The Concept of Force in Population Dynamics

John Hayward, Paul Roach

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The area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations. This paper describes a method for understanding the influence one variable exerts on another in such models as a force, with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour. Using the stock/flow form of a model, a symbolic notation is developed that identifies the forces with the causal pathways of the model's feedback loops. A force is measured by its impact, defined as the ratio of acceleration to rate of change, computed by differentiation along its associated pathway between variables. Different phases of force dominance are determined to enhance the standard stability analysis of the models, providing an explanation of model behaviour in Newtonian mechanical terms. The concepts developed are applied to well-known models from mathematical biology: the Spruce Budworm model, where force dominance identifies scenarios that give clarity to intervention points; and the Lotka-Volterra predator-prey model where the analysis highlights the importance of dissipative forces in achieving stability. Conclusions are drawn on the explanatory power of this approach, with suggestions made for future work.
Original languageEnglish
Article numberPHYSA-19247
Number of pages24
JournalPhysica A: Statistical Mechanics and its Applications
Issue numberOctober 2019
Publication statusPublished - 10 Jun 2019


  • Sociophysics
  • Population models
  • Differential equations
  • Force
  • System dynamics
  • Feedback


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