Presentation on theme: "Pedro Ribeiro de Andrade Gilberto Câmara"— Presentation transcript:
1 Pedro Ribeiro de Andrade Gilberto Câmara Predator-Prey ModelsPedro Ribeiro de AndradeGilberto Câmara
2 Acknowledgments and thanks Many thanks to the following professors for making slides available on the internet that were reused by usAbdessamad Tridane (ASU)Gleen Ledder (Univ of Nebraska)Roger Day (Illinois State University)
6 hare and lynx populations (Canada) Note regular periodicity, and lag by lynx population peaks just after hare peaks
7 Predator-prey systems The principal cause of death among the prey is being eaten by a predator.The birth and survival rates of the predators depend on their available food supply—namely, the prey.
8 Predator-prey systems Two species encounter each other at a rate that is proportional to both populations
9 predator population decreases predator population increases Predator-prey cyclesnormal prey populationprey populationincreasesprey populationincreasespredator population decreasesas less foodpredator population increasesas more foodprey population decreasesbecause of more predators
10 Generic Model f(x) prey growth term g(y) predator mortality term h(x,y) predation terme - prey into predator biomass conversion coefficientMTBI summer 2008
11 Lotka-Volterra Model r - prey growth rate : Malthus law m - predator mortality rate : natural mortalitya and b predation coefficients : b=eae prey into predator biomass conversion coefficientMTBI summer 2008
12 Predator-prey population fluctuations in Lotka-Volterra model
13 Predator-prey systems Suppose that populations of rabbits and wolves are described by the Lotka-Volterra equations with: k = 0.08, a = 0.001, r = 0.02, b =The time t is measured in months.There are 40 wolfes and 1000 rabbits
14 Phase planeVariation of one species in relation to the other
15 Phase trajectories: solution curve A phase trajectory is a path traced out by solutions (R, W) as time goes by.
16 Equilibrium pointThe point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80.