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!