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Finite-dimensionality of 2-D micropolar fluid flow with periodic boundary conditions

Piotr Szopa — 2007

Applicationes Mathematicae

This paper is devoted to proving the finite-dimensionality of a two-dimensional micropolar fluid flow with periodic boundary conditions. We define the notions of determining modes and nodes and estimate their number. We check how the distribution of the forces and moments through modes influences the estimate of the number of determining modes. We also estimate the dimension of the global attractor. Finally, we compare our results with analogous results for the Navier-Stokes equation.

Growth of heterotrophe and autotrophe populations in an isolated terrestrial environment

Piotr Paweł Szopa;  Monika Joanna Piotrowska — 2011

Applicationes Mathematicae

We consider the model, proposed by Dawidowicz and Zalasiński, describing the interactions between the heterotrophic and autotrophic organisms coexisting in a terrestrial environment with available oxygen. We modify this model by assuming intraspecific competition between heterotrophic organisms. Moreover, we introduce a diffusion of both types of organisms and oxygen. The basic properties of the extended model are examined and illustrated by numerical simulations.

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