Data Science, Forecasting and Modelling with Python.

David Horgan

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.

Photo by Victor Xok on Unsplash
UK Department Market Share 2007–2017

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.

Department Store Percentage Market Share in 2017

Using integrable nonautonomous Lotka–Volterra (LV) models

the market share dynamics can be modelled using Python.

Code snippet to calculate the utility function
UK Department Utility Functions 2007–2017

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

Quadratic polynomial fit for Departmental Store Utility Functions

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

Competitive Roles for UK Department Stores

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.

The Competitive Interactions between UK Department Stores are Mutualistic

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.

UK Department Market Share Forecast 2017–2025 using nonautonomous Lotka-Volterra Equations


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,, DOI: 10.13140/RG.2.2.23377.68965

David Horgan

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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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