Martin Boer prepared this three-dimensional state space plot with Povray. The three
axes show the prey (x_{1}), predator
(x_{2}), and top-predator (x_{3}) population sizes in a chemostat, for a
particular choice of throughput rate and concentration of substrate in
the feed. Substrate (x_{0}), on which the
prey lives, is not shown.

The red worm is the chaotic attractor of the food chain, which is
wrapped around the unstable equilibrium (yellow dot). The yellow dots
in the plane x_{3} = 0, represent
unstable equilibria of the reduced prey-predator system (without
top-predator); the red curve in that plane represents a stable limit
cycle. The blue blanket is the separatrix; trajectories that start
inside the separatrix converge to the chaotic attractor, the others
converge to the limit cycle (of the reduced system). The blanket is
cut, to reveal the contents, but in reality it is cut by a plane that
is defined through the mass balance. The green curve lying on the
separatrix is a limit cycle of the saddle type. All orbits starting on
the separatrix are asymptotic to the saddle cycle.

More detailed information:

- Description of the food chain model
- Heteroclinic orbit between an equilibrium and a saddle cycle
- Homoclinic orbit of a saddle cycle

Martin presented a full analysis of tri-trophic chains in his thesis:

Boer, M. P. 2000. *The dynamics of tritrophic food chains.*
Vrije Universiteit, Amsterdam.

Yes, Martin likes to play chess ...

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