Geometrical Methods and Numerical Computations for Prey-Predator Systems

The goal of this paper is the geometrical and numerical study of the main sizes of the mathematical models of prey-predator interactions which are important in determining long-time dynamics, based on the application of various notions from the theory of dynamical systems to the numerical approximation of initial value problems over long-time intervals. The numerical methods are widely used for the study of complicated temporal behavior of dynamical systems, in order to approximate different types of invariants sets or invariant manifolds and also to extract statistical information on the dynamical behavior in the computation of natural invariant measures or almost invariants sets. The present study is a interplay between dynamical systems geometrical theory and computational calculus of dynamical systems, knowing that the theory provides a framework for interpreting numerical observations and foundations for efficient numerical algorithms.