Spatial Lotka-Volterra Systems

Joe WildenbergDepartment of Physics, University of Wisconsin

Madison, Wisconsin 53706 USA

Lotka-Volterra Equation

N

jjijiii xaxrx

1

)1( Nonlinear ri are growth rates – set to 1 (Coste et. al)

aij are interactions Widely used – chemistry, biology,

economics, etc.

Interaction Matrix

NNNN

N

N

ij

aaa

aaa

aaa

a

21

22221

11211

)(

Rows effect species i Columns show how species i effects

others Not necessarily symmetric!

jiij aa

Spatial Dependence

Structure of equations contains no spatial dependence

Why include?

Real-world systems have it!

Spatial Interaction Matrices

1000

1000

0100

0010

0001

0001

)(

6561

5651

4543

3432

2321

1612

aa

aa

aa

aa

aa

aa

aij

Rows are permutations of each other

All species are identical

Circulant Matrices

1000

1000

0100

0010

0001

0001

)(

11

11

11

11

11

11

aa

aa

aa

aa

aa

aa

aij

1,1 jiij aa

Case “Z”

00110100

Ring Mathematically simple

i i+1i-2

)1( 12 iiiii xxxxx

Z’s Eigenvalues

Z’s Eigenvalues

Case “Y”

Goals Ring Interactions decrease with distance

00100 2112 iiii aaaa

i

Bees can only fly so far from their hive

Interactions with other bees depends on distance

Can be influenced by far hives if their neighbors are affected

Buzz

Case “Y”

00237.0852.01505.0451.000

Goals Ring Interactions decrease with distance Chaotic Populations above 10-6 (Ovaskainen and Hanski, 2003)

i 00100 2112 iiii aaaa

01.011.0 ND fit

Lyapunov Functions An “energy”

function Always positive Equilibrium point

has value of zero Value decreases

along all orbits If one exists, no

periodicity or chaos is possible

Ring Lotka-Volterra Lyapunov Function

Requirements (Zeeman, 1997)

Circulant interaction matrix (all species identical)

Real part of the eigenvalues positive

NN

jj

N

ii

x

xxV

1

1)(

Ring Lotka-Volterra Lyapunov Function (cont.)

Eigenvalues:

Lyapunov function exists if:

N

j

jkjk c

1

)1)(1( N

i

e

2

0)Re( k2

0N

k

Case “Z” revisited

c1 = 1, c2 = b = 1, cN-1 = a = 1 all others zero

N

kNi

N

ki

k aebe)2(22

1

)4

cos()2

cos(1)Re(N

ka

N

kbk

00110100 i i+1i-2

0100011

Case “Z” revisited (cont.)

Largest LE Lyapunov Function

Case “Y” revisited

.

505.0451.000237.0852.01

N

ki

N

ki

N

kNi

N

kNi

k eeee

42)1(2)2(2

237.0852.0505.0451.01

)2

cos()()4

cos()688.0(1)Re( 11 N

kaa

N

kk

N

ki

N

ki

N

Nki

N

kNi

eeee 442)2(2

Case “Y” revisited (cont.)

Largest LE Lyapunov Function

Line Systems

Not restricted by Lyapunov function Most likely others

Real-world examples exist Many ways to create boundary

conditions

Boundary Conditions

Simply sever ring (remove entries in lower left and upper right of A)

Hold ends fixed “Mirror” – strengthen connections

on opposite side

1000

1000

0100

0010

0001

0001

)(

6561

5651

4543

3432

2321

1612

aa

aa

aa

aa

aa

aa

aij

10000

1000

0100

0010

0001

00001

)(

65

5651

4543

3432

2321

12

a

aa

aa

aa

aa

a

aij

2 1

2

“Mirror”

1

111

1 … …

Mirror Y

Similar spatio-temporal patterns More restrictive parameter space

Mirror Y (cont.)

Line Ring

Line Eigenvalues

Line Eigenvalues (cont.)

Line Eigenvalues (cont.)

Future Work

Understand eigenvalues of line systems

Determine Lyapunov function(s) Apply results to real-world systems

Thank You!