| The blend of geometry and numerical analysis that is taking place in the present material produces a subject with can gives a lots of detail and richness for the case of dynamical models arising in mathematics, science and engineering. The goal of this paper is the geometrical and numerical study of the main sizes of the mathematical models of the multispecies 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. We perform a computational study for the case of four important examples: prey-predator 2D Lotka-Volterra system, Bailey model for the evolution of epidemics, classical Kermack-McKendrick model of evolution of epidemics and the prey-predator 3D Lotka-Volterra system. |
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