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Predator-Prey Models Pedro Ribeiro de Andrade Gilberto Câmara.

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Presentation on theme: "Predator-Prey Models Pedro Ribeiro de Andrade Gilberto Câmara."— Presentation transcript:

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 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 normal prey population prey population increases prey population increases predator population increases as more food predator population decreases as less food prey population decreases because of more predators Predator-prey cycles

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 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

14 Phase plane Variation 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 point The point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80.

17 Hare-lynx data

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