Market share dynamics using Lotka–Volterra models

• We develop a novel approach based on LV systems to model market share dynamics.
• We rely a nonautonomous LV model because firms change competitive roles in time.
• The proposed LV model has a strong connection with the logit model.
• The analytical solutions only depend on the utility functions of the competing firms.
• We do not need to fit LV coefficients, because we know the analytical solutions.

Although competition in the marketplace is inherently dynamic and firms change their competitive behavior over time, firms' competitive struggle is generally described using autonomous Lotka–Volterra (LV) models. A great limitation of autonomous LV systems is that the interaction coefficients are constant, and hence firms are assumed to have constant competitive strategies. Also, the solutions of LV models are generally unknown. To address these shortcomings, we introduce a class of integrable nonautonomous LV models. Our LV models present some relevant advantages. First, the analytical solutions of this system are known, therefore we no longer need to fit the LV coefficients. Second, the analytical solutions only depend on the utility functions of the competing firms. Third, our model has a strong connection with the logit model. As mainstream economics extensively use the logit model to describe market demand, our approach has solid economic foundations. In the second part of the article, we test the performance of our approach by studying two cases in competition economics. We find that our model has a better ability to describe and forecast market evolution than the LV autonomous models proposed in the literature.