# Department Store Dynamics Using the Lokta-Volterra Equations

In an earlier post, I looked at the market share dynamics of UK supermarkets using Lotka-Volterra — link here. In this post, I am going to look at the dynamics of department store market shares.

It is quite a turbulent time for retailers in the UK at present and department stores have been hit hard by the impact of online shopping and Brexit. Even though I am examining the top retailers in the UK, two of the six have recently been through administration.

The market share of UK department stores is quite dynamic and rankings regularly change over time. In 2017 the situation was that John Lewis had overtaken Marks & Spencer in terms of market share and House of Fraser had fallen to sixth place behind Harrods and Selfridges.

Using integrable nonautonomous Lotka–Volterra (LV) models

the market share dynamics can be modelled using Python.

The utility function can then be fitted using a quadratic polynomial,

The market shares are then calculated from the utility function by using,

The department store market share time evolution is related to the stores’ utility functions. The nonautonomous Lotka-Volterra model allows the prediction and explanation of the market dynamics using the signs of the time-dependent coefficients g(t)in the Lotka-Volterra equations.

The competitive roles are deduced from the signs of g(t) which is obtained from the derivative of the fitted quadratic polynomial. In the case of department stores, the interactions are - - which is mutualism in which both stores involved benefit to some extent with neither being damaged.

The community matrix for the mutual competitive interactions of department stores is shown below.

Next, a 5-year forecast was produced, assuming that the competitive strategies of the department stores remained on a business as usual basis. The initial analysis indicates that Harrods will consistently increase its market share. At the same time Marks and Spencer, if it continues as it does at present, would rapidly lose sharemarket.

It would be expected that department stores will make adaptations to their competitive strategies in response to the evolutionary pressures arising from the increase in online shopping. This forecast needs to be seen within that context, the size of the instore retail market is decreasing, so a rise in market share will not necessarily translate to an overall increase in sales. That process will be the topic of a further article.

References

Marasco, A. & Picucci, A. & Romano, A., 2016. “Determining firms׳ utility functions and competitive roles from data on market shares using Lotka–Volterra models,” Data in Brief, Volume 7, pages 709–713.

Marasco, A. & Picucci, A. & Romano, A., 2016. “Market share dynamics using Lotka–Volterra models,” Technological Forecasting and Social Change, Elsevier, vol. 105(C), pages 49–62.

Horgan, D., 2019. “UK retail market share dynamics using Lotka-Volterra models,” online, https://www.researchgate.net/publication/334769813/, DOI: 10.13140/RG.2.2.23377.68965

Written by

## David Horgan

#### I am a theoretical physicist with a data science background. At present, I am developing a UK retail market using ABM, ML and computational econometrics.

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