Abstracts

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

There is the consideration of the isolated population dynamics model which is a modification of well-known Verhulst’s model. In the boundaries of the model there is the fixed time lag in a birth rate:
, (1)
where is a density of population at time moment , is a Malthusian parameter of reproduction, is a coefficient of natural mortality, is a coefficient of self-regulation, is a value of time lag, is initial function.

Analysis of the model allowed to obtain the folowing results:
1. Under all nonnegative initial functions solutions are limited and nonnegative.
2. If the following condition is satisfied
(2)
there is one stable trivial point in phase space (the population degenerates under all initial conditions). If the inverse inequality in (2) is realized then two equilibria are in a phase space.
3. With the help of Neimark’ method the boundaries of stability zones (in parameter’s space) were determined for all stationary points.
4. As it was obtained under the numerical analysis (with the help of Adams-Bethford’s method) in a model there are three various types of population behaviour: degeneration of population under all initial conditions, stabilization at a unique nonnegative level and stable oscillations (period of oscillations is equal to ).

Note. Abstracts are published in author's edition

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