Math 19. Lecture 28Periodic SolutionsT. JudsonFall 20051 An Improved Predator-Prey ModelIf p(t) be the number of prey and q(t) is the number of predators at time t,then we can mo del a predator-prey system asdpdt=23p1 −p4−pq1 + p(1)dqdt= sq1 −qp, (2)where s > 0.• If there are no predators, our system (1) is just logistic growth:dpdt=23p1 −p4.We have a stable equilibrium at p = 4.• The existence of predators decreases dp/dt by pq/(1 + q).– When p is small,pq1 + p≈ pq.This tells us that the dp/dt is dependent on predator-prey inter-action.1– When p is large,pq1 + p≈ q.In other words, food is abundant and the death rate is only de-pendent on the number of predators.• In (2), we have the standard logistic equation if p is constant:dqdt= sq1 −qp.A lion can only eat s o much! Thus, this equation models the fact t hatthe carrying capacity for the predator is proportional to the number ofprey.2 The Phase Plane• The p null clines are p = 0 and q = (2/3)(1 − p/4)(1 + p).• The q null clines are q = 0 and q = p.• The equilibrium points for p > 0 are(p, q) = (1, 1)(p, q) = (4, 0).3 Stability• At p = 1, q = 1,A =1/12 −1/2s −s.In this case,tr(A) =112− sdet(A) =512s.This point is a stable equilibrium point if s > 1/12 and unstable ifs < 1/12.2• At p = 4, q = 0,A =−2/3 −4/50 s.In this case, det A = −2 s/3 < 0. Therefore, this point is not stable.4 A Repelling Equilibrium Poi ntAn equilibrium point is a repelling equilibrium point if whenever a non-equilibrium solution is close to the equilibrium solution at t, it moves furtheraway as t increases. The equilibrium point p = 1 and q = 1 is repelling ifs < 1/12. If p(t) and q(t) are both near 1, thenpqis almost a solution toddtp − 1q − 1=1/12 −1/2s −sp − 1q − 1=(p − 1 )/12 − (q − 1)/12s(p − 1) − s(q − 1).These solutions grow exponentially with time. An equilibrium point is re-pelling if tr(A) > 0 and det(A) > 0.5 Basin of Attractio nA basin o f attraction or a trapping region is a region V in the (p, q)-planewhere no solutionp(t)q(t)of our predator-system that enters V ever leaves V . We claim that the squareregionV = {(p, q) : 0 < p < 4, 0 < q < 4}is a basin of attraction.36 Poincar´e-Bendixs on The oremConsider the systemdpdt= f(p, q)dqdt= g(p, q),and supp ose that a region V is a basin of at t raction in the (p, q)-plane. If Vcontains a single equilibrium point that is repelling, then t he system has aperiodic solution that is inside V for all t.7 Periodic SolutionsBy the Poincar´e-Bendixson Theorem, our predator-prey system has a peri-odic solution if s < 1/12.8 StabilityOur periodic solution is stable in the following sense. Starting inside the pe-riodic solution, a trajectory will spiral out towards the stable orbit. Startingoutside the periodic solution, a trajectory will spiral in towards the stableorbit.Readings and References• C. Taubes. Modeling Differe ntial Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 23.• “Snowshoe Hare Populations: Squeezed from Below and Above,” pp.382–385• “Impact of Food and Predation on the Snowshoe Hare Cycle,”
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