On the Integral Equation Arising in the Biological Model After the Power-3 Closure

Abstract

This article is devoted to the nonlinear integral equation arising in the biological model of Ulf Dieckmann and Richard Low. A brief review of the model of foreign authors Individual-based model is made, the meaning and necessity of introducing spatial moments is described. The following is the derivation of the nonlinear equation (for the equilibrium state) from the system of dynamics of spatial moments, after the Power-3 closure. As it was supposed earlier, as a result of this closure the integral equation with nonlinear convolution is deduced. The resulting equation is transformed to a form convenient for the application of the numerical method based on the Neumann’s series. The authors developed a stable numerical method for solving the obtained integral equation. At the end of the article, a large number of examples of the use of the constructed numerical method and numerical modeling are given: a surface in the parameter space for the integral nuclei of motion and competition is constructed; the dependence of the solution of the nonlinear integral equation depending on the area of integration (the solution for these manipulations goes to the asymptote, and does not change significantly); numerical study of the integral equation for the parameter of natural mortality equal to zero. An interesting result is the existence of a nontrivial solution, the investigated nonlinear integral equation for the natural mortality parameter, d > 0. This essentially distinguishes the derived integral equation from its linear analogue, widely studied in previous works.

Keywords

• mathematical biology
• individual-based model
• nonlinear integral equations
• numerical methods