PredatorPrey Models Pedro Ribeiro de Andrade Gilberto Cmara

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 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