The estimates of the convergence of linear systems solutions with aftereffect in the neighbourhood of point of the rest.

*(English)*Zbl 1374.93174Summary: The study of many mathematical models of population dynamics is reduced to a qualitative research and the construction of the phase portrait of a system of ordinary differential equations which are nonlinear and have multiple equilibria. The singular trajectories of the system are found, which consists of the rest points and cycles, they are “stapled”. The phase portrait of the system as a whole is constructed. The peculiarity of the population model is the account of the factor-effect, which leads to the study of delay systems. Simple models are systems of differential-difference equations with a constant delay. In the present paper, we consider a system of nonlinear differential equations on the plane with delay. Specific points of the system are the same as for the system without delay, and they are found by solving system of two equations in algebraic form. If the equilibrium position of the linearized system is “rough”, then the original nonlinear system has similar properties in a sufficiently small neighbourhood of the equilibrium position. The methods of stability analysis of the system are used. The second Lyapunov method with a quadratic function is selected to obtain the convergence estimates.

##### MSC:

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93C10 | Nonlinear systems in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

92D25 | Population dynamics (general) |