Chernyshenko V. Topological analysis of global stability of generalized Volterra models.

The objects are nonlinear processes, which take place in ecological and other real systems. The objective is development of new models of mathematical ecology, which belong to generalized Volterra type, and methods of topological analysis of their stability. The methods are topology and invariant analysis, methods of qualitative investigation of dynamic systems, bifurcation analysis, computer simulation. The main models are following: models of genetically non-homogeneous populations; a competition model taking into account energetic aspects of the process; models with parameters, which depend on environmental conditions. New algorithms, based on methods of topological and invariant analyses of models' phase space, are proposed for investigation of their dynamic properties. Global stability of the general form of quadratic differential equations on a cone has been studied. For two-dimensional systems initial and basic invariants have been made. On the base of them a general criterion of boundedness ofequations' solution are developed. For the generalized Lotka-Volterra systems (quadratic and fractionally rational) complete topological classifications for regular cases are realised. Practical usefulness of the methods for real ecological problems is demonstrated. Application spheres are ecology and training courses.