Presentation on theme: "Pedro Ribeiro de Andrade Gilberto Câmara"— Presentation transcript:
1Pedro Ribeiro de Andrade Gilberto Câmara Predator-Prey ModelsPedro Ribeiro de AndradeGilberto Câmara
2Acknowledgments 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)
6hare and lynx populations (Canada) Note regular periodicity, and lag by lynx population peaks just after hare peaks
7Predator-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.
8Predator-prey systems Two species encounter each other at a rate that is proportional to both populations
9predator 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
10Generic Model f(x) prey growth term g(y) predator mortality term h(x,y) predation terme - prey into predator biomass conversion coefficientMTBI summer 2008
11Lotka-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
12Predator-prey population fluctuations in Lotka-Volterra model
13Predator-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
14Phase planeVariation of one species in relation to the other
15Phase trajectories: solution curve A phase trajectory is a path traced out by solutions (R, W) as time goes by.
16Equilibrium pointThe point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80.