Moving of points on a metric graph associated with the problem of dynamics and statistics of Gaussian packets on a spatial network is under consideration. For an arbitrary tree graph we obtain a representation for the number of points arising from the initial vertex. For a certain graph we find the number of moving points on the graph as the sum of the number of solutions of linear inequalities. We find the first term of the difference of numbers of points moving on two graphs, obtained by permutation of edges. Also we find the leading term for a symmetrical difference of the number of moving points.
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 Schroeodinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situa- tion 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 a final compact graph packets almost always are distributed evenly. The formula for the leading coefficient of the asymp- totic behavior of the number of packets with an increasing time is obtained. The situation is also discussed where the edge travel times are not linearly independent over the rationals.