one that is unstable? Exercise 7: Use Sage to plot the two parts of Equation 3.16 (increases and decreses) for 3 values of a. Find the resulting equilibria by seeing where the positive parts equal to the negative part., and determine their stability FE1: We saw that in the model of logistic growth with an Allee effect, X0 = rX 1 - KX XA - 1 A, the growth threshold, becomes a stable equilibrium point and K, the carrying capacity, becomes an unstable one when A > K. Does this make biological sense? For what ranges of parameter values does the model behave reasonably? Exercise 3: Why can h act as a half-saturation density? In other words, what is the consumption rate when N = h and what does this mean biologically? Exercise 4: Find the equilbria for this model. (Hint: Work with the second equation first.) Exercise 5: Verify that for values of n less than 8, the system goes to a stable equilibrium, but as n passes 8, the equilibrium point becomes unstable and a stable oscillation is created. FE 2: Does a trajectory that approaches a limit cycle attractor ever reach the attractor? Explain. [Show More]

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