Transcript REU 2004 - Pennsylvania State University

REU 2005
Predator Prey Population
Models
REU ’05
• 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
Predator Prey
• Today we have 2 species; one predator
y(t) (e.g. wolf) and one its prey x(t) (e.g.
hare)
Actual Data
Lotka – Volterra 2- species
model
• Want DE to model situation
(1920’s A.Lotka & V.Volterra)
• dx/dt = ax-bxy
dy/dt = -cx+dxy
a → growth rate for x
c → death rate for y
b → inhibition of x in presence of y
d → benefit to y in presence of x
Model Simplification
• Could get rid of
how?
•
3
__________constants—
dx/dt= ax-bxy = x(a-by)
dy/dt=-cx+dxy = y(-c+dx)
Called Lotka-Volterra Equations,
Lotka & Volterra independently
studied this post WW I.
• Fixed points: (0,0),
(c/d,a/b)
Solution vs time
The Phase Portrait (x vs. y)
No logistic term
• Note in 1 species model, all orbits
grow unbounded.
• In 2-species model, all orbits (most
anyway) stay bounded.
• What happens in 3 or 4 species?!
What are you going to do?
• Try to use analysis to argue that this
is indeed the phase portrait.
Poincare’-Bendixson**
•
dx/dt= f(x,y)
dy/dt= g(x,y); f & g nice
If (x(t),y(t)) is bounded then as t increases, (x(t),y(t))
must approach:
1. a fixed point
2. a periodic orbit
3. a cycle of one or more fixed point(s) connected by
hetero/homoclincs.
** only works in the plane!
OK what now?
• 3 species food chain!
– x = worms; y= robins;
z= eagles
dx/dt = ax-bxy
dy/dt= -cy+dxy-eyz
dz/dt= -fz+gyz
Analysis – 2000 REU Penn
State Erie
• For ag=bf ; get invariant surfaces, numerical
solutions are periodic.
Invariant Surfaces
{
(1)
dx/dt= f(x,y,z)
dy/dt= g(x,y,z)
dz/dt= h(x,y,z); f,g, and h nice
If S is a smooth surface with normal
vector n at (x,y,z) and
dot (n,<f,g,h>)=0 at all (x,y,z) then
the surface S is invariant w.r.t. (1).
3d time plot for ag=bf
Linearization of 3d system
(informal stable manifold Thm)
If at fixed point P, Jac(P) has:
3 + real part evals – unstable
3 - real part evals – stable
a mix– a generalized saddle
0 real parts–
(pure imaginary)
no good
Cases ag ≠ bf
Trapping Surfaces
{
(1)
dx/dt= f(x,y,z)
dy/dt= g(x,y,z)
dz/dt= h(x,y,z); f,g, and h nice
If S is a smooth surface with normal
vector n at (x,y,z) ‘upward’
dot (n,<f,g,h>)>0 at all (x,y,z) then
all trajectories of (1) pass through S
upward.
ag<bf
ag>bf
3 species Open Question
(research opportunity)
• When ag > bf
what is the behavior of y as t →∞?
Critical analysis
• ag > bf
• ag < bf
• ag = bf
→ unbounded orbits
→ species z goes extinct
→ periodicity
• Highly unrealistic model!! (vs. 2species)
• Result: A nice pedagogical tool
• Adding a top predator causes
possible unbounded behavior!!!!
What are you going to do?
• Verify many of the statements made
using:
1. trapping regions
2. invariant sets
3. linearization
4-species model
dw/dt = aw-bxw
=w(a-bx)
dx/dt= -cx+dwx-exy =x(-c+dw-ey)
dy/dt= -fy+gxy - hyz =y(-f+gx-hz)
dz/dt= -iz+jyz
=z(-i+jy)
2004 REU analysis
• Orbits bounded again as in n=2
• Quasi periodicity
• ag<bf gives death to top 2
• ag=bf gives death to top species
• ag>bf gives quasi-periodicity
In case ag > bf; invariant
surface [Volterra]
K = x- x0 ln(x)
+b/d y – y0 ln(y)
+ be/dg z – be/dg z0 ln(z)
+ beh/dgj w – e/j w0 ln (w)
These are closed surfaces so long as
ag >bf:
Moral: NO unbounded orbits!!
For ag > bf: this should be
verifiable!
• Someone give me a 4-species
historical population time series!,
“RESEARCH PROJECT”!
(Try to fit such data to our “surface”.
ag=bf
• 4th species goes
extinct!
• Limits to 3-species
ag=bf case
ag< bf death to y and z—
back to 2d
Equilibria
• (0,0,0,0)
• (c/d,a/b,0,0)
• ((cj+ei)/dj,a/b,i/j,(ag-bf)/hb)
J(0,0,0,0): 3 -, 1 + eigenvalues (saddle)
J(c/d,a/b,0,0): 2 pure im; 1 -, 1 ~ ag-bf
J((cj+ei)/dj,a/b,i/j,(ag-bf)/hb)
4 pure im!
• Each pair of pure imaginary evals
corresponds to a rotation: so we
have 2 independent rotations θ and
φ
θ
φ
A torus is S^1 x S^1
(ag>bf)
Quasi-periodicity
Grand finale: Even vs odd
disparity
• Hairston Smith Slobodkin in 1960
(biologists) hypothesize that
(HSS-conjecture)
• Even level food chains (world is brown)
(top- down) (n=2, n=4 orbits bounded)
• Odd level food chains (world is green)
(bottom –up) (n=1, 3 orbits
unbounded)
• Taught in ecology courses.
Project #1
• Prove the asymptotic behavior of y in
the 3-species model.
Project #2
• Obtain 4 species data and fit the
data to our model.
Project #3
• Prove that in the n-species cases (n
even) that the Jacobian about P
always has imaginary eigenvalues.
(n=4 is done)
• Examine the odd cases where
unbounded orbits exist, completely
characterize these cases.
Project #4
• In ag>bf n=4, it appears that all orbits
stay on a torus (the linearization is not
enough)
(a) amass numerical evidence to verify
(b) prove it (!?)
This may involve series solutions, trying to
find an invariant set for special
parameters, or some very sophisticated
math (KAM theory)