Abstracts

The perspectives of applications of mathematics in industry, economics, etc.

The modification of well-known Verhulst model of the isolated population dynamics in a case when there is the time lag in a birth process can be presented in the form: $$mathop ylimits^ ullet = Ay(t - 1)e^{ - B - intlimits_{t - 1}^t {ydt} } - By - y^2 eqno{(1)}$$ with the following initial data: $$y(t) = ho (t)geq 0, t in [ - 1,0], ho (t) in C_{[ - 1,0]}.eqno{(2)}$$ where $y(t)$ is a population size at moment $t$, $A,B equiv const > 0$ are the parameters of equation (1), $ ho (t)$ is a non-negative function. The basic properties of model (1)-(2) can be described by the following theorem: Theorem: 1. All solutions of the system (1)-(2) are non-negative and bounded. 2. If $Aleq Be^B $ the origin is unique global stable state. 3. If $A > Be^B $ in phase space there is unique non-trivial stable state. In spite of the fact that the non-trivial state is stable under all respective values of parameters in the given equation (1)-(2) fading fluctuations near the stable state can take place (in a Verhulst model the monotonous stabilizations of population number are possible only).

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