FURTHER NOTE: The model discussed on this page appears not to be a CA in the strict sense, and therefore some conclusions on this page might not be true. Please study the paper in detail if you are interested in this model!

  • Add section on discreteness and attofox?: Durrett, R. and S.A. Levin (1994b) The importance of being discrete (and spatial). Theor. Pop. Biol. 46:363-394 DOI

external image Yellow_daffodils_-_floriade_canberra.jpg

Daffodils and vanishingly small parameter regimes?

Langton's analysis shows that some types of model behaviour (i.e. Class IV) may occur in vanishingly small parameter regimes. One question that follows from such a finding is whether such behaviour should therefore be considered to be of importance. We discuss this issue using an example of CA models made of daffodils.

In a study of daffodils, Barkham & Hance (1982) measured the population density of daffodils under different conditions, i.e. they characterized the biology of daffodils using an observable "population" and not individual states. In an attempt to make sense of their data they constructed a CA model using their real data. However, when implemented in the CA, the model could not generate a viable population (i.e. all CA daffodils died out). Moreover tweaking with the parameters only allowed for either a full CA field filled with daffodils or extinction and not the intermediate conditions observed in the field. The ecologists were therefore quite unsatisfied with the CA formalism.

A study by Levin and Durrett (1994a) addressed the issue of birth and death parameters in CAs. Their results show that intermediate population densities indeed only occurred for a vanishingly small parameter set (see Fig 7). Moreover this finding seems to be a general result in space. Without space (e.g. ODEs) one also gets infinity of extinction with only birth and death, however in CA there is always competition for space where empty squares depend on the population size (N). Birth on the other hand is a constant process and seen in 1D shows long transients and branching growth with low predictability of survival when a species is persisting with borderline survival (see Fig 4). This is analogous to Class IV CA type behaviour.

Thus for a local birth process we get a sharp transition between empty and full field. The quantitative similarity attempted by Barkham and Hance (1982) was in fact not sensible because obviously one cannot model daffodils in space in this way without taking other species and spatial inhomogeneities into account. Fitting CAs to data is therefore an extremely difficult enterprise. In contrast it is relatively easy to make a quantitative fit between an ODE and data, however this does not imply that a good fit is equivalent to a good model. Both these insights may give us some idea about how not to make a model.

Interestingly, it appears that there is parallel between intermediate population densities and Class IV type behaviour: long transients, qualitative behaviour unpredictable. Moreover, both phenomena occur for vanishingly small parameter sets, and this gets smaller with more states in the system. Seen in this light these cases appear to be very unlikely. However they may play an important role. On the one hand they are conceptually important in order for us to realize that such complex unpredictable behaviour can arise from simple local rules. Moreover, as we shall see later, sometimes the dynamics of the system may drive a system to such unlikely cases.


Barkham JP & Hance CE (1982) Population dynamics of the wild daffodil (Narcissus pesudonaricussus) III. Implications of a computer model of 1000 years of population change. J. Ecol. 70:323-344. Download PDF
Durrett R and Levin SA (1994a) Stochastic Spatial models: a users guide for ecological applications. Phil. Trans. Roy. Soc. B. 343:1047-166. Download PDF


Case Study: Daffodils => population size is an 'observable' not a 'state'
In an attempt to study the reasons for particular Daffodil density a CA was constructed using real data. When implemented in CA, the study could not generate a viable population (died out). Parameter tweaking also only lead to a full field or death and not to an intermediate condition. Are CAs therefore a bad modelling formalism?
Levin and Duret: did a parameter sweep of birth and death rates and found that intermediate densities occured only for a vanishingly small set of parameter conditions (picture) and this appears to be a general finding in space.
Without space (ODE) on also gets infinity or death with only birth and death. However in CA there is always spatial competition where empty sqaures depend on N. But birth is a contact process: seen in 1D shows long transients and branching growth with low predictability of survival.
This analysis shows parallel of intermediate pop densities to CA Class IV: long transients, qualititive behaviour unpredictable.
Also the iV region is vanishingly small and get smaller with more states: is it therefore important?
Class IV is a unlikely case: 1) it is important in order to recognize such complex behaviour in simple local rules
2) in some cases the dynamics of the system drives the system to such unlikely cases!