Chernyshev

This article deals with a discrete problem of points moving along a metric graph connected with studying the statistic of Gaussian packets in a three-dimensional network. This problem arises when considering the Cauchy problem for the non-stationary Schrödinger equation. For an arbitrary finite compact graph-tree, representation for the number of points occurring in the initial vertex was obtained. By the example of one particular graph, the number of points moving along this graph was obtained as a sum of solutions for slack linear inequalities. The authors found the highest term of difference of the number of points moving along the graphs which were obtained from various assemblies of edges; the highest term of symmetric difference of the number of moving points was also obtained. 

The paper considers the possibility of organizing the spatial patterns of plankton's population densities defined solely by biological factors, under the condition of homogeneous environment. To simulate self-organization of two plankton's species populations a mathematical model of the "predator-prey" type with regard to the effect of limited seeking by the predator was studied. The simulation results showed the fundamental possibility of spatial structures organization through limited self-movement of plankton.

The description of the statistical behavior of Gaussian packets on a metric graph is considered. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrodinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for the final compact graph packets almost always are distributed evenly. The formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The situation is also discussed where the times of passage over the edges are not linearly independent over the rationals.