In the modern era, mathematical and computer models of cities have been in development for around sixty years – not surprisingly, mirroring the growth of computing power. Much has been achieved. We are pretty good at modelling the flows of people in cities for a variety of purposes and loading these trips on to transport networks; we are pretty good at estimating some activity totals at locations such as retail revenue or demand for housing. Much of this has been stitched together into comprehensive models, embracing demography and input-output economics. We have the beginnings of understanding of urban dynamics and the associated implications of nonlinearity – particularly, path dependence and phase changes. There have been good applications of many of these models in planning – both public and commercial. However, we are, or should be, a long way away from complacency. Computing power continues to increase and we can use some of this with contemporary models. But I will argue that there is potential for accelerated progress. To achieve this, we need some new mathematical inputs. I suggest there are at least four dimensions for potential development:

- dynamics;
- high dimensionality;
- integrating across competing methods – beyond the silos, chains of causality and conditional probabilities; and
- engaging effectively with other disciplines.

These ideas were developed in a failed ERC application – one of the referees managed to kill it, I suspect by arguing that “it wasn’t economics” – even though it was never intended to be. The ideas are now offered here in the hope that others might take them up – have me as a member of the team if you wish! – and be more successful with their grant bids. Let us take each area in turn.

Dynamics.

We have made progress by using the competition-for-resources version of the Lotka-Volterra equations, and adding space. In the retail case, retailers and developers compete for consumers and the Lotka-Volterra mechanism seems to work well. However, the models are very much at the proof-of-concept stage and much more detailed work is needed – see ‘history in the last section. Further development is likely to involve new models. Francesca Medda, for example, using Turing’s morphogenesis equations, has used diffusion equations and one obvious challenge, is to fully integrate diffusion and L-V competition. One advantage of the ‘L-V with space’ approach is that it does generate structures, some as a result of phase changes – e.g. of retail centres – but there must be other systems of differential or difference equations that do this and so there is scope for mathematical exploration here. In the study of dynamics, we constantly emphasise the importance of path dependence and phase changes. An interesting research question is the extent to which the dynamics, in particular cases, are ‘more’ dependent on the sequence of initial conditions – the development path – as the form of the equations themselves.

High dimensionality.

A number of years ago, I introduced a mathematician to the standard retail model and its dynamics, looking for help. His immediate reaction was to reduce it to a two-zone model so that some of the problems could be tackled analytically. This sort of simplification may occasionally produce something interesting, but the real interest lies in the high dimensionality. This usually means that we have to proceed through simulation rather than analytically. However, there is another aspect to this challenge: that is, we need to handle even higher dimensions than we are accustomed to. For example, when Joel Dearden and I were exploring retail dynamics, we backed away from London, with its 200+ centres and reverted to South Yorkshire, with 19. Even then, we had a 19-dimensional phase space for the dynamics, though Joel found a way of presenting the results using ‘parallel coordinates’. I once estimated that for a medium sized city with relative coarse granularity, I would like to have 10^{13} variables. In some recent work, still under way, with Camilo Vargas, we are trying to model movers including simultaneous house-job moves. For a 1000-zone system, this involves 10004 [possible interactions, and the possible need for super computers becomes evident. Handling all this needs new mathematics and, I suspect, new computing skills. Micro simulation kicks in here, as an efficient way of handling storage.

Competing models.

Particular styles of urban modelling are, or have been, fashionable at different times. There is a current fashion for agent-based modelling (ABM) for example. Then think of entropy-maximising models, logit models and various ‘economic’ models. Interesting thinking develops if we recall the fact that in many if not most of these cases, attempts are being made to model the same system. Intuitively, this suggests that at some deeper level, we might find some equivalences. This has more or less been done with entropy-maximising and logit models and relatively recently, by Joel Dearden and myself, with spatial-interaction retail models and similar ABMs. The challenge in that case is to devise the behavioural rules for the ABMs to make the two model approaches equivalent. This is more or less done, but not yet complete. Again, some new maths needed here?

Many of the differences between approaches simply turn on model design decisions – continuous space or discrete zones, scales, what is endogenous etc, but the common ground will turn on real hypotheses about directions of causality and about the conditional probabilities that represent these. There may be methods such as ‘probabilistic graphical models’ that can be brought to bear here.

Engaging effectively with other disciplines.

In urban modelling, we have by now a reasonably well-defined set of research challenges. It may well be the case that some of our methods can be translated to other disciplines and this may then demand modifications to the models that amount to ‘new maths’. Or vice versa: are the diffusion methods used b y James Steele and his colleagues in UCL Archaeology applicable in urban modelling? (cf. Venturing into other disciplines)

Concluding comment.

Reflection shows that, as ever, these four themes are interdependent!

Alan Wilson

June 2015