Researching the phase portrait for a differential equations system modelling competitive interaction

Abstract

Background. Applying parametric systems of ordinary differential equations to the dynamic modelling of competitive interaction is of current interest. There are lots of publications dedicated to two-dimensional systems in this connection. Investigation of higher-dimensional system often demands the methods of numerical analysis; actual scientific literature represents some difficulties in applying classic methods of qualitative theory to higher-dimensional systems. The purpose of this article is to study a 3-dimentional system being applied for modelling of three competing groups interaction, just by the qualitative theory method. Materials and methods. A review of publications about differential equations in competition dynamics modelling is done. We consider a 6-parametric 3- dimentional system on the invariant triangle of frequencies. Definitions of approaching and retiring areas of singular points of the system are formulated. Through consideration of level surfaces of specially constructed functions, we demonstrate how to find out whether an arbitrary point of the frequency triangle belongs to approaching or retiring areas relatively to all singular points located on the triangle sides (but not at its apexes). Results. The equations of the bounds of these areas have been derived. Some theorems describing mutual disposition of those curves (approaching-retiring areas bounds) and singular points of system are proved. A numeric example is given to illustrate the theoretical results. This example is based on linguistic problem data. Conclusions. The developed and theoretically grounded method allows to precise phase portrait of the system under consideration without solving analytically nor numerically.

Keywords

• ordinary differential equations
• аutonomous 3 -dimensional system
• dynamic modelling of competition
• phase portrait