Abstracts

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

The modification of Lotka-Volterra predator-prey system dynamics for the situation when the birth process for preys is realized with time lag can be presented in the following form: $$egin{array}{l} mathop xlimits^ ullet = ax(t - au )e^{ - alpha au - intlimits_{t - au }^t {(eta x + gamma z)dt} } - alpha x - eta x^2 - gamma xz mathop zlimits^ ullet = - alpha _1 z + gamma _1 xz end{array}eqno{(1)}$$ where $x(t)$ and $z(t)$ are the numbers of preys and predators at time $t$ respectively. All parameters in (1) are positive. In (1) there is the assumption that self-regulative mechanisms in a population of predators are absent. After the transformation of variables $t = au t'$ , $eta au x = u$ , $gamma au z = v$ system (1) can be presented in another form: $$egin{array}{l} mathop ulimits^ ullet = Au(t' - 1)e^{ - B - intlimits_{t' - 1}^{t'} {(u + v)dt} } - Bu - u^2 - uv mathop vlimits^ ullet = - Cv + Duv end{array}eqno{(2)}$$ where $A = a au $ , $B = alpha au $ , $C = alpha _1 au $ , $D = gamma _1 au $ . For system (2) the initial data are the following: $$u(t') = ho _u (t')geq 0 , v(t') = ho _v (t')0 , t' in [ - 1,0] , ho _u (t'), ho _v (t') in C_{[ - 1,0]} .eqno{(3)}$$ The basic properties of model (2)-(3) can be described by the theorem: Theorem: 1. All solutions of system (2) under the conditions (3) are bounded and non-negative. 2. If $Aleq Be^B $ all solutions of system (2)-(3) goes to origin. 3. If $A > Be^B $ there is the stationary state on Ou axes which is asymptotically stable if the following inequality is realized: $$Be^B < A < left( {B + frac{C}{D}} ight)e^{left( {B + frac{C}{D}} ight)} eqno{(4)}$$ 4. If the inverse inequality is realized in (4) then there exists the non-trivial stable stationary state on the plane Ouv.

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