Planktons discrete-time dynamical systems

U. A. Rozikov; Sobirjon Shoyimardonov; R. Varro

U. A. Rozikov V.I.Romanovskiy Institute of MathemV.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciencesnces
Sobirjon Shoyimardonov V.I.Romanovskiy Institute of Mathematics
R. Varro Institut MontpelliÅ½rain Alexander Grothendieck,\\ UniversitÅ½ de Montpellier, CNRS, Place EugÂne Bataillon - 35095 Montpellier, France.

Abstract

In this paper, we initiate the study of a discrete-time dynamical system modeling a trophic network connecting the three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator $V$ associated with this dynamical system is of type 4-Volterra quadratic stochastic operator (QSO) with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional
standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this
operator. For each situations we give some biological interpretations.