REU 2004 Population Models Day 1 Competing Species

REU’04—Day 1 Often know how populations change over time (e.g. birth rates, predation, etc.), as opposed to knowing a ‘population function’ Differential Equations! Knowing how population evolves over time w/ initial population  population function

Example – Hypothetical rabbit colony lives in a field, no predators. Let x(t) be population at time t; Want to write equation for dx/dt Q: What is the biggest factor that affects dx/dt? A: x(t) itself! more bunnies  more baby bunnies

1 st Model—exponential, Malthusian Solution: x(t)=x(0)exp(at)

Critique Unbounded growth Non integer number of rabbits Unbounded growth even w/ 1 rabbit! Let’s fix the unbounded growth issue dx/dt = ????

Logistic Model dx/dt = ax(1-x/K) K-carrying capacity we can change variables (time) to get dx/dT = x(1-x/K) Can actually solve this DE

Solutions: Critique: –Still non-integer rabbits –Still get rabbits with x(0)=.02

Suppose 2 species x(t) rabbits and y(t) deer compete for the same food source. dx/dt = dy/dt = Ax(1-x/K) By(1-y/W) -Cxy -Dxy Or…. (after changes of coordinates…) dx/dt = x(1-x-ay) dy/dt = y(b-by-cx)

Analysis of one case dx/dt = x(1-x-2y) dy/dt = y(2-2y-5x) Equlibria: (0,0), (0,1), (1,0), (1/4,3/8) Jacobian:

Jacobians J(0,0) = J(1,0) = J(0,1) = J(1/4,3/8)= Evals 1, 2 so unstable node!—evect : [1,0], [0,1] Evals both negative– stable node evect: [1,0], [1,1] Evals both negative– stable node evects: [0,1],[1,-5] Evals, -3/2,1/2 –saddle evects: uns [1,-1.5], stab [1,2.5]

Nullcline Analysis

XPP Phase portrait

3 competing species A first step->May-Leonard model dx/dt= x(1-x-ay-bz) dy/dt= y(1-bx-y-az) dz/dt= z(1-ax-by-z) for all parameter values, characterize the behavior of all solutions.

What is the goal? Goal is to complete a phase portrait for all parameter values In higher dimension, phase portraits won’t work, so we want to describe be able to describe the fate of a solution with a given initial condition, (e.g. goes to a fixed point, goes to a periodic, etc.)

Some tools u.edu/~jpp/talk /hirsch.pdfhttp://vortex.bd.ps u.edu/~jpp/talk /hirsch.pdf Stable/Center manifold Theorem

Project 1 Describe the phase portrait for all values of the general 3 competing species model and possibly 4 competing species More tricks to come! Stokes’ Theorem!, Invariant sets!