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Analyzing the Lotka-Volterra Predator-Prey Model: Insights and Applications

This study delves into the dynamics of predator-prey interactions using the Lotka-Volterra equations. We examine two species: a predator (e.g., wolves) and its prey (e.g., hares). The mathematical model describes population changes over time and presents fixed points critical for understanding population stability. The project encompasses analysis techniques to visualize phase portraits and their implications in ecological systems. Additionally, we will explore multi-species interactions within food chains, including worms, robins, and eagles, enhancing our grasp of ecological balance.

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Analyzing the Lotka-Volterra Predator-Prey Model: Insights and Applications

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  1. REU 2004 Population Models Day 2Predator Prey

  2. 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)

  3. Actual Data

  4. Model • Want a DE to describe this situation • dx/dt= ax-bxy = x(a-by) dy/dt=-cx+dxy = y(-c+dx) 3 • Could get rid of __________constants

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

  6. The ANSWER:

  7. Solution vs time

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

  9. OK what now? • 3 species food chain! • x = worms; y= robins; z= eagles dx/dt = ax-bxydy/dt= -cy+dxy-eyzdz/dt= -fz+gyz

  10. New tools • Invariant sets & trapping regions!

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