The theme of this work is to translate the quadratic and monotone structures of the model into global geometric tools for predicting species coexistence, oscillations or extinction. Much of the work was inspired by visualization of the carrying simplex and dynamics in three dimensions. See the visualizations below which are joint with Michael Drew LaMar.
Zeeman's 1993 paper classifies the dynamics of 3-D Lotka Volterra systems, modulo periodic orbits, using a geometric analysis of the nullclines of the system. Zeeman and LaMar have written software to visualise Carrying Simplices and their dynamics, using the Geometry Center's visualisation package Geomview.
The following tour through parameter space shows 21 example systems, one in each row. First the nullcline configuration of the system; then a face view of the carrying simplex with representative orbits; and finally a side view of the carrying simplex showing it's curvature. The tour includes birurcations at the boundary of the carrying simplex, and at the interior fixed point.
The coloring system for the axes is: red, green and blue (RGB) for x, y and z respectively. the coloring system for the fixed points (represented by colored dots) is red, green, blue and white for repelling, saddle, attracting and neutral fixed points respectively. Note that the fixed points are colored to represent their dynamics restricted to the carrying simplex.
Click on the images for an enlarged view and a caption. In the viewer mode, click on the image or use the N and P keys to view the Next and Previous photos.
1. b1=12 a11=12 a12=12 a13=12
b2=12 a21= 6 a22= 6 a23= 6
b3=12 a31= 3 a32= 3 a33= 3
2. Perturb a13 to 6
3. Perturb a13 to 4
1. Perturb a13 to 3
2. Perturb a13 to 2
3. Perturb a13 to 1
So we're at: 12 12 12 1
12 6 6 6
12 3 3 3
1. Perturb a31 to 9
2. Perturb a31 to 10
3. Perturb a31 to 11
1. Perturb a31 to 12
2. Perturb a31 to 13
3. Perturb a31 to 14
1. Perturb a31 to 15
2. Perturb a31 to 18
3. Perturb a31 to 24
OK, now we'll make a Hopf Bifurcation happen, giving rise to an
attracting periodic orbit, with a repelling fixed point inside.
Start from Page 4, number2: 12 12 12 1
12 6 6 6
12 13 3 3
1. Multiply the 1st row by .5
2. Multiply the 1st row by 1/3 (ie: 4 4 4 1/3)
3. Multiply the 1st row by 1/10 (ie: 1.2 1.2 1.2 .1)
Another Hopf Bifurcation. This time giving rise to a repelling
periodic orbit, with an attracting fixed point inside.
1. Start from: 12 12 12 1
12 6 6 6
12 16 3 3
2. Now Multiply the 2nd row by 1/2 (1e: 6 3 3 3)
3. Multiply 2nd row by 1/3 (ie: 4 2 2 2)
Next we exhibit an example of E.C. Zeeman, which has two isolated periodic orbits. The inside one attracting, and the outside one repelling.
These are the params for E.C. Zeeman's 2 periodic orbits example.
There is a neutral (weakly attracting) internal fixed point for the system
a11=5.3 a12=10.3 a13=1.73
a21=4.5 a22=7.5 a23=11.07
a31=10 a32=2 a33=7
With b's being the sums of the a's.
i.e. b1=17.33, b2=23.07 and b3=19
The boundary of this system also attracts, and hence there is a repelling
periodic orbit in between.
Now we perturb along the diagonal, subtracting .001 from each diagonal
entry, and from each b, to keep the b's as the sum of the a's. The
internal fixed point becomes repelling, giving rise to a small attracting
orbit through a Hopf Bifurcation. The boundary continues to attract,
throughout the Hopf bifurcation, so we now have two periodic orbits.
One repelling and one attracting.
The attracting fixed point passes through (approx) 0.117, 0.602, 0.289
The repelling fixed point passes through (approx) 0.694 0.054 0.237