Investigation in general of the behaviour of the trajectories of a predator – prey system

By the methods of the classical qualitative theory of dynamical systems on the plane, the problem of constructing a phase portrait of Kolmogorov’s predator – prey system has been solved in general. Possible topological structures of this model are considered for six cases of coefficient conditions with positive values of three parameters. The phase portraits in the Poincare disk are constructed by dividing the set of values of one of the parameters into intervals. The values of this parameter are found at which the self-oscillation mode is possible in the system. It is shown that a weak focus of order 1 (multiplicity 1) is not surrounded by closed trajectories. Based on the analysis of the location of the main isoclines of the system on the entire phase plane, exclusively geometrically, the topological structure of a complex equilibrium state at infinity is established without relying on known analytical (more time-consuming) methods.