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This is a model to study cyclic catalytic support between species, as described as a "hypercycle" by Eigen and Schuster in 1978. In 1991 we developed a cellular automaton model to study the spatial pattern formation of hypercycles (see Boerlijst & Hogeweg, 1991). It turns out that hypercycles consisting of 5 or more members create so-called spiral waves, where each species sits directly behind its catalytic supporter. This coincides with the non-spatial system displaying limit cycle behaviour. It turns out that the spatial pattern formation into spiral waves provides stability to a so-called “parasitic” species, that is a species that receives catalysis but does not return this favour to any of the species in the hypercycle.


The model can be started with "random start", which seeds the field with 90% coverage with equal numbers of each species (except for the parasite). The number of species in the hypercycle can be varied between 3 and 8. A parasitic species can be manually added by mouse clicking in the field. The parasite gets catalysis from species 1 (the red species). Firthermore, “diffusion” can be added.


In the model a variable number of species can be simulated, who form one large cycle of competitive dominance, where 1 > 2 > 3 > ... > n > 1. During each tick, all patches are updated. During an update patches that are occupied with an individual can become empty with 1% probability. An empty patch chooses a random neighbour (out of the 4 direct neighbours), and if this neighbour patch contains an individual it can create an offspring with 5% probability. However, if there is a catalyst of the species in the 8-cell neighbourhood, the probability of growth is increased tenfold. The parasitic species (black) receives catalysis from species 1 (red) and this catalysis is 50% increased compared to the catalysis orange gets from red.
Diffusion is simulated by swapping the contents of adjacent patches. The diffusion rate indicates which fraction of the patches is altered after each tick of the model. These patches are randomly selected, and a patch can be altered multiple times within one tick of the model. If the diffusion rate is set to "mean-field" the field is completely randomized after each tick, effectively simulating the non spatial dynamics of the model.


Notice that the cyclic dominance is reflected in a spatial pattern formation that slowly is established, and where each species sits directly behind the species from which it gets catalysis. The pattern is dynamic and it forms so-called "spiral waves", which tend to organize in pairs that rotate clockwise and anti-clockwise.


Start a hypercycle with 8 species and let the system run until a few large spiral waves are established. You can speed up this process by switching “view updates” off, which will cause the model to run at maximum speed. After some spiral waves have established (around tick 20,000) you can try a local invasion of some parasites. The invasion is most effective if you start in a region where the red species dominates. Try first what happens if you start the invasion well outside the centre of the spirals. Also investigate what happens if the parasite invasion is in or near the centre of a single spiral.

Now investigate what happens if you add some diffusion (random local movement of individuals) to the model. How does this alter the patterns? Does this alter the stability against the parasite?

You could further adapt to model to add a variable that controls the amount of catalysis that the parasite gets from species 1 (this is indicated in the code). You could also implement more complex interactions between the species, e.g. multiple hypercycle paths in the species interactions (for instance a 3-cycle competing with an overlapping 4-cycle).


Boerlijst, M.C.; Hogeweg, P. , Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites. (1991) Physica D, 48: 17-28.

Eigen, M., Schuster, P. The Hypercycle. Naturwissenschaften 65, 7–41 (1978).

Please cite the NetLogo software as:

* Wilensky, U. (1999). NetLogo. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.


The model was adapted from the basic Prisoners Dilemma model from the Netlogo Models Library (Copyright 2002 Uri Wilensky). Code adapted by M.C. Boerlijst

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