Transcript Scavenger on Predator-Prey

Hopf Bifurcations on a
Scavenger/Predator/Prey
System
Malorie Winters
Dr. Joseph Previte
Lotka-Volterra
• dx/dt = x(1 – bx – y)
• dy/dt = y(-c + x)
• Everything spirals in to
(c, 1 – bc)
• Introduce scavenger
onto this system
The Model
• dx/dt = x(1 – bx – y – z)
• dy/dt = y(-c + x)
• dz/dt = z(-e + fx + gy + hxy – βz)
• Lotka-Volterra with logistic on x
• Scavenger z with linear and quadratic
death
z
z
z
resource limitations
Owl
Crow
Rabbit
Fixed points & Stability
1. (0, 0, 0) – saddle point
2. (1/b, 0, 0) – saddle/stable point
 eβ
f  be 
 - stable/unstable point
, 0,
3. 
f  bβ 
 f  bβ
4. c, 1  bc, 0 - stable/unstable point
- when stable, scavenger becomes extinct
Interior Fixed Point
 e  fc  β  bcβ g  hc  e  fc  bcg  bc 2 h 
 c,

,
g  hc  β
g  hc  β


• Linearization of system at fixed point
• Eigenvalues of Jacobian are messy
• Characteristic polynomial of Jacobian
A 3  B 2  C  D
– Routh Hurwitz Analysis
Routh Hurwitz Method
• Categorizes real parts of roots of a polynomial
n
n1
an  an1  ...  a1  a0
• Array
• To finish array
• Count number of sign changes in first column
– Gives number of roots with positive real part
Example
4s  2s  s  3  0
3
Row 1
Row 2
4
2

?

?
1

3
?

?
2
4 1 4 1
2 3 2 3

21  3  4  
 5 0  5 35  00 2
2


 
?   35 0 
?
2 sign changes means 2 roots with pos. real part
Example
4s  2s  s  3  0
3
2
RH array predicted 2 roots with positive real
part.
Actual solutions:
s  1
1
11
s 
4
4
1
11
s 
4
4
Results
• Routh Hurwitz analysis on characteristic
polynomial A 3  B 2  C  D
– For our system A = 1
• CB – D > 0, stable
– no sign changes, all eigenvalues neg. real part
• CB – D < 0, unstable
– two sign changes, two eigenvalues pos. real part
• CB – D = 0, special case
CB – D
2
2
2 2
2
2
2
c ( h c b  b c  e h cf c gb e  g b  g b  f c he f  e f gh  gf g  e h  e h ge h c
2
2
2
2
2
2
2 2
2
2 2
h c  f c  f c ge  e ge hf g g  b c f  hb c g  hf c g b he g b c he h c b
2
2 2 2
2 2 2
2
2
2
f c  h c   g b c  f c g bh c b  h c b  b c g  b c g  b c f  f c h ge f c h
2 2 2
2
2
b e  g2 f c h  b c h  ) ( g b cg  g b c  b c  h c be  h c b  f c  h c  )
3
2
2 2
2
2
3
( gh c ) c ( g  f c ge gg e b c2 g b c  g f c bg b c  2 e f cf c he b c hf c b h
2 3
2
2
2 2 2
b c  h2 h c b  e h ce b c  h c  f c  f c b  f c e e  )( gh c )
Hopf Bifurcations
• Suppose you have a family of systems of ODEs
parameterized by s and  xs , ys , zs  is a fixed point,
then a Hopf bifurcation occurs when a pair of
complex conjugate eigenvalues of the Jacobian
cross the imaginary axis non-tangentially and not at
zero.
• Movie
• Hopf Bifurcation Theorem
– Guarantees a limit cycle
Idea
If there is a path in parameter space
  s   b  s  , c  s  , e  s  , f  s  , g  s  , h  s  ,   s 
and
C  s  B  s   D  s   0 at s*
with
d
C  s  B  s   D  s   s  0
*
ds
then the interior fixed point has a Hopf
bifurcation at s* .
Example
Let
2
 2
91897
s6
s
 10 
3
2
  s    s  0.01,sin s  2, 4s 
, cos s 
,  s  1 , 4s  s  1, 3

11243
3
s

1


g s
f s
hs
b s
e s
 s
c  s
Then at s1  0.05151 and s2  0.00018
C  s B  s  D s  0
and
d
C  s  B  s   D  s   s , s  0.
1 2
ds
Results
• Bifurcation diagram
0.00018
-0.05151
Region 1: Before Bifurcation
Trajectories
begin red
and travel in
toward purple
(stable node).
All species
survive.
Stable Node
Region 2: Stable limit cycle
Trajectories
begin red and
travel toward
purple (stable
limit cycle).
Oscillatory
behavior in all
species.
Unstable Node
Stable Limit Cycle
Region 3: Multiple limit cycles
Unstable Limit Cycle
It depends on the
initial conditions
whether oscillations
occur or not.
Stable Limit Cycle
Stable Center
Region 4: After Limit Cycles
Trajectories
begin red
and travel in
toward blue
(stable node).
All species
survive.
Stable Node
Biology
• Eventual behavior dependent upon
parameters and initial conditions for some
of these systems
• Biologically viable and interesting?
• Does it exist in nature?
Notes/Questions
• Bifurcations possible by varying all
parameters except b
• Do unbounded orbits exist?
Acknowledgements
Thanks to
• Dr. Joseph Previte
• Behrend REU 2006
• NSF Award 0552148