Transcript REU 2004 - Pennsylvania State University

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
3
__________constants
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!