Transcript REU 2004 - Pennsylvania State University

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
1st Model—exponential,
Malthusian
dx
 ax
dt
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 = Ax(1-x/K) -Cxy
dy/dt = By(1-y/W) -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)
 2x 
1  2 x  2 y
Jacobian: 


5
y
2

4
y

5
x


Jacobians
•
•
•
•
1 0 
J(0,0) = 
Evals 1, 2 so unstable

node!—evect : [1,0], [0,1]
0 2 
J(1,0) =  1  2 Evals both negative– stable node
 0  3 evect: [1,0], [1,1]


 1 0 
both negative– stable node
J(0,1) =  5  2 Evals

 evects: [0,1],[1,-5]
  1 / 4  1 / 2  Evals , -3/2,1/2 –saddle
J(1/4,3/8)= 
evects: uns [1,-1.5],

 15 / 8  3 / 4 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
• http://vortex.bd.ps
u.edu/~jpp/talk102-2003/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!