Transcript REU 2004 - Pennsylvania State University

2008 REU
ODE and Population
Models
Intro
• 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
Example:
dx/dt = x(1-x/7)
• Solutions:
• Critique:
– Still non-integer
rabbits
– Still get rabbits with
x(0)=.02
Fixed Points (equilibria)
• In Previous example:
x=0 and x=7 are fixed points
• Fixed Point: dx/dt = 0 (so it’s fixed!)
• Stability: stable – near solutions tend to
fixed point
unstable = not stable
Stability
• Note: near x=7
d/dx ( du/dt) <0
(stable)
Stability
• Note: near x=0
d/dx ( du/dt) > 0
(unstable)
Taylor series at x*
• dx/dt=f(x) (no dependence on t)
• dx/dt = f(x)= c0+c1(x-x*)+c2(xx*)^2+ ….
(c0 = 0)
If c1≠0, we can tell stability.
Moral:
• If
dx/dt = f(x) and f(x*)=0
1) d/dx( f(x)) <0 at x* then x* is
stable.
2) d/dx( f(x) ) >0 at x* then x* is
unstable.
x’ versus x
• For first order autonomous
equations, plotting x’ versus x
encapsulates all this info
x’ positive
(unstable)
x’ negative
(stable)
Reality check
• Find and classify all equilibria of
dx/dt = sin (x(t))
• Firefly example (tomorrow)
Rabbit vs. Deer
http://www.dcnr.state.pa.us/polycomm/pres
srel/presqueislesp1100.htm
• Let 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/Fixed Points: (0,0) , (0,1),
(1,0), (1/4,3/8)
Q: How do we know if these are
stable or unstable?
A: Linear approximation (derivative)
Linear Systems
• dx/dt= Ax
(given by matrix mult)
• Fixed Point(s)?
What’s an eigenvalue again?
• Ax = λx
(λ,x) are eigenvalue eigenvector
pair
• Who cares?
Think about:
x(t) = exp (λt)x
(Handout/Maple)
Other Tools
• Trapping regions
• Poincare Bendixson
• Nullclines
• Series solutions ,etc.
• Invariant Sets
• Bifurcations
Suppose 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=-cy+dxy = y(-c+dx)
• Let’s look at:
dx/dt= x(1-y)
dy/dt=y(-1+x)
Called Lotka-Volterra Equation,
Lotka & Volterra independently
studied this post WW I.
• Fixed points: (0,0),
(c/d,a/b) (in example (1,1)).
Phase portrait
y
(1,1)
x
A typical portrait:
a ln y – b y + c lnx – dx=C
Solution vs time
Critiques
• Nicely captures periodic nature of
data
• Orbits are all bounded, so we do not
need a logistic term to bound x.
• Periodic cycles not seen in nature
Previte’s Population Projects
• 3-species chains • 3 Competing Species
• 4-species chains • Adding a scavenger
2000 REU
2002/3 REU
2004/5 REUs
2005/7 REUs
• (other interactions possible!)
3-species model (REU 2000)
3 species food chain!
 x = worms; y= robins; z= eagles
dx/dt = ax-bxy
dy/dt= -cy+dxy-eyz
dz/dt= -fz+gyz
=x(a-by)
=y(-c+dx-ez)
=z(-f+gy)
Critical analysis of 3-species chain
ag > bf
ag < bf
ag = bf
→ unbounded orbits
→ species z goes extinct
→ periodicity
ag ≠ bf
ag=bf
2000 REU and paper
Tools used in analysis
Linearization
Trapping regions
Invariant sets
Liapunov functions (“energy” functions)
One open conjecture
ag>bf
y tends to a limit as time increases
all numerical evidence shows this, but
no analytic proof.
4-species model
dw/dt = aw-bxw
dx/dt= -cx+dwx-exy
dy/dt= -fy+gxy - hyz
dz/dt= -iz+jyz
=w(a-bx)
=x(-c+dw-ey)
=y(-f+gx-hz)
=z(-i+jy)
2004/5 REU did analysis
Orbits bounded again as in n=2
Quasi periodicity (next slide)
ag<bf gives death to top 2
ag=bf gives death to top species
ag>bf gives quasi-periodicity
Quasi-periodicity
Previte’s doughnut conjecture (ag>bf)
This is wide open
Project never finished
Proof seems too hard, may involve deep
topics such as
KAM theory, Hamiltonian systems
Simple Scavenger Model
lynx
beetle
hare
Semi-Simple scavenger– Ben Nolting 2005
x'  x  xy
y'  cy  xy
z '  ez  fxyz  gxz  hyz  z
2
Know (x,y) -> (c, 1-bc) use this to see
fc+gc+h=e every solution is periodic
fc+gc+h<e implies z goes extinct
fc+gc+h>e implies z to a periodic on the cylinder
Ben Nolting and his poster in San
Antonio, TX
Scavenger Model with feedback
(Malorie Winters & James Greene
2006/7)
x'  ax  bx  xy  xz
2
y '  cy  dxy
z '  ez  fxyz  gyz  hxz   z
2
Biological Example (crowding prey)
Rainbow Trout (predator)
crayfish
Scavenger of trout
carcasses
Mayfly nymph (Prey)
Predator of mayfly
nymph
Crayfish are scavenger & predator
Analysis (Malorie Winters)
Regions of periodic behavior and
Hopf bifurcations and stable coexistence.
Regions with multi stability and dependence
on initial conditions
Malorie Winters, and in New Orleans, LA
REU 2007
James Greene finds a model that exhibits chaos
2007 scavenger system
dx/dt=x(1-bx-y-z)
dy/dt=y(-c+x)
dz/dt=z(-e+fx+gy-βz)
b, c, e, f, g, β > 0
Period Doubling cascade and attractor
TO DO
Finish up the analysis from 2007
Including Hopf Bifurcation analysis,
boundedness of orbits, and compare onset
of chaos with other models
Crowd the predator