О замыкании пространственных моментов в биологической модели, и интегральных уравнениях, к которым оно приводит
This article is devoted to the mathematical formulation and numerical study of the spatial model of stationary biological communities by U. Dickman and R. Low. The main idea of this model is to find a "projection" of the simulated biological process on certain characteristics, the dynamics of which can be written analytically. Such "characteristics" in the Dickman’s and Low’s model are the so-called "spatial moments". A system of integro-differential equations is described describing the spatial dynamics in this model. The problem of moment’s closure of the third spatial moment, which relieves the system of an infinite number of equations is posed. Examples of closures of the third moment are written out, two of which, the so-called asymmetric and symmetric closures of the second degree, are studied in more detail. The problem is posed with a linear integral equilibrium’s equation, which is obtained by using an asymmetric closure. The shortcomings of the obtained model are explained, in connection with its biological interpretation, and also with the application of this closure in the study of the two-species model. Next, we formulate a nonlinear problem that arises when a symmetric parametric closure of the second degree is used, and its numerical study is car.