Bifurcation Lab, MAP 2302 Honors Differential Equations

Last Updated: Wednesday, January 24, 2007 at 15:36

Goal

Assignment

• Parameters in logistic growth
  1. Investigate the dependence between the growth rate coefficient r and the "relaxation time" T (the time when the solution appears to "reach" an equilibrium). Try different carrying capacities K and different initial populations N₀.
  2. At what size does the population grow the fastest? How does your answer depend on r, K, and N₀?
  3. For most values of the parameters, the solution does not appear like a "textbook" S-shaped curve. Find the parameter values that produce the nicest, most symmetrical "textbook" S-shaped curve.
• Phase line in logistic growth
• Bifurcation in logistic harvest
  1. What are the values of K and r here?
  2. Assuming that N and K are measured in 1000 fish (or "kilofish"), at what size does the population grow the fastest? How does your answer depend on the harvesting rate h (at least, for 0 <= h < 0.25)?
  3. For h > 0.25, at what size does the population decay the slowest?
  4. Set h to two near-by values either side of 0.25, such as 0.24 and 0.26, and plot some solutions. How are the overall pictures (slope fields and solutions) similar? Different?
  5. If h=0.26, there are no equilibria. Yet, one can observe the "ghost node" at N=1/2. How? (Hint: plot some solutions and describe their behavior in your own words.)
  6. In this example, the bifurcation value for h appears to be around 1/4. Prove that it is exactly 0.25 by observing that the highest point on the parabola occurs exactly halfway between the equilibria 0 and 1, so its height is (0.5)(1 - 0.5) = 0.25.
  7. Repeat the arguments above to find the bifurcation value for h in the general logistics harvest model dx/dt = rx(1-x/K) - h. Before you begin the computations, predict how increasing r and increasing K will affect this value.
  8. After years of being harvested at a rate h=0.3, which is a bit over the bifurcation value h^*=0.25, the fish population fell to near zero (say, x=0.1). Scared of losing their business, the fishermen decide to cut the harvesting rate in half. After some more experimenting with the Bifurcation in logistic harvest tool, explain why cutting the harvesting rate to h=0.15 is not good enough and why fishing must be banned completely (h=0) for a while.
• Consider the following bifurcation diagrams. Describe the qualitative change (i.e., what happens) as the parameter passes the bifurcation value 0 in each case:
  • Saddle-node
    1. What are the types of the two equilibria? Can a DE with continuous right-hand side have two sinks and no other equilibria?
    2. Suppose that you can perform a physical experiment described by this model, where you can control r via a knob and measure x. Describe your "observations" as you turn the r-knob through 0. (Hint: use the tool to plot some solutions.) Which equilibrium will you measure?
    3. As the parameter r approaches its bifurcation value, what happens to the "speed" with which the equilibria get closer to each other?
    4. Why does the bifurcation diagram (on the bottom right) look like the plot of the right-hand side (top left)? Do you expect to see this similarity in all bifurcation diagrams?
  • Trans-critical
    1. Does this bifurcation diagram (on the bottom right) look like the plot of the right-hand side (top left)? Revise your last answer for the previous diagram.
    2. In terms of logistic growth, what is the meaning of r here?
    3. What does the color (red/yellow) represent here? What about the circle shape (solid/hollow)? What about the line style (solid/dotted) in the bifurcation diagram?
    4. Suppose again that you can perform a physical experiment described by this model. What do you observe "physically" as you turn the r-knob through 0? What do you observe mathematically?
    5. As the parameter r approaches its bifurcation value, what happens to the "speed" with which the equilibria get closer to each other?
  • Pitchfork
    1. Can a system with a cubic right-hand side have no equilibria?
    2. Historically, this situation is the very reason for the name bifurcation. But wait, this bifurcation diagram looks like a horizontal trident. Why, then, the phenomenon was not called "trifurcation"? (Seriously, think what type of equilibria are "physically observable" here.)
    3. What is the shape of the curved part of the bifurcation diagram? Try to derive a formula for it?
    4. There is another version of the pitchfork bifurcation that looks very similar geometrically (just a mirror image) but is very different "physically". What is the difference?

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