Hypercycles+in+space

Prev: Hypercycles Next: Multi-level evolution


 * TODO List**
 * more detail on why / how spirals arise
 * REF: rule 54 Crutchfield
 * more detail on how parasites are purged from spirals

=Hyper-cycles in space=

The assumptions of infinite and well-mixed populations automatically disappear when we use a spatial modelling formalism. Here we discuss the study of hyper-cycles in a CA by [|Boerlijst and Hogeweg (1991)]. In this model the only assumptions are: decay, replication (local) and local catalysis of replication. With these assumptions, the CA rules are a "translation" of the ODE assumptions into the CA formalism.

The question then is: __//how does the system behave?//__ A very general result of this model is that the system organizes itself into a particular form of mesoscale patterns, namely [|spiral] waves. In these spirals, bands of individuals that receive catalysis from the band preceding their band are formed. So if A catalyzes B catalyzes C catalyzes D catalyzes A, we will see patterns in which we see a bands of A - B - C - D - A etc. Just like we saw in the hypercycle ODEs, the dynamics in the CA depend on the number of species (i.e. length of the cycle):

-> attofox problems! || Spiral waves -> Very stable, up to a large number of species ||
 * **Cycle length** || **ODE behaviour** || **CA behaviour** ||
 * n <= 3 || Stable fixed point || No pattern, seemingly random distribution ||
 * n = 4 || Stable spiral || Chaotic waves ||
 * n >= 5 || Limit cycle (increasing amplitude)

So, we see that the stability of the dynamics is very different in the CA than in the ODE. In the ODE, for systems with many species we found limit cycles with high amplitude and hence we would expect extinction in the oscillations (when correcting for attofoxes). In the CA, this "problem" is solved: the stable spiral wave patterns ensure that a large number of species can be sustained.

A second important thing about the CA dynamics is that the spirals grow from their core. All individuals that are part of a spiral are offspring of one of the few core individuals. Hence, long-term fitness is determined by location: only individuals in the spiral core are "at the right place at the right time" and will have non-zero long-term fitness!

The next question then is __//whether these spirals have any meaning for the biological system in question?//__ Well unlike the rule 54 CA studied by Crutchfield (1989? 1995?), there is a model relation to something in biology, i.e. entities in the CA represent molecules. Hence, the mesoscale pattern has some biological meaning in terms of those entities.

The next question we can ask is __//whether such meaningful patterns are also important, or just look nice?//__ This question becomes clearer when we study the systems properties, in particular its resistance to parasites. Unlike the ODE model, we find that the CA system can recover even from introduction of very severe parasites. We see that parasites can grow locally, but eventually die out locally as well because they have exhausted their catalyst. Obviously, you could argue that if we were "lucky" not to introduce any mutant parasites into the spiral cores, we should expect that the parasites should be expelled from the system, because the spirals are generated and 'refreshed' from their cores, washing out all parasites in the spiral. However, even if we specifically introduce parasites into the spiral core and these manage to take over a whole spiral, they are still purged from the system by other non-infected spirals. (Note that we here take a worst case scenario in terms of infection in order to show how resistant the system is!)



So given that we find such a difference in results with the ODE model: __//which is the best model?//__ This is actually a bad question! Instead it is more fruitful to view both models as two extremes in terms of **local** interactions versus **well-mixed** populations. One can then ask what happens in intermediate situations: i.e. some level of **diffusion**. In the CA we can add this diffusion. Then, results show that with no diffusion waves have quite a small scale and stochasticity plays a large role. As diffusion increases, the spatial scale of waves tends to increase making them more resistant to parasite invasion. On the other hand, fewer waves fit in the field making the system more sensitive to invasion if a wave would be taken over by parasites. In a sense these results indicate an irony in that in going towards the more well-mixed state (with no patterns), the power of mesoscale patterning is actually increased, because stochasticity is reduced. This illustrates that the ODE result is the limit of extreme diffusion in an infinite domain. With "mixed" space on the other hand, spatial resolution dominates!

Next: Multi-level evolution

**Boerlijst MC & Hogeweg P** (1991) Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites. //Physica D//, **48**: 17-28. [|DownLoad PDF].  **Hogeweg P** (2007) From population dynamics to ecoinformatics: Ecosystems as multilevel information processing systems.//Ecological Informatics//, **2**: 103-111. [|DownLoad PDF].
 * References**