REU 2004 Population Models Day 2 Predator Prey

REU’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 Data

Model 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 __________constants 3

Called Lotka-Volterra Equation, Lotka & Volterra independently studied this post WW I. Fixed points: (0,0), (c/d,a/b)

The ANSWER:

Solution vs time

What are you going to do? Try to use analysis to argue that this is indeed the phase portrait.

OK what now? 3 species food chain! – x = worms; y= robins; z= eagles dx/dt = ax-bxy dy/dt= -cy+dxy-eyz dz/dt= -fz+gyz

New tools Invariant sets & trapping regions!