On some types of dynamical behavior in models of reproductive networks

An approximate method of description of the dynamical behavior in $n$-dimensional nonlinear systems of ordinary differential equations, possessing a closed separatrix orbit in the phase space, is proposed. In the case of a simplest reproductive network, by means of such a method it is possible to obtain the condition under which the phase trajectory of the system will approach increasingly closer the separatrix orbit (so called “absorbtion into the separatrix”). It is also possible to calculate the parameters of a limit cycle which appears in the system when this condition is violated. The results of the numerical simulation confirm the correctness of the qualitative description of the system dynamics.