Abstracts

Evolution of Species and Ecosystems: Theoretical Analysis and Computer-Assisted Modeling

The publication is devoted to research of parametrical model of dynamics of the isolated population with sexual structure. Within the limits of the model it is supposed, that the birth rate in population has discrete character and the occurrence of new species occurs in the fixed moments of time , and mortality has continuous character: at time intervals there is only monotonous reduction of numbers. At intervals of time dynamics of number of population is described by the following system of the differential equations: (1)

where - number of male, - number of female species at the moment . At the moments of occurrence of new species the parities are performed:

, where (2)

and , , .

Without reducing generality, it is possible to consider and .

Model (1) - (2) has the following properties:

1) The solutions of problem (1), (2) with the positive initial data are limited and not negative, i.e. in stable invariant compact set exists.

2) During performance of one of conditions:

, point (0,0) – globally stable equilibrium.

3) If и , then (0,0) – unstable equilibrium.

А) If inequalities (3)

are performed therewith, then only one nontrivial globally stable equilibrium exists in at any values of remaining parameters.

Б) If conditions (3) are not performed, then a classical picture of bifurcations of doubling of oscillation’s period and emerging of chaos arises with magnification of parameters and (at a constant ratio ).

Note. Abstracts are published in author's edition

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