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Tdiscrete & continuous nonlinear dynamics 32144p(t)p22P(t)00 200 400t00 1 2PFigure 12.15 The populations of prey p and predator P from the Lotka–Volterra model. Left:The time dependences of the prey p(t) (solid) and the predators P(t) (dashed). Right: Preypopulation p versus predator population P. The different orbits in this “phase space plot”correspond to different initial conditions.where ɛ is a constant that measures the efficiency with which predators convertprey interactions into food.Equations (12.55) and (12.57) are two simultaneous ODEs and are our firstmodel. We solve them with the rk4 algorithm of Chapter 9, “Differential EquationApplications”, after placing them in the standard dynamic form,dy/dt = f(y,t),y 0 = p, f 0 = ay 0 − by 0 y 1 ,y 1 = P, f 1 = ɛby 0 y 1 − my 1 .(12.58)A sample code to do this is PredatorPrey.java and is given on the CD. Results fromour solution are shown in Figure 12.15. On the left we see that the two populationsoscillate out of phase with each other in time; when there are many prey, the predatorpopulation eats them and grows; yet then the predators face a decreased foodsupply and so their population decreases; that in turn permits the prey populationto grow, and so forth. On the right in Figure 12.15 we plot a phase space plot (phasespace plots are discussed in Unit II) of P (t) versus p(t). A closed orbit here indicatesa limit cycle that repeats indefinitely. Although increasing the initial numberof predators does decrease the maximum number of pests, it is not a satisfactorysolution to our problem, as the large variation in the number of pests cannot becalled control.12.19.1 LVM with Prey LimitThe initial assumption in the LVM that prey grow without limit in the absence ofpredators is clearly unrealistic. As with the logistic map, we include a limit on prey−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 321