Analysis of the Lotka―Volterra competition equations as a technological substitution model

摘要

This paper provides insight into the dynamics of the Lotka-Volterra competition (LVC) equations, a much used competition model, and compares the dynamics of LVC competitive substitution to that of several well-known substitution models. The behavior of the LVC equations is analyzed for the special case of a dominant competitor at equilibrium being replaced after the introduction of a small population of an invading competitor with a competitive advantage. Expressions are derived that describe the growth of the invading competitor and that growth is shown to be of four classes: left asymmetric, logistic, right asymmetric with 1 ― ε_2 asymptote and right asymmetric withγ asymptote. It is shown that the LVC model reverts to logistic substitution in a market of fixed size, a result with important implications. The LVC equations are fitted to the Gompertz, Bass, Non-Symmetrical Responding Logistic (NSRL) and Sharif―Kabir substitution models and compared using a novel graphical technique. The LVC equations can reasonably mimic the full range of curve shapes exhibited by each of these models.