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Predator-Prey Models

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)

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Predator-Prey Models

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  1. Predator-Prey Models Pedro Ribeiro de Andrade Gilberto Câmara

  2. 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)

  3. “nature red in tooth and claw”

  4. One species uses another as a food resource: lynx and hare.

  5. The Hudson’s Bay Company

  6. hare and lynx populations (Canada) Note regular periodicity, and lag by lynx population peaks just after hare peaks

  7. 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

  8. Two species encounter each other at a rate that is proportional to both populations Predator-prey systems

  9. 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 Predator-prey cycles normal prey population

  10. Generic Model • f(x) prey growth term • g(y) predator mortality term • h(x,y) predation term • e - prey into predator biomass conversion coefficient

  11. 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

  12. Predator-prey population fluctuations in Lotka-Volterra model

  13. 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 Predator-prey systems

  14. Variation of one species in relation to the other Phase plane

  15. A phase trajectory is a path traced out by solutions (R, W) as time goes by. Phase trajectories: solution curve

  16. The point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80. Equilibrium point

  17. Hare-lynx data

  18. Hare-lynx data

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