Question

3. (10pts) Consider the follow population curves that are solutions for the worm-robins from a predictor-prey system 1.8 1.6 1.4 1.2 Rabins 05 1 15 2 3 0.8 0.6 0.4 0.2 2 2.4S Problem 3 10 15 25 Time t (a) There are two trajectories drawn on the left. Which one seems to correspond to the solutions? Indicate the initial point and its moving direction. Explain. the rate of population of either warm or robins is zero. system to complete one cycle? (b) Indicate the points on the correct trajectory corresponding to (c) If time unit is month, how many months does it take for the

Answer 1

P2 Ps 2 (a) The initial conditions ie· the population of worms and obins at t-0 are 2.2 and 1 millions so, This initial condi tion is met only on the inner trajectory, as determined by the intersechion of the lines W 2.2 and r-1 on the trajectvry (b) he ate of Popula tion or worms and rabit is zero when :0 and du.:0 which can be dLtermined dt using r,w vs t plots, and, 2.2 , W-Ỉ ís another point (1, 2.2) und, B4,W 0.9 i.e. (o.9, 0.4) W-2.2, Y - is one such point or (2.2, 1 So, the points on the trajectory corresponding to the rate of Populaion of either worms or robins is zero are a opp bximotely B( 2.2, 1 ) , ( 1, 2.2), P3 ( 0.4, 1 ), and,

(0.9,0. 4). Note, the rate of population of warms or robins is 2co hen their pots attain maximum or minimum in and hW s t' plots and at those points tangent is paralel t t - axis C) The time it takes por the System to complete one ycle is 5 months, appoximately