REU 2004REU 2004

Population Models Day 2Population Models Day 2Predator PreyPredator Prey

REU’04—Day 2REU’04—Day 2

• Today we have 2 species; one predator y(t) (e.g. wolf) and one its prey x(t) (e.g. hare)

Actual DataActual Data

ModelModel

• Want a DE to describe this situation

• dx/dt= ax-bxy = x(a-by) dy/dt=-cx+dxy = y(-c+dx)

• Could get rid of __________constants3

Called Lotka-Volterra Equation, Lotka & Volterra independently studied this post WW I.

• Fixed points: (0,0), (c/d,a/b)

The ANSWER:

Solution vs timeSolution vs time

What are you going to do?What are you going to do?

• Try to use analysis to argue that this is indeed the phase portrait.

OK what now?OK what now?

• 3 species food chain!– x = worms; y= robins; z= eagles

dx/dt = ax-bxydy/dt= -cy+dxy-eyzdz/dt= -fz+gyz

New toolsNew tools

• Invariant sets & trapping regions!