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324 chapter 12400200pPopulationpP00 t40000Pt0 400tFigure 12.17 Lotka–Volterra model with predation efficiency and prey limitations. From left toright: overdamping, b = 0.01; damped oscillations, b = 0.1, and limit cycle, b = 0.3.Model a b E m K RLVM-I 0.2 0.1 1 0.1 0 —LVM-II 0.2 0.1 1 0.1 20.0 —LVM-III 0.2 0.1 — 0.1 500.0 0.22. For each of the three models, constructa. a time series for prey and predator populations,b. phase space plots of predator versus prey populations.3. LVM-I: Compute the equilibrium values for the prey and predator populations.Do you think that a model in which the cycle amplitude depends on theinitial conditions can be realistic? Explain.4. LVM-II: Calculate numerical values for the equilibrium values of the preyand predator populations. Make a series of runs for different values of preycarrying capacity K. Can you deduce how the equilibrium populations varywith prey carrying capacity?5. Make a series of runs for different initial conditions for predator and preypopulations. Do the cycle amplitudes depend on the initial conditions?6. LVM-III: Make a series of runs for different values of b and reproduce thethree regimes present in Figure 12.17.7. Calculate the critical value for b corresponding to a phase transition betweenthe stable equilibrium and the limit cycle.12.19.4 Two Predators, One Prey (Exploration)1. Another version of the LVM includes the possibility that two populations ofpredators P 1 and P 2 may “share” the same prey population p. Investigate thebehavior of a system in which the prey population grows logistically in the−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 324