Hopping

We have implemented a termite bound building simulation as an example of emergence in complex systems. We have also implemented a PDE integrator and used Bonabeau's PDE termite model as a test (with a lot of help from Matt at Imperial College). The graphs show active materials at 60000 time steps in the pillar formation experiment. The simulation takes quite a long time to run.

In order to simulate and analyse the construction process of natural social agents, such as termites, there are a number of approaches available, each with specific strengths and weaknesses. At the top-end is a fully embodied simulation of all physical interactions and thought processes. Naturally, for a complex system such as a termite colony, this is infeasible with current technology.

Individual-based models are related to cellular automata and feature simplified agents, who move in a simplified (generally discretised) approximation of their real environment. These models allow simulation of individual agents' decision making processes; but because of the need to make the simulation tractable the agent and environment are often over-simplified, with parameters set for convenience over realism, making the model only valid as a proof-of-concept at some high level of description.

In order to more accurately represent spatial/temporal scales and natural processes, such as diffusion, whilst still maintaining a high enough level of description to make a simulation tractable, we can turn to partial differential equation (PDE) models. PDE simulation models operate in a continuum, with the distribution of a mass of substance modelled, as opposed to the individual particles/items making up the mass. This means that individual agent behaviour is not modelled; rather the overall behaviour of the agents in an area. Also, physicality is hard to represent, especially in such a complex system as termite construction, so enforcing physical constraints is tricky.However PDE models can, as stated, represent scale much more easily and can include vector/scalar field-based physical processes,such as diffusion, naturally.

As a first step towards a PDE simulation of termite construction we turn to the system of equations presented in (Bonabeau et al., 1998) for investigating the initiation of basic structures involved in termite mound construction.Three 'substances'are modelled: cement pheromone, loaded termites and active material.

For the cement pheromone at position x at time t, H(x,t):

(1)

Where: k₂, k₄ and D_H are constants and P is the amount of 慳ctive� (cement carrying) material.

Only loaded termites (termites carrying soil) are modelled, denoted by C(x,t):

(2)

Where: Φ, k₁, D_C and γ are constants.

Finally, we only model the active material, P(x,t):

(3)

In order to implement this system of equations we must numerically integrate with respect to time over a given area of space. To do this we first discretise the space by partitioning an area of continuous space into a discrete lattice of a given resolution, with each cell containing the 'smoothed-out' mass of substance within that area. Then we asynchronously update each cell using the equations above multiplied by some (small) time-step. This is the basic methodology of numerical integration. Several expanded algorithms exist to improve accuracy, but all at the cost of simulation speed, which is already at a premium in a system such as this.

The below screenshots show the state of active material in a 2D environment at t=0 and t=100. The simulation was run on a 180x180 lattice with a cell spacing of 0.01 and a time-step of 0.0003. All values were initialised to 0, except for P, which was initialised to a random value such that 0 < P(x,0) < 3. Other parameters were set as follows:

As can be seen from the graphs, the initial random distribution of material becomes a series of smooth 憄iles� as statistical fluctuations mean certain sites attract more termites and are positively reinforced, whereas others decay. There is a more-or-less regular distribution of these proto-pillars, which is an emergent property of the system created by the competition between neighbouring sites.

This system works in 2D, uses a minimal set of substances, does not account for physical constraints or take into consideration many facets of termite construction that may have a serious impact on the system. However, the result is still encouraging, and with little modification the equations can be used to explain the initiation of many other base structures in termite construction (see Bonabeau et al., 1998). Further development into a 3D system with more factors taken into account could potentially be very rewarding, though it would likely need the use of a great deal of computing power. The possibility also exists of improving system efficiency algorithmically and hybridizing the system with other simulation methodologies (e.g. particle systems) to expand the capabilities of the approach.

The results for Population behave as expected, the population will grow (or shrink if it starts above a certain size) until it hits the ceiling value, it then fluctuates around that level, generally as the population of one type of agent decreases, the other increases. Though they will always rebalance due to the limited resources.

The distribution of reproduction probability starts off Gaussian distributed, but mutations lead to a slow upward trend. This is due to the fact that those with higher probabilities are more likely to reproduce, this trait is passed onto the young, and so the overall probability increases. It is important to note that the average probability for the system approaches 1.

The distribution of resource production probability fluctuates around the same point that is starts. There is no advantage or disadvantage to individual agents if this value increases or decreases. So no real evolution takes place. It is important to note that because this is random it can end up well above or below its starting value, purely by chance.

You can see in the graph for resources that the resources grow exponentially to begin with. But because of the imposed limit (due to waste) they fluctuate very wildly around the maximum. The graph is actually very similar to the graph for population but growth rate is faster. This causes the fluctuations to be bigger and closer together.

A final important observation is that after some time systems lose stability and the population falls to zero. The reason for this seems to be that reproduction probability gets very high, and resource production probability gets very low. We have already seen that reproduction probability grows, so this is not surprising. But that the production probability decreases is surprising. The best explanation is that because the resource production probability fluctuates randomly it will sometimes fall much lower than its average. If there are enough agents in the system then they will consume all the stockpiled resources and die out before the probability has a chance to stabilise. This is supported by the fact that lower mutation rates cause this to happen less quickly on average. and that resource levels fall very low just before the agents die out.

We present a review of the attempt within the Tangled Nature model to understand the effect of evolution and interaction of ecological and evolutionary observables. We report on the relation between the interaction structure in genotype space and the resulting Species Abundance Distribution (SAD). Ecological relevant SADs are only obtained if the genotype space allow for a potential high connectivity between species. We also study the relation between the degree of genotype interaction and species diversity. Furthermore we include spatial degrees of freedom to investigate the Species Area Relation from an evolutionary perspective. Comparison with observed SARs is favourable and suggests that evolution may be a fundamental factor in understanding the observed power law-like SARs.

The random walker as a mean-field SOC sandpile - the
fun you can have with analytical solutions.

Self-organized critical (SOC) sandpile models are close relatives of fixed energy sandpiles displaying absorbing-state phase transitions (AS) at a given energy density. A recipe for designing equivalent AS models for given SOC models was furnished by Dickman and coworkers in 1998. However, finite size effects are not taken into account in their work. Including these opens up a new series of questions about the exact nature of the link between AS and SOC. I will present the AS mechanism for attaining SOC and explain the puzzling questions raised by including finite-size effects. Preliminary results from recent numerical work addressing these questions will also be presented.

SMAS is a three days mini-workshop intended to strengthen the collaboration between people working on topics closely related to the goals of the STOCHDYN programme.

The main issue to be addressed in the mini-workshop will be agent-based models with applications in different fields (Biology, Economy, Social Sciences,...) with a special focus on the role played by fluctuations in those systems and the proper way to study it analytically, specifically how evolutionary game theory can be extended to include fluctuation effects.

The European Conference on Mathematical and Theoretical Biology (ECMTB), Dresden, Germany, July 18-22, 2005

The ECMTB conference aims to communicate the best of recent advances in mathematical modelling approaches in the life sciences and to identify key targets of future work.

Isaac Newton Institute for Mathematical Sciences - Pattern Formation in Large Domains

1 - 5 August: Training Course: Pattern Formation in Large Domains (David Feltell received financial support)