Institute of Medical a nd Biological Cybernetics, Russian Academy of Science, Siberian Branch, Timakova ul. 2, Novosibirsk 630117, Russia

University of New Jersey, Albucerk, USA

Abstract

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that in population mortality of individuals is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassell models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassell model is realized. Modification of basic model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilize at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.