Analysis of the Lotka-Volterra competition equations as a technological substitution model |
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Authors: | Steven A MorrisAuthor Vitae |
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Institution: | Electrical and Computer Engineering, 202 Engineering South, Oklahoma State University, Stillwater, OK 74078, USA |
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Abstract: | 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−ε2 asymptote and right asymmetric with γ 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. |
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Keywords: | Lotka-Volterra Competition model Substitution model Gompertz Fisher-Pry Bass NSRL Non-symmetrical responding logistic Floyd Sharif-Kabir S-curve Growth curve |
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