Abstract
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.
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ACKNOWLEDGMENTS
The authors thank U. Dieckmann for formulating the problem and discussing our results. We also thank M.V. Nikolaev for his helpful advice on this work.
Funding
This work was supported by the Science Fund of the National Research University Higher School of Economics in 2020–2021 (project no. 20-04-021); and by the state 5-100 program for the support of leading universities of the Russian Federation.
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Translated by E. Oborin
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Gadzhiev, S.R., Galkin, E.G. & Nikitin, A.A. Analytical and Modeling Approaches to Studying the Integral Equation Appearing after a Power-3 Closure. MoscowUniv.Comput.Math.Cybern. 45, 53–59 (2021). https://doi.org/10.3103/S0278641921020023
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DOI: https://doi.org/10.3103/S0278641921020023