Download REU 2004 - Pennsylvania State University

2007 Math
Biology
Seminar
ODE 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
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
Generalizations of L.V.
• 3-species chains • 4-species chains • Adding a scavenger
2000 REU
2004/5 REUs
2005/6 REUs
• (other interactions possible!)
3-species model
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
Highly unrealistic model!! (vs. 2-species)
Adding a top predator causes possible
unbounded behavior!!!!
ag ≠ bf
ag=bf
2000 REU and paper
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 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
Even vs odd disparity
Hairston Smith Slobodkin in 1960
(biologists) hypothesize that
(HSS-conjecture)
Even level food chains (world is brown)
(top- down)
Odd level food chains (world is green)
(bottom –up)
Taught in ecology courses.
Quasi-periodicity
Previte’s doughnut conjecture (ag>bf)
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
Dynamics trapped on cylinders
Several trajectories
Ben Nolting and his poster in San
Antonio, TX
Scavenger Model with feedback
(Malorie Winters 2006/7)
x'  ax  bx  xy  xz
y '  cy  dxy
2
z '  ez  fxyz  gyz  hxz  z
2
Scavenger Model w/ scavenger prey crowding
owl
opossum
hare
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
Lots more to do!!
Competing species
Different crowding
Previte’s doughnut
How do I learn the necessary tools?
Advanced ODE techniques/modeling
course
Work independently with someone
Graduate school
REU?
R.E.U.?
Research Experience for Undergraduates
Usually a summer
100’s of them in science (ours is in math
biology)
All expenses paid plus stipend $$$!
Competitive
Good for resume
Experience doing research