Analytical and Modeling Approaches to Studying the Integral Equation Appearing after a Power-3 Closure

Moscow University Computational Mathematics and Cybernetics. 2021. Vol. 45. No. 2. P. 53–59.

Gadzhiev S., Галкин Е. Г., Nikitin A. A.

A study is performed of the nonlinear integral equation that arises in the biological
model of Dieckmann and Law. A brief outline of the model is given, and the meaning and need
to introduce spatial moments are described. The nonlinear equation (for the state of equilibrium)
is derived from the system of dynamics of spatial moments after power-3 closure. The obtained
equation is transformed to the form appropriate for applying an iterative numerical approach based
on a Neumann series. A numerical way of solving the derived integral equation is developed, and
an example of using the numerical approach and numerical modeling is provided. It is shown
there exists a nontrivial solution to the considered nonlinear integral equation at a strictly positive
parameter of natural mortality. This considerably differs the derived integral equation from its linear
analog widely used in earlier works.

Keywords: интегральные уравнения, математическая биология, integral equations

Publication based on the results of: Дифференциальные уравнения и численные методы. Методы решения и практические приложения (2021)