Ecolab, Webworld and self-organisation

Ecolab and Webworld are both models of evolution produced by adding evolution to ecological equations. They differ primarily in the form of the ecological equations. Both models are self-organised to a state where extinctions balance speciations. However, Ecolab shows evidence of this self-organised state being critical, whereas Webworld does not. This paper examines the self-organised states of these two models and suggest the likely cause of the difference. Also the lifetime distribution for a mean field version of Ecolab is computed, showing that the fat tail of the distribution is due to coevolutionary adaption of the species. Introduction In models of evolving ecologies, a “drip feed” of mutated species are added to a simulation of ecological dynamics. As new species are incorporated into the ecology, they create new links in the food web, perturbing the system dynamics. When enough links are added, feedback loops will form, and the simulated ecology will suffer a mass extinction. Over time, the system self organises to a state where the introduction of new species will be balanced by extinctions, and the system diversity fluctuates around some mean value. But what is this state that the system self organises to? The first suggestion was a critical state (Bak and Sneppen, 1993), characterised by long range influences of a species extinction over others in the food web. The original model of Bak and Sneppen used to illustrate this idea is no more than a cartoon. The interactions between species in this model had no relation to biological interactions. The first attempt to use some real biologically inspired dynamics was probably Ecolab (Standish, 1994), which employed the well known Lotka-Volterra equations, for which a quite a bit of theoretical information is available. This model clearly self organises to a state where speciation is balanced by extinction of average (Standish, 1999), although a variation of the model (incorporating a mechanism of specialisation) produces unbounded growth in diversity (speciation exceeding extinction) (Standish, 2002). So is this state a critical state? One problem is that criticality in self-organised systems is only achieved in the limit of zero driving rate — in this case zero mutation rate. Sole et al. (Solé et al., 2002) prefer the term self-organised instability. Whilst I am sympathetic to this notion, I would also like to point out that stability is very precise term in dynamical systems theory, referring to the behaviour of the linearised system around an equilibrium point. Unstable ecosystems do not have to fall apart — the classical LotkaVolterra (Maynard Smith, 1974) limit cycle is a case in point. Rather the notion of an ecosystem persisting in time without falling apart is captured by permanence, for which a few modest results are known for Lotka-Volterra systems (Law and Blackford, 1992). So perhaps self-organised impermanence would be a more accurate description. Self organised critical systems are characterised by a power law distribution of extinction avalanches, and also a power law distribution of lifetimes. Traditionally, the presence of power law signatures in a self-organising system is taken as evidence of self-organised criticality. Newman (1997) developed another toy evolutionary model that exhibited power law spectra, with neither self-organisation nor criticality in sight. However, when the artificial constant diversity restriction is lifted in the obvious way, selforganisation reappears (Standish, 1999), and the model can also be understood as a mean field approximation of coevolutionary system that potentially admits critical behaviour. Ecolab demonstrates power law spectra of lifetimes (Standish, 1999), with an exponent of -1. However, it has proven very difficult to measure the distribution of extinctions, as extinction avalanches overlap in Ecolab due to the finite rate of speciation. Conversely, studies of a similar model called Webworld claim an absence of any power law signatures (Drossel et al., 2001). I have implemented the Webworld model using the EcoLab (Standish, ) simulation system. I was similarly unable to see evidence of power law signatures, and propose a possible explanation. In this paper, I show that the Fourier transform of the diversity time series is related to the lifetime distribution. Furthermore, in the limiting case of infinitesimal speciation, this transform is the distribution of extinction avalanches (extinction frequency). Ecolab model We start with a generalised form of the Lotka-Volterra equation ṅi = rini −ni ∑