A new approach to the study of the stability of non-autonomous discrete systems of the Lotka- Volterra type

The problems connected with asymptotic behavior of solutions of nonautonomous third-order discrete system of Lotka - Volterra type are considered. This system describes the dynamics of infectious disease in a heterogeneous group which consists of three populations. On the basis of new methods of limiting equations theory and limiting Lyapunov functions the conditions of asymptotic stability are obtained which provide full convalescence of all populations. The presented approach allows one to carry out the investigation of asymptotic stability of Lotka-Volterra system with arbitrary number of populations. The additional examples are considered which show that asymptotic stability conditions obtained on the basis of degenerative Lyapunov function are not only sufficient, but also necessary from point of view of classical stability conditions by linear approximation.