Two competing populations: simulations with differential equations

By Hans Rudolf Schneebeli1, Claudio Marsan1

1. Switzerland

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Suppose two populations are competing with each other. How can they then develop? We offer insights with simulation calculations. In the model is the evolution of the biomass of the two populations described with differential equations. A fundamental building block is the logistic growth equation.

One can use a numeric toolbox, e.g., Matlab or Octave, Scilab, to rener graphics and simulations as well as illustrate differential equations as flows in phase space. Each differential equation corresponds to a vector field, with ​​direction and strength of a flow represented. Streamlines that fit this velocity field are solutions. Fixed points or closed streamlines are particularly interesting solutions. For population models the stability of solutions are also important because the values ​​of the model parameters may be roughly estimated.

Note from the author:

"We offer modelling predator prey competition in a way more convincing than Lotka and Volterra did. Here the motivation is that the models of Lotka did not concern the behaviour of animals, but came from chemistry and thermodynamics and were basically combinatorial. Hence we model the flow of biomass between a unnamed substrate, a vegetarian and a carnivorous species. The resulting model is similar to Holling type variation of predator prey models, but not a remake of these models."

This paper is in German under the title, "Zwei Populationen im Wettstreit."

Hans R. Schneebeli was born in 1946 in Zurich, Switzerland and obtained the following degrees: MSc Mathematics ETH Zuerich 1972 and PhD Mathematics ETH Zurich 1977 in homological methods in group theory.

He taught mathematics from 1978-2011 at the pre-university level in Switzerland with a special interest in relevant applications accessible in High School/College, motivated by publications such as UMAP-modules. He is retired and is now a potter.

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Researchers should cite this work as follows:

  • Hans Rudolf Schneebeli; Claudio Marsan (2018), "Two competing populations: simulations with differential equations,"

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