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Predator-Prey Models Pedro Ribeiro de Andrade Gilberto Câmara Acknowledgments and thanks Many thanks to the following professors for making slides available on the internet that were reused by us Abdessamad Tridane (ASU) Gleen Ledder (Univ of Nebraska) Roger Day (Illinois State University) “nature red in tooth and claw” One species uses another as a food resource: lynx and hare. The Hudson’s Bay Company hare and lynx populations (Canada) Note regular periodicity, and lag by lynx population peaks just after hare peaks 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. Predator-prey systems Two species encounter each other at a rate that is proportional to both populations Predator-prey cycles normal prey population prey population increases prey population increases predator population decreases as less food predator population increases as more food prey population decreases because of more predators Generic Model • f(x) prey growth term • g(y) predator mortality term • h(x,y) predation term • e - prey into predator biomass conversion coefficient Lotka-Volterra Model r - prey growth rate : Malthus law m - predator mortality rate : natural mortality a and b predation coefficients : b=ea e prey into predator biomass conversion coefficient Predator-prey population fluctuations in Lotka-Volterra model 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 = 0.00002 The time t is measured in months. There are 40 wolfes and 1000 rabbits Phase plane Variation of one species in relation to the other Phase trajectories: solution curve A phase trajectory is a path traced out by solutions (R, W) as time goes by. Equilibrium point The point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80. Hare-lynx data Hare-lynx data